1 What is Image Segmentation? There are many definitions. Three common ones are Image Segmentation is the process of isolating objects of interest from.

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11

What is Image Segmentation

There are many definitions Three common There are many definitions Three common

ones areones are

Image Segmentation is the process of Image Segmentation is the process of

isolating objects of interest from the rest isolating objects of interest from the rest

of the scene (of the scene (Castleman Castleman ))

Image segmentation is the process of Image segmentation is the process of

partitioning partitioning (( 分 割分 割 ) ) an image into non-an image into non-

intersecting region such that each region intersecting region such that each region

is homogeneous is homogeneous (( 相似的相似的 ) ) and the union of and the union of

no two adjacent regions is homogeneous no two adjacent regions is homogeneous

((Pal Pal ))

22

What is Image Segmentation

Image segmentation is to divide an image Image segmentation is to divide an image

into parts that have a strong correlation into parts that have a strong correlation

with objects or areas of the real world with objects or areas of the real world

contained in the image (contained in the image (Watt Watt ))

In brief sIn brief segmentation is to subdivide an egmentation is to subdivide an

imageimage into its constituentinto its constituent ( ( 组成的组成的 )) regions regions

or objectsor objects

―Segmentation should stop when theSegmentation should stop when the

objects of interest in an application objects of interest in an application

havehave been isolatedbeen isolated

33

Image Processing Flow based onImage Processing Flow based on

Image SegmentationImage Segmentation

Image Segmentation

InputInputImageImage

ObjectObjectImageImage

FeatureFeatureVectorVector

ObjectObjectTypeType

FeatureExtraction

Classification

44

Why is it difficult

In generalIn general autonomousautonomous ( ( 自 主 的自 主 的 ))

segmentation is one segmentation is one of of the most difficult the most difficult

tasks in image processing It tasks in image processing It isis difficult difficult

because of because of many reasonmany reasonss Here are some Here are some

typical obstaclestypical obstacles Non-uniform illuminationNon-uniform illumination

No control of the environmentNo control of the environment

Inadequate model of the object of interestInadequate model of the object of interest

NoiseNoise

etcetc

55

Segmentation methods can be divided into three groups according to the dominant features they employ

Segmentation based on global knowledge about an image

―The knowledge is usually represented by a histogram of image features

Edge-based segmentations

―Utilizing edge detection processes to find a closed boundary so that an inside and an outside can be defined

Region-based segmentations

―This techniques proceed by dividing the image into regions that exhibit similar properties

Principal approachesPrincipal approaches

66

2 basis properties of intensity 2 basis properties of intensity valuesvalues

Segmentation algorithms generally are baseSegmentation algorithms generally are based on one of 2 basis properties of intensity vad on one of 2 basis properties of intensity valueslues DiscontinuityDiscontinuity

―to partition an image based on abrupt (to partition an image based on abrupt ( 突然突然的的 ) changes in intensity (such as edges)) changes in intensity (such as edges)

SimilaritySimilarity

―to partition an image into regions that are simto partition an image into regions that are similar according to a set of predefined criteriailar according to a set of predefined criteria

77

Detection of DiscontinuitiesDetection of Discontinuities

detect the three basic types of gray detect the three basic types of gray

level discontinuitieslevel discontinuities

points lines edgespoints lines edges

the common way is to run a mask the common way is to run a mask

through the imagethrough the image

88

ContentsContents

ThresholdingThresholding

Point DetectionPoint Detection

Line DetectionLine Detection

Edge-based SegmentationEdge-based Segmentation

Region-based SegmentationRegion-based Segmentation

99

71 Thresholding71 Thresholding

Thresholding is a labeling operation Thresholding is a labeling operation

on a gray scale image that on a gray scale image that

distinguishes pixels of a higher distinguishes pixels of a higher

intensity from pixels with a lower intensity from pixels with a lower

intensity valueintensity value

The output of thresholding usuallyThe output of thresholding usually is is a a

binary imagebinary image

This technique is particularly useful This technique is particularly useful

for scenes which contain for scenes which contain solid objects solid objects

on a uniform contrasting backgroundon a uniform contrasting background

1010

Classification of ThresholdingClassification of Thresholding

Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

T T x y p x y f x y

1111

Classification of ThresholdingClassification of Thresholding

When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

1048638 1048638 Global Global thresholdingthresholding

both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

Local Local thresholdingthresholding

x and y (in addition)x and y (in addition)

Dynamic Dynamic thresholdingthresholding

T T x y p x y f x y

1212

Basic Global ThresholdingBasic Global Thresholding

Original imageOriginal image HistogramHistogram

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

SolutionSolution use T midway between the max and use T midway between the max and

min gray levelsmin gray levels

See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

1313

Basic Global ThresholdingBasic Global Thresholding

Let light objects in dark backgroundLet light objects in dark background

To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

e backgrounde background

ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

A thresholded imageA thresholded image

1

0

if f x y T backgroundg x y

if f x y T foreground

1414

Heuristic Global ThresholdingHeuristic Global Thresholding

11 Select an initial estimate for TSelect an initial estimate for T

22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

1515

Basic Adaptive ThresholdingBasic Adaptive Thresholding

subdivide original image into small areassubdivide original image into small areas

utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

1616

Multilevel ThresholdingMultilevel Thresholding

a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

another object class if f(xy) another object class if f(xy) gtgt T T22

to background if f(xy) le Tto background if f(xy) le T11

1717

The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

1 1 2 2

1 2 1

p z P p z P p z

P P

1818

Optimal ThresholdingOptimal Thresholding

If the form of the If the form of the

densities is densities is

known or known or

assumed in assumed in

terms of terms of

minimum error minimum error

determining an determining an

optimal optimal

threshold for threshold for

segmenting the segmenting the

image is image is

possiblepossible

1 1 2 2

1 2 1

p z P p z P p z

P P

1919

Optimal ThresholdingOptimal Thresholding

Probability of erroneouslyProbability of erroneously

2020

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

find T which makesfind T which makes

2121

Optimal ThresholdingOptimal Thresholding

Minimum errorMinimum error

Specially if Specially if PP11 = P = P22 then the optimum then the optimum

threshold is where the curve pthreshold is where the curve p11(z) and (z) and

pp22(z) intersect(z) intersect

2222

Optimal ThresholdingOptimal Thresholding

For exampleFor example

Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

pp22(z)(z)

where μwhere μ11 and σ and σ1122 are the mean and are the mean and

variance of the Gaussian density of one variance of the Gaussian density of one

objectobject

μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

the Gaussian density of the other objectthe Gaussian density of the other object

2323

Optimal ThresholdingOptimal Thresholding

Quadratic equation (Quadratic equation (二次方程二次方程 ))

2424

Problems of ThresholdingProblems of Thresholding

Original imageOriginal image Thresholded imageThresholded image

2525

Problems of ThresholdingProblems of Thresholding

(a)(a) Exact threshold Exact threshold

segmentationsegmentation

(b)(b) Threshold too lowThreshold too low

(c)(c) Threshold too Threshold too

highhigh

2626

72 Point Detection72 Point Detection

a point has been detected at the a point has been detected at the

location on which the mark is location on which the mark is

centered ifcentered if

|R|geT|R|geT

where T is a nonnegative thresholdwhere T is a nonnegative threshold

R is the sum of products of the R is the sum of products of the

coefficients with the gray levels contained coefficients with the gray levels contained

in the region encompassed by the markin the region encompassed by the mark

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

2727

72 Point Detection72 Point Detection

Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

2828

ExampleExample

2929

73 Line Detection73 Line Detection

Horizontal mask will result with max Horizontal mask will result with max

response when a line passed through the response when a line passed through the

middle row of the mask with a constant middle row of the mask with a constant

backgroundbackground

the similar idea is used with other masksthe similar idea is used with other masks

Note the preferred direction of each mask Note the preferred direction of each mask

is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

than other possible directionsthan other possible directions

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

1 1 1 1 1 2 1 2 1 2 1 1

2 2 2 1 2 1 1 2 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

45 45Horizontal Vertical

3030

Idea 1 of Line DetectionIdea 1 of Line Detection

Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

if at a certain point in the imageif at a certain point in the image

|Ri||Ri|gtgt|Rj||Rj|

for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

3131

Idea 2 of Line DetectionIdea 2 of Line Detection

Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

3232

ExampleExample

3333

74 Edge-based 74 Edge-based SegmentationSegmentation

Edge-based segmentations rely on edges found in an image by edge detecting operators

these edges mark image locations of discontinuities in gray level

Edge detection is the most common approach for detecting meaningful discontinuities in gray level

There are a large group of methods based on information about edges in the image

3434

What is edgeWhat is edge

Edge is where change occurs Change is measured by derivative in 1D

―Biggest change derivative has maximum magnitude

Or 2nd derivative is zero we discuss approaches for implementing

―first-order derivative (Gradient operator)

―second-order derivative (Laplacian operator)

―we have introduced both derivatives in chapter 3

―Here we will talk only about their properties for edge detection

3535

What is edgeWhat is edge

In other wordsIn other words an edge is a set of an edge is a set of

connected pixelsconnected pixels

that lie on the boundary between two that lie on the boundary between two

regions with relatively distinct gray-level regions with relatively distinct gray-level

propertiesproperties

Note edge vs boundaryNote edge vs boundary

―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

―whereas a region boundary owing to whereas a region boundary owing to

the way it is defined is a more global the way it is defined is a more global

ideaidea

3636

Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

because of because of

optics optics

sampling sampling

image image

acquisition acquisition

imperfectionimperfection

3737

Thick and Thin EdgeThick and Thin Edge

The slope of the ramp is inversely The slope of the ramp is inversely

proportional to the degree of blurring in the proportional to the degree of blurring in the

edgeedge

Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

pathpath

Instead an edge point now is any point Instead an edge point now is any point

contained in the ramp and contained in the ramp and an edge would an edge would

then be a set of such points that are then be a set of such points that are

connectedconnected

The thickness is determined by the length of the The thickness is determined by the length of the

rampramp

The length is determined by the slope which is in The length is determined by the slope which is in

turn determined by the degree of blurringturn determined by the degree of blurring

Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

edges tend to be thinedges tend to be thin

3838

First and Second derivatives (First and Second derivatives ( 导数导数 ))

the signs of the the signs of the

derivatives would be derivatives would be

reversed for an edge reversed for an edge

that transitions from that transitions from

light to darklight to dark

First First derivatderivatee

SeconSecond d derivatderivatee

Gray-Gray-level level profileprofile

3939

Second derivativesSecond derivatives

an undesirable featurean undesirable feature

produces 2 values for every edge in an produces 2 values for every edge in an

imageimage

zero-crossing propertyzero-crossing property

an imaginary straight line joining the an imaginary straight line joining the

extreme positive and negative values of extreme positive and negative values of

the second derivative would cross zero the second derivative would cross zero

near the midpoint of the edgenear the midpoint of the edge

quite useful for locating the centers of quite useful for locating the centers of

thick edgesthick edges

4040

Basic idea of edge detectionBasic idea of edge detection

A profile is defined perpendicularly to A profile is defined perpendicularly to

the edge direction and the results are the edge direction and the results are

interpretedinterpreted

The magnitude of the first derivative is The magnitude of the first derivative is

used to detect an edge (if a point is on a used to detect an edge (if a point is on a

ramp)ramp)

The sign of the second derivative can The sign of the second derivative can

determine whether an edge pixel is on the determine whether an edge pixel is on the

dark or light side of an edgedark or light side of an edge

4141

Review of First DerivateReview of First Derivate

Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

Sobel operators 3Sobel operators 333

6 5 8 5x yG z z G z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

1 2 3

4 5 6

7 8 9

z z z

z z z

z z z

9 5 8 6x yG z z G z z 1 0 0 1

0 1 1 0

1 0 0 1

0 1 1 0

7 8 9 1 2 3

3 6 9 1 4 7

2 2

2 2

x

y

G z z z z z z

G z z z z z z

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

x yf G G

4242

Edge direction and strengthEdge direction and strength

Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

2 2x yf G G

4343

Gradient MasksGradient Masks

1 0 0 1

0 1 1 0

Roberts

1 0 0 1

0 1 1 0

Roberts

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 2 1 1 0 1

0 0 0 2 0 2

1 2 1 1 0 1

Sobel

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

1 1 1 1 0 1

0 0 0 1 0 1

1 1 1 1 0 1

Prewitt

4444

Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

0 1 1 1 1 0

1 0 1 1 0 1

1 1 0 0 1 1

Prewitt

4545

Review of Second DerivateReview of Second Derivate

Laplacian OperatorLaplacian Operator

21 1

1 1 4

f x y f x yf

f x y f x y f x y

0 1 0

1 4 1

0 1 0

0 1 0

1 4 1

0 1 0

LaplacianLaplacian

MaskMask

1 1 1

1 8 1

1 1 1

1 1 1

1 8 1

1 1 1

4646

Example of edge detectionExample of edge detection

See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

derivative)derivative)

produces double edgesproduces double edges

unable to detect edge directionunable to detect edge direction

4747

Canny edge detectorCanny edge detector

The most powerful edge-detection The most powerful edge-detection

method method

It differs from the other edge-It differs from the other edge-

detection methods in that detection methods in that

it uses two different thresholds (to detect it uses two different thresholds (to detect

strong and weak edges) strong and weak edges)

and includes the weak edges in the and includes the weak edges in the

output only if they are connected to output only if they are connected to

strong edges strong edges

This method is therefore less likely This method is therefore less likely

than the others to be fooled by than the others to be fooled by

noise and more likely to detect true noise and more likely to detect true

weak edgesweak edges

4848

Laplacian of GaussianLaplacian of Gaussian

Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

2 2 22

4 2

2 2 2

2exp

r rh

r x y

determines the degrdetermines the degree of blurring that occee of blurring that occursurs

4949

Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

0 0 1 0 0

0 1 2 1 0

1 2 16 2 1

0 1 2 1 0

0 0 1 0 0

The coefficient must sum to The coefficient must sum to

zerozero

5050

Edge Detection and Edge Detection and SegmentationSegmentation

Image resulting from edge detection cannot be used as a segmentation result

Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

5151

75 Region-based 75 Region-based SegmentationSegmentation

GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

5252

Two methods of Region Two methods of Region SegmentationSegmentation

Region GrowingRegion Growing

Region SplittingRegion Splitting

Region growing is the opposite of the Region growing is the opposite of the

split and merge approachsplit and merge approach

5353

Region GrowingRegion Growing

The objective of segmentation is to The objective of segmentation is to

partition an image into regionspartition an image into regions

A region is a connected component with A region is a connected component with

some uniformity (say gray-levels or some uniformity (say gray-levels or

texture)texture)

In region growing we start with a set In region growing we start with a set

of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

appending to each seedrsquos neighbor appending to each seedrsquos neighbor

pixels if they have pixels if they have similar propertiessimilar properties

such as specific ranges of gray level such as specific ranges of gray level

and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

Need initialization Need initialization similarity similarity

criterioncriterion

5454

Steps of Region GrowingSteps of Region Growing

Start by choosing an arbitrary seed Start by choosing an arbitrary seed

pixel andpixel and compare it with neighbor compare it with neighbor

ppixelsixels

When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

is grown from the seed pixel by is grown from the seed pixel by

addingadding neighboneighborr pixel pixelss

When the growth of one region stopsWhen the growth of one region stops

choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

againagain

5555

Region Region growing growing

An initial set of small An initial set of small

areas are iterativelyareas are iteratively

merged according to merged according to

similarity constraintssimilarity constraints

5656

Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

We need initial seed points to groWe need initial seed points to grow into regionsw into regions

On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

SeedSeed pointspoints

5757

CriterionCriterion

There is a valley at around 190 in the There is a valley at around 190 in the

histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

to be considered as a part of region to the to be considered as a part of region to the

seed pointseed point

The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

to at least one pixel in that regionto at least one pixel in that region

Result of region growing and boundaries of Result of region growing and boundaries of

defectsdefects

5858

Region SplittingRegion Splitting

The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

ption that the entire image is homogeneousption that the entire image is homogeneous

If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

Need homogeneity criterion split ruleNeed homogeneity criterion split rule

5959

Region SplittingRegion Splitting

DisadvantageDisadvantage

they create regions that may be adjacent they create regions that may be adjacent

and homogeneous but not mergedand homogeneous but not merged

6060

Region Splitting and MergingRegion Splitting and Merging

ProcedureProcedure

11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

6161

Region Splitting and Merging

Quadtree

(四叉树 )

6262

PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

mmii is the mean gray level of that region is the mean gray level of that region

σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

ExampleExample

Original Original

imageimageThresholded imageThresholded image Result of Result of

Splitting and Splitting and

MergingMerging

  • Slide 1
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    22

    What is Image Segmentation

    Image segmentation is to divide an image Image segmentation is to divide an image

    into parts that have a strong correlation into parts that have a strong correlation

    with objects or areas of the real world with objects or areas of the real world

    contained in the image (contained in the image (Watt Watt ))

    In brief sIn brief segmentation is to subdivide an egmentation is to subdivide an

    imageimage into its constituentinto its constituent ( ( 组成的组成的 )) regions regions

    or objectsor objects

    ―Segmentation should stop when theSegmentation should stop when the

    objects of interest in an application objects of interest in an application

    havehave been isolatedbeen isolated

    33

    Image Processing Flow based onImage Processing Flow based on

    Image SegmentationImage Segmentation

    Image Segmentation

    InputInputImageImage

    ObjectObjectImageImage

    FeatureFeatureVectorVector

    ObjectObjectTypeType

    FeatureExtraction

    Classification

    44

    Why is it difficult

    In generalIn general autonomousautonomous ( ( 自 主 的自 主 的 ))

    segmentation is one segmentation is one of of the most difficult the most difficult

    tasks in image processing It tasks in image processing It isis difficult difficult

    because of because of many reasonmany reasonss Here are some Here are some

    typical obstaclestypical obstacles Non-uniform illuminationNon-uniform illumination

    No control of the environmentNo control of the environment

    Inadequate model of the object of interestInadequate model of the object of interest

    NoiseNoise

    etcetc

    55

    Segmentation methods can be divided into three groups according to the dominant features they employ

    Segmentation based on global knowledge about an image

    ―The knowledge is usually represented by a histogram of image features

    Edge-based segmentations

    ―Utilizing edge detection processes to find a closed boundary so that an inside and an outside can be defined

    Region-based segmentations

    ―This techniques proceed by dividing the image into regions that exhibit similar properties

    Principal approachesPrincipal approaches

    66

    2 basis properties of intensity 2 basis properties of intensity valuesvalues

    Segmentation algorithms generally are baseSegmentation algorithms generally are based on one of 2 basis properties of intensity vad on one of 2 basis properties of intensity valueslues DiscontinuityDiscontinuity

    ―to partition an image based on abrupt (to partition an image based on abrupt ( 突然突然的的 ) changes in intensity (such as edges)) changes in intensity (such as edges)

    SimilaritySimilarity

    ―to partition an image into regions that are simto partition an image into regions that are similar according to a set of predefined criteriailar according to a set of predefined criteria

    77

    Detection of DiscontinuitiesDetection of Discontinuities

    detect the three basic types of gray detect the three basic types of gray

    level discontinuitieslevel discontinuities

    points lines edgespoints lines edges

    the common way is to run a mask the common way is to run a mask

    through the imagethrough the image

    88

    ContentsContents

    ThresholdingThresholding

    Point DetectionPoint Detection

    Line DetectionLine Detection

    Edge-based SegmentationEdge-based Segmentation

    Region-based SegmentationRegion-based Segmentation

    99

    71 Thresholding71 Thresholding

    Thresholding is a labeling operation Thresholding is a labeling operation

    on a gray scale image that on a gray scale image that

    distinguishes pixels of a higher distinguishes pixels of a higher

    intensity from pixels with a lower intensity from pixels with a lower

    intensity valueintensity value

    The output of thresholding usuallyThe output of thresholding usually is is a a

    binary imagebinary image

    This technique is particularly useful This technique is particularly useful

    for scenes which contain for scenes which contain solid objects solid objects

    on a uniform contrasting backgroundon a uniform contrasting background

    1010

    Classification of ThresholdingClassification of Thresholding

    Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

    where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

    T T x y p x y f x y

    1111

    Classification of ThresholdingClassification of Thresholding

    When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

    1048638 1048638 Global Global thresholdingthresholding

    both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

    Local Local thresholdingthresholding

    x and y (in addition)x and y (in addition)

    Dynamic Dynamic thresholdingthresholding

    T T x y p x y f x y

    1212

    Basic Global ThresholdingBasic Global Thresholding

    Original imageOriginal image HistogramHistogram

    SolutionSolution use T midway between the max and use T midway between the max and

    min gray levelsmin gray levels

    SolutionSolution use T midway between the max and use T midway between the max and

    min gray levelsmin gray levels

    See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

    1313

    Basic Global ThresholdingBasic Global Thresholding

    Let light objects in dark backgroundLet light objects in dark background

    To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

    e backgrounde background

    ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

    A thresholded imageA thresholded image

    1

    0

    if f x y T backgroundg x y

    if f x y T foreground

    1414

    Heuristic Global ThresholdingHeuristic Global Thresholding

    11 Select an initial estimate for TSelect an initial estimate for T

    22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

    33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

    44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

    55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

    1515

    Basic Adaptive ThresholdingBasic Adaptive Thresholding

    subdivide original image into small areassubdivide original image into small areas

    utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

    since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

    See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

    1616

    Multilevel ThresholdingMultilevel Thresholding

    a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

    another object class if f(xy) another object class if f(xy) gtgt T T22

    to background if f(xy) le Tto background if f(xy) le T11

    1717

    The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

    1 1 2 2

    1 2 1

    p z P p z P p z

    P P

    1818

    Optimal ThresholdingOptimal Thresholding

    If the form of the If the form of the

    densities is densities is

    known or known or

    assumed in assumed in

    terms of terms of

    minimum error minimum error

    determining an determining an

    optimal optimal

    threshold for threshold for

    segmenting the segmenting the

    image is image is

    possiblepossible

    1 1 2 2

    1 2 1

    p z P p z P p z

    P P

    1919

    Optimal ThresholdingOptimal Thresholding

    Probability of erroneouslyProbability of erroneously

    2020

    Optimal ThresholdingOptimal Thresholding

    Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

    ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

    find T which makesfind T which makes

    2121

    Optimal ThresholdingOptimal Thresholding

    Minimum errorMinimum error

    Specially if Specially if PP11 = P = P22 then the optimum then the optimum

    threshold is where the curve pthreshold is where the curve p11(z) and (z) and

    pp22(z) intersect(z) intersect

    2222

    Optimal ThresholdingOptimal Thresholding

    For exampleFor example

    Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

    pp22(z)(z)

    where μwhere μ11 and σ and σ1122 are the mean and are the mean and

    variance of the Gaussian density of one variance of the Gaussian density of one

    objectobject

    μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

    the Gaussian density of the other objectthe Gaussian density of the other object

    2323

    Optimal ThresholdingOptimal Thresholding

    Quadratic equation (Quadratic equation (二次方程二次方程 ))

    2424

    Problems of ThresholdingProblems of Thresholding

    Original imageOriginal image Thresholded imageThresholded image

    2525

    Problems of ThresholdingProblems of Thresholding

    (a)(a) Exact threshold Exact threshold

    segmentationsegmentation

    (b)(b) Threshold too lowThreshold too low

    (c)(c) Threshold too Threshold too

    highhigh

    2626

    72 Point Detection72 Point Detection

    a point has been detected at the a point has been detected at the

    location on which the mark is location on which the mark is

    centered ifcentered if

    |R|geT|R|geT

    where T is a nonnegative thresholdwhere T is a nonnegative threshold

    R is the sum of products of the R is the sum of products of the

    coefficients with the gray levels contained coefficients with the gray levels contained

    in the region encompassed by the markin the region encompassed by the mark

    1 1 1

    1 8 1

    1 1 1

    1 1 1

    1 8 1

    1 1 1

    2727

    72 Point Detection72 Point Detection

    Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

    The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

    1 1 1

    1 8 1

    1 1 1

    1 1 1

    1 8 1

    1 1 1

    0 1 0

    1 4 1

    0 1 0

    0 1 0

    1 4 1

    0 1 0

    2828

    ExampleExample

    2929

    73 Line Detection73 Line Detection

    Horizontal mask will result with max Horizontal mask will result with max

    response when a line passed through the response when a line passed through the

    middle row of the mask with a constant middle row of the mask with a constant

    backgroundbackground

    the similar idea is used with other masksthe similar idea is used with other masks

    Note the preferred direction of each mask Note the preferred direction of each mask

    is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

    than other possible directionsthan other possible directions

    1 1 1 1 1 2 1 2 1 2 1 1

    2 2 2 1 2 1 1 2 1 1 2 1

    1 1 1 2 1 1 1 2 1 1 1 2

    45 45Horizontal Vertical

    1 1 1 1 1 2 1 2 1 2 1 1

    2 2 2 1 2 1 1 2 1 1 2 1

    1 1 1 2 1 1 1 2 1 1 1 2

    45 45Horizontal Vertical

    3030

    Idea 1 of Line DetectionIdea 1 of Line Detection

    Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

    izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

    if at a certain point in the imageif at a certain point in the image

    |Ri||Ri|gtgt|Rj||Rj|

    for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

    3131

    Idea 2 of Line DetectionIdea 2 of Line Detection

    Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

    After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

    3232

    ExampleExample

    3333

    74 Edge-based 74 Edge-based SegmentationSegmentation

    Edge-based segmentations rely on edges found in an image by edge detecting operators

    these edges mark image locations of discontinuities in gray level

    Edge detection is the most common approach for detecting meaningful discontinuities in gray level

    There are a large group of methods based on information about edges in the image

    3434

    What is edgeWhat is edge

    Edge is where change occurs Change is measured by derivative in 1D

    ―Biggest change derivative has maximum magnitude

    Or 2nd derivative is zero we discuss approaches for implementing

    ―first-order derivative (Gradient operator)

    ―second-order derivative (Laplacian operator)

    ―we have introduced both derivatives in chapter 3

    ―Here we will talk only about their properties for edge detection

    3535

    What is edgeWhat is edge

    In other wordsIn other words an edge is a set of an edge is a set of

    connected pixelsconnected pixels

    that lie on the boundary between two that lie on the boundary between two

    regions with relatively distinct gray-level regions with relatively distinct gray-level

    propertiesproperties

    Note edge vs boundaryNote edge vs boundary

    ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

    ―whereas a region boundary owing to whereas a region boundary owing to

    the way it is defined is a more global the way it is defined is a more global

    ideaidea

    3636

    Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

    because of because of

    optics optics

    sampling sampling

    image image

    acquisition acquisition

    imperfectionimperfection

    3737

    Thick and Thin EdgeThick and Thin Edge

    The slope of the ramp is inversely The slope of the ramp is inversely

    proportional to the degree of blurring in the proportional to the degree of blurring in the

    edgeedge

    Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

    pathpath

    Instead an edge point now is any point Instead an edge point now is any point

    contained in the ramp and contained in the ramp and an edge would an edge would

    then be a set of such points that are then be a set of such points that are

    connectedconnected

    The thickness is determined by the length of the The thickness is determined by the length of the

    rampramp

    The length is determined by the slope which is in The length is determined by the slope which is in

    turn determined by the degree of blurringturn determined by the degree of blurring

    Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

    edges tend to be thinedges tend to be thin

    3838

    First and Second derivatives (First and Second derivatives ( 导数导数 ))

    the signs of the the signs of the

    derivatives would be derivatives would be

    reversed for an edge reversed for an edge

    that transitions from that transitions from

    light to darklight to dark

    First First derivatderivatee

    SeconSecond d derivatderivatee

    Gray-Gray-level level profileprofile

    3939

    Second derivativesSecond derivatives

    an undesirable featurean undesirable feature

    produces 2 values for every edge in an produces 2 values for every edge in an

    imageimage

    zero-crossing propertyzero-crossing property

    an imaginary straight line joining the an imaginary straight line joining the

    extreme positive and negative values of extreme positive and negative values of

    the second derivative would cross zero the second derivative would cross zero

    near the midpoint of the edgenear the midpoint of the edge

    quite useful for locating the centers of quite useful for locating the centers of

    thick edgesthick edges

    4040

    Basic idea of edge detectionBasic idea of edge detection

    A profile is defined perpendicularly to A profile is defined perpendicularly to

    the edge direction and the results are the edge direction and the results are

    interpretedinterpreted

    The magnitude of the first derivative is The magnitude of the first derivative is

    used to detect an edge (if a point is on a used to detect an edge (if a point is on a

    ramp)ramp)

    The sign of the second derivative can The sign of the second derivative can

    determine whether an edge pixel is on the determine whether an edge pixel is on the

    dark or light side of an edgedark or light side of an edge

    4141

    Review of First DerivateReview of First Derivate

    Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

    Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

    Sobel operators 3Sobel operators 333

    6 5 8 5x yG z z G z z

    1 2 3

    4 5 6

    7 8 9

    z z z

    z z z

    z z z

    1 2 3

    4 5 6

    7 8 9

    z z z

    z z z

    z z z

    9 5 8 6x yG z z G z z 1 0 0 1

    0 1 1 0

    1 0 0 1

    0 1 1 0

    7 8 9 1 2 3

    3 6 9 1 4 7

    2 2

    2 2

    x

    y

    G z z z z z z

    G z z z z z z

    1 2 1 1 0 1

    0 0 0 2 0 2

    1 2 1 1 0 1

    1 2 1 1 0 1

    0 0 0 2 0 2

    1 2 1 1 0 1

    x yf G G

    4242

    Edge direction and strengthEdge direction and strength

    Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

    α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

    The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

    The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

    2 2x yf G G

    4343

    Gradient MasksGradient Masks

    1 0 0 1

    0 1 1 0

    Roberts

    1 0 0 1

    0 1 1 0

    Roberts

    1 2 1 1 0 1

    0 0 0 2 0 2

    1 2 1 1 0 1

    Sobel

    1 2 1 1 0 1

    0 0 0 2 0 2

    1 2 1 1 0 1

    Sobel

    1 1 1 1 0 1

    0 0 0 1 0 1

    1 1 1 1 0 1

    Prewitt

    1 1 1 1 0 1

    0 0 0 1 0 1

    1 1 1 1 0 1

    Prewitt

    4444

    Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

    0 1 1 1 1 0

    1 0 1 1 0 1

    1 1 0 0 1 1

    Prewitt

    0 1 1 1 1 0

    1 0 1 1 0 1

    1 1 0 0 1 1

    Prewitt

    4545

    Review of Second DerivateReview of Second Derivate

    Laplacian OperatorLaplacian Operator

    21 1

    1 1 4

    f x y f x yf

    f x y f x y f x y

    0 1 0

    1 4 1

    0 1 0

    0 1 0

    1 4 1

    0 1 0

    LaplacianLaplacian

    MaskMask

    1 1 1

    1 8 1

    1 1 1

    1 1 1

    1 8 1

    1 1 1

    4646

    Example of edge detectionExample of edge detection

    See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

    Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

    derivative)derivative)

    produces double edgesproduces double edges

    unable to detect edge directionunable to detect edge direction

    4747

    Canny edge detectorCanny edge detector

    The most powerful edge-detection The most powerful edge-detection

    method method

    It differs from the other edge-It differs from the other edge-

    detection methods in that detection methods in that

    it uses two different thresholds (to detect it uses two different thresholds (to detect

    strong and weak edges) strong and weak edges)

    and includes the weak edges in the and includes the weak edges in the

    output only if they are connected to output only if they are connected to

    strong edges strong edges

    This method is therefore less likely This method is therefore less likely

    than the others to be fooled by than the others to be fooled by

    noise and more likely to detect true noise and more likely to detect true

    weak edgesweak edges

    4848

    Laplacian of GaussianLaplacian of Gaussian

    Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

    2 2 22

    4 2

    2 2 2

    2exp

    r rh

    r x y

    determines the degrdetermines the degree of blurring that occee of blurring that occursurs

    4949

    Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

    0 0 1 0 0

    0 1 2 1 0

    1 2 16 2 1

    0 1 2 1 0

    0 0 1 0 0

    0 0 1 0 0

    0 1 2 1 0

    1 2 16 2 1

    0 1 2 1 0

    0 0 1 0 0

    The coefficient must sum to The coefficient must sum to

    zerozero

    5050

    Edge Detection and Edge Detection and SegmentationSegmentation

    Image resulting from edge detection cannot be used as a segmentation result

    Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

    5151

    75 Region-based 75 Region-based SegmentationSegmentation

    GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

    Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

    5252

    Two methods of Region Two methods of Region SegmentationSegmentation

    Region GrowingRegion Growing

    Region SplittingRegion Splitting

    Region growing is the opposite of the Region growing is the opposite of the

    split and merge approachsplit and merge approach

    5353

    Region GrowingRegion Growing

    The objective of segmentation is to The objective of segmentation is to

    partition an image into regionspartition an image into regions

    A region is a connected component with A region is a connected component with

    some uniformity (say gray-levels or some uniformity (say gray-levels or

    texture)texture)

    In region growing we start with a set In region growing we start with a set

    of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

    appending to each seedrsquos neighbor appending to each seedrsquos neighbor

    pixels if they have pixels if they have similar propertiessimilar properties

    such as specific ranges of gray level such as specific ranges of gray level

    and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

    Need initialization Need initialization similarity similarity

    criterioncriterion

    5454

    Steps of Region GrowingSteps of Region Growing

    Start by choosing an arbitrary seed Start by choosing an arbitrary seed

    pixel andpixel and compare it with neighbor compare it with neighbor

    ppixelsixels

    When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

    is grown from the seed pixel by is grown from the seed pixel by

    addingadding neighboneighborr pixel pixelss

    When the growth of one region stopsWhen the growth of one region stops

    choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

    againagain

    5555

    Region Region growing growing

    An initial set of small An initial set of small

    areas are iterativelyareas are iteratively

    merged according to merged according to

    similarity constraintssimilarity constraints

    5656

    Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

    X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

    We need initial seed points to groWe need initial seed points to grow into regionsw into regions

    On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

    Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

    SeedSeed pointspoints

    5757

    CriterionCriterion

    There is a valley at around 190 in the There is a valley at around 190 in the

    histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

    to be considered as a part of region to the to be considered as a part of region to the

    seed pointseed point

    The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

    to at least one pixel in that regionto at least one pixel in that region

    Result of region growing and boundaries of Result of region growing and boundaries of

    defectsdefects

    5858

    Region SplittingRegion Splitting

    The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

    ption that the entire image is homogeneousption that the entire image is homogeneous

    If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

    This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

    Need homogeneity criterion split ruleNeed homogeneity criterion split rule

    5959

    Region SplittingRegion Splitting

    DisadvantageDisadvantage

    they create regions that may be adjacent they create regions that may be adjacent

    and homogeneous but not mergedand homogeneous but not merged

    6060

    Region Splitting and MergingRegion Splitting and Merging

    ProcedureProcedure

    11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

    22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

    33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

    6161

    Region Splitting and Merging

    Quadtree

    (四叉树 )

    6262

    PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

    where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

    mmii is the mean gray level of that region is the mean gray level of that region

    σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

    ExampleExample

    Original Original

    imageimageThresholded imageThresholded image Result of Result of

    Splitting and Splitting and

    MergingMerging

    • Slide 1
    • Slide 2
    • Slide 3
    • Slide 4
    • Slide 5
    • Slide 6
    • Slide 7
    • Slide 8
    • Slide 9
    • Slide 10
    • Slide 11
    • Slide 12
    • Slide 13
    • Slide 14
    • Slide 15
    • Slide 16
    • Slide 17
    • Slide 18
    • Slide 19
    • Slide 20
    • Slide 21
    • Slide 22
    • Slide 23
    • Slide 24
    • Slide 25
    • Slide 26
    • Slide 27
    • Slide 28
    • Slide 29
    • Slide 30
    • Slide 31
    • Slide 32
    • Slide 33
    • Slide 34
    • Slide 35
    • Slide 36
    • Slide 37
    • Slide 38
    • Slide 39
    • Slide 40
    • Slide 41
    • Slide 42
    • Slide 43
    • Slide 44
    • Slide 45
    • Slide 46
    • Slide 47
    • Slide 48
    • Slide 49
    • Slide 50
    • Slide 51
    • Slide 52
    • Slide 53
    • Slide 54
    • Slide 55
    • Slide 56
    • Slide 57
    • Slide 58
    • Slide 59
    • Slide 60
    • Slide 61
    • Slide 62

      33

      Image Processing Flow based onImage Processing Flow based on

      Image SegmentationImage Segmentation

      Image Segmentation

      InputInputImageImage

      ObjectObjectImageImage

      FeatureFeatureVectorVector

      ObjectObjectTypeType

      FeatureExtraction

      Classification

      44

      Why is it difficult

      In generalIn general autonomousautonomous ( ( 自 主 的自 主 的 ))

      segmentation is one segmentation is one of of the most difficult the most difficult

      tasks in image processing It tasks in image processing It isis difficult difficult

      because of because of many reasonmany reasonss Here are some Here are some

      typical obstaclestypical obstacles Non-uniform illuminationNon-uniform illumination

      No control of the environmentNo control of the environment

      Inadequate model of the object of interestInadequate model of the object of interest

      NoiseNoise

      etcetc

      55

      Segmentation methods can be divided into three groups according to the dominant features they employ

      Segmentation based on global knowledge about an image

      ―The knowledge is usually represented by a histogram of image features

      Edge-based segmentations

      ―Utilizing edge detection processes to find a closed boundary so that an inside and an outside can be defined

      Region-based segmentations

      ―This techniques proceed by dividing the image into regions that exhibit similar properties

      Principal approachesPrincipal approaches

      66

      2 basis properties of intensity 2 basis properties of intensity valuesvalues

      Segmentation algorithms generally are baseSegmentation algorithms generally are based on one of 2 basis properties of intensity vad on one of 2 basis properties of intensity valueslues DiscontinuityDiscontinuity

      ―to partition an image based on abrupt (to partition an image based on abrupt ( 突然突然的的 ) changes in intensity (such as edges)) changes in intensity (such as edges)

      SimilaritySimilarity

      ―to partition an image into regions that are simto partition an image into regions that are similar according to a set of predefined criteriailar according to a set of predefined criteria

      77

      Detection of DiscontinuitiesDetection of Discontinuities

      detect the three basic types of gray detect the three basic types of gray

      level discontinuitieslevel discontinuities

      points lines edgespoints lines edges

      the common way is to run a mask the common way is to run a mask

      through the imagethrough the image

      88

      ContentsContents

      ThresholdingThresholding

      Point DetectionPoint Detection

      Line DetectionLine Detection

      Edge-based SegmentationEdge-based Segmentation

      Region-based SegmentationRegion-based Segmentation

      99

      71 Thresholding71 Thresholding

      Thresholding is a labeling operation Thresholding is a labeling operation

      on a gray scale image that on a gray scale image that

      distinguishes pixels of a higher distinguishes pixels of a higher

      intensity from pixels with a lower intensity from pixels with a lower

      intensity valueintensity value

      The output of thresholding usuallyThe output of thresholding usually is is a a

      binary imagebinary image

      This technique is particularly useful This technique is particularly useful

      for scenes which contain for scenes which contain solid objects solid objects

      on a uniform contrasting backgroundon a uniform contrasting background

      1010

      Classification of ThresholdingClassification of Thresholding

      Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

      where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

      T T x y p x y f x y

      1111

      Classification of ThresholdingClassification of Thresholding

      When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

      1048638 1048638 Global Global thresholdingthresholding

      both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

      Local Local thresholdingthresholding

      x and y (in addition)x and y (in addition)

      Dynamic Dynamic thresholdingthresholding

      T T x y p x y f x y

      1212

      Basic Global ThresholdingBasic Global Thresholding

      Original imageOriginal image HistogramHistogram

      SolutionSolution use T midway between the max and use T midway between the max and

      min gray levelsmin gray levels

      SolutionSolution use T midway between the max and use T midway between the max and

      min gray levelsmin gray levels

      See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

      1313

      Basic Global ThresholdingBasic Global Thresholding

      Let light objects in dark backgroundLet light objects in dark background

      To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

      e backgrounde background

      ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

      A thresholded imageA thresholded image

      1

      0

      if f x y T backgroundg x y

      if f x y T foreground

      1414

      Heuristic Global ThresholdingHeuristic Global Thresholding

      11 Select an initial estimate for TSelect an initial estimate for T

      22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

      33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

      44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

      55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

      1515

      Basic Adaptive ThresholdingBasic Adaptive Thresholding

      subdivide original image into small areassubdivide original image into small areas

      utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

      since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

      See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

      1616

      Multilevel ThresholdingMultilevel Thresholding

      a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

      another object class if f(xy) another object class if f(xy) gtgt T T22

      to background if f(xy) le Tto background if f(xy) le T11

      1717

      The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

      1 1 2 2

      1 2 1

      p z P p z P p z

      P P

      1818

      Optimal ThresholdingOptimal Thresholding

      If the form of the If the form of the

      densities is densities is

      known or known or

      assumed in assumed in

      terms of terms of

      minimum error minimum error

      determining an determining an

      optimal optimal

      threshold for threshold for

      segmenting the segmenting the

      image is image is

      possiblepossible

      1 1 2 2

      1 2 1

      p z P p z P p z

      P P

      1919

      Optimal ThresholdingOptimal Thresholding

      Probability of erroneouslyProbability of erroneously

      2020

      Optimal ThresholdingOptimal Thresholding

      Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

      ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

      find T which makesfind T which makes

      2121

      Optimal ThresholdingOptimal Thresholding

      Minimum errorMinimum error

      Specially if Specially if PP11 = P = P22 then the optimum then the optimum

      threshold is where the curve pthreshold is where the curve p11(z) and (z) and

      pp22(z) intersect(z) intersect

      2222

      Optimal ThresholdingOptimal Thresholding

      For exampleFor example

      Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

      pp22(z)(z)

      where μwhere μ11 and σ and σ1122 are the mean and are the mean and

      variance of the Gaussian density of one variance of the Gaussian density of one

      objectobject

      μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

      the Gaussian density of the other objectthe Gaussian density of the other object

      2323

      Optimal ThresholdingOptimal Thresholding

      Quadratic equation (Quadratic equation (二次方程二次方程 ))

      2424

      Problems of ThresholdingProblems of Thresholding

      Original imageOriginal image Thresholded imageThresholded image

      2525

      Problems of ThresholdingProblems of Thresholding

      (a)(a) Exact threshold Exact threshold

      segmentationsegmentation

      (b)(b) Threshold too lowThreshold too low

      (c)(c) Threshold too Threshold too

      highhigh

      2626

      72 Point Detection72 Point Detection

      a point has been detected at the a point has been detected at the

      location on which the mark is location on which the mark is

      centered ifcentered if

      |R|geT|R|geT

      where T is a nonnegative thresholdwhere T is a nonnegative threshold

      R is the sum of products of the R is the sum of products of the

      coefficients with the gray levels contained coefficients with the gray levels contained

      in the region encompassed by the markin the region encompassed by the mark

      1 1 1

      1 8 1

      1 1 1

      1 1 1

      1 8 1

      1 1 1

      2727

      72 Point Detection72 Point Detection

      Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

      The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

      1 1 1

      1 8 1

      1 1 1

      1 1 1

      1 8 1

      1 1 1

      0 1 0

      1 4 1

      0 1 0

      0 1 0

      1 4 1

      0 1 0

      2828

      ExampleExample

      2929

      73 Line Detection73 Line Detection

      Horizontal mask will result with max Horizontal mask will result with max

      response when a line passed through the response when a line passed through the

      middle row of the mask with a constant middle row of the mask with a constant

      backgroundbackground

      the similar idea is used with other masksthe similar idea is used with other masks

      Note the preferred direction of each mask Note the preferred direction of each mask

      is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

      than other possible directionsthan other possible directions

      1 1 1 1 1 2 1 2 1 2 1 1

      2 2 2 1 2 1 1 2 1 1 2 1

      1 1 1 2 1 1 1 2 1 1 1 2

      45 45Horizontal Vertical

      1 1 1 1 1 2 1 2 1 2 1 1

      2 2 2 1 2 1 1 2 1 1 2 1

      1 1 1 2 1 1 1 2 1 1 1 2

      45 45Horizontal Vertical

      3030

      Idea 1 of Line DetectionIdea 1 of Line Detection

      Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

      izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

      if at a certain point in the imageif at a certain point in the image

      |Ri||Ri|gtgt|Rj||Rj|

      for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

      3131

      Idea 2 of Line DetectionIdea 2 of Line Detection

      Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

      After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

      3232

      ExampleExample

      3333

      74 Edge-based 74 Edge-based SegmentationSegmentation

      Edge-based segmentations rely on edges found in an image by edge detecting operators

      these edges mark image locations of discontinuities in gray level

      Edge detection is the most common approach for detecting meaningful discontinuities in gray level

      There are a large group of methods based on information about edges in the image

      3434

      What is edgeWhat is edge

      Edge is where change occurs Change is measured by derivative in 1D

      ―Biggest change derivative has maximum magnitude

      Or 2nd derivative is zero we discuss approaches for implementing

      ―first-order derivative (Gradient operator)

      ―second-order derivative (Laplacian operator)

      ―we have introduced both derivatives in chapter 3

      ―Here we will talk only about their properties for edge detection

      3535

      What is edgeWhat is edge

      In other wordsIn other words an edge is a set of an edge is a set of

      connected pixelsconnected pixels

      that lie on the boundary between two that lie on the boundary between two

      regions with relatively distinct gray-level regions with relatively distinct gray-level

      propertiesproperties

      Note edge vs boundaryNote edge vs boundary

      ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

      ―whereas a region boundary owing to whereas a region boundary owing to

      the way it is defined is a more global the way it is defined is a more global

      ideaidea

      3636

      Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

      because of because of

      optics optics

      sampling sampling

      image image

      acquisition acquisition

      imperfectionimperfection

      3737

      Thick and Thin EdgeThick and Thin Edge

      The slope of the ramp is inversely The slope of the ramp is inversely

      proportional to the degree of blurring in the proportional to the degree of blurring in the

      edgeedge

      Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

      pathpath

      Instead an edge point now is any point Instead an edge point now is any point

      contained in the ramp and contained in the ramp and an edge would an edge would

      then be a set of such points that are then be a set of such points that are

      connectedconnected

      The thickness is determined by the length of the The thickness is determined by the length of the

      rampramp

      The length is determined by the slope which is in The length is determined by the slope which is in

      turn determined by the degree of blurringturn determined by the degree of blurring

      Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

      edges tend to be thinedges tend to be thin

      3838

      First and Second derivatives (First and Second derivatives ( 导数导数 ))

      the signs of the the signs of the

      derivatives would be derivatives would be

      reversed for an edge reversed for an edge

      that transitions from that transitions from

      light to darklight to dark

      First First derivatderivatee

      SeconSecond d derivatderivatee

      Gray-Gray-level level profileprofile

      3939

      Second derivativesSecond derivatives

      an undesirable featurean undesirable feature

      produces 2 values for every edge in an produces 2 values for every edge in an

      imageimage

      zero-crossing propertyzero-crossing property

      an imaginary straight line joining the an imaginary straight line joining the

      extreme positive and negative values of extreme positive and negative values of

      the second derivative would cross zero the second derivative would cross zero

      near the midpoint of the edgenear the midpoint of the edge

      quite useful for locating the centers of quite useful for locating the centers of

      thick edgesthick edges

      4040

      Basic idea of edge detectionBasic idea of edge detection

      A profile is defined perpendicularly to A profile is defined perpendicularly to

      the edge direction and the results are the edge direction and the results are

      interpretedinterpreted

      The magnitude of the first derivative is The magnitude of the first derivative is

      used to detect an edge (if a point is on a used to detect an edge (if a point is on a

      ramp)ramp)

      The sign of the second derivative can The sign of the second derivative can

      determine whether an edge pixel is on the determine whether an edge pixel is on the

      dark or light side of an edgedark or light side of an edge

      4141

      Review of First DerivateReview of First Derivate

      Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

      Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

      Sobel operators 3Sobel operators 333

      6 5 8 5x yG z z G z z

      1 2 3

      4 5 6

      7 8 9

      z z z

      z z z

      z z z

      1 2 3

      4 5 6

      7 8 9

      z z z

      z z z

      z z z

      9 5 8 6x yG z z G z z 1 0 0 1

      0 1 1 0

      1 0 0 1

      0 1 1 0

      7 8 9 1 2 3

      3 6 9 1 4 7

      2 2

      2 2

      x

      y

      G z z z z z z

      G z z z z z z

      1 2 1 1 0 1

      0 0 0 2 0 2

      1 2 1 1 0 1

      1 2 1 1 0 1

      0 0 0 2 0 2

      1 2 1 1 0 1

      x yf G G

      4242

      Edge direction and strengthEdge direction and strength

      Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

      α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

      The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

      The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

      2 2x yf G G

      4343

      Gradient MasksGradient Masks

      1 0 0 1

      0 1 1 0

      Roberts

      1 0 0 1

      0 1 1 0

      Roberts

      1 2 1 1 0 1

      0 0 0 2 0 2

      1 2 1 1 0 1

      Sobel

      1 2 1 1 0 1

      0 0 0 2 0 2

      1 2 1 1 0 1

      Sobel

      1 1 1 1 0 1

      0 0 0 1 0 1

      1 1 1 1 0 1

      Prewitt

      1 1 1 1 0 1

      0 0 0 1 0 1

      1 1 1 1 0 1

      Prewitt

      4444

      Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

      0 1 1 1 1 0

      1 0 1 1 0 1

      1 1 0 0 1 1

      Prewitt

      0 1 1 1 1 0

      1 0 1 1 0 1

      1 1 0 0 1 1

      Prewitt

      4545

      Review of Second DerivateReview of Second Derivate

      Laplacian OperatorLaplacian Operator

      21 1

      1 1 4

      f x y f x yf

      f x y f x y f x y

      0 1 0

      1 4 1

      0 1 0

      0 1 0

      1 4 1

      0 1 0

      LaplacianLaplacian

      MaskMask

      1 1 1

      1 8 1

      1 1 1

      1 1 1

      1 8 1

      1 1 1

      4646

      Example of edge detectionExample of edge detection

      See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

      Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

      derivative)derivative)

      produces double edgesproduces double edges

      unable to detect edge directionunable to detect edge direction

      4747

      Canny edge detectorCanny edge detector

      The most powerful edge-detection The most powerful edge-detection

      method method

      It differs from the other edge-It differs from the other edge-

      detection methods in that detection methods in that

      it uses two different thresholds (to detect it uses two different thresholds (to detect

      strong and weak edges) strong and weak edges)

      and includes the weak edges in the and includes the weak edges in the

      output only if they are connected to output only if they are connected to

      strong edges strong edges

      This method is therefore less likely This method is therefore less likely

      than the others to be fooled by than the others to be fooled by

      noise and more likely to detect true noise and more likely to detect true

      weak edgesweak edges

      4848

      Laplacian of GaussianLaplacian of Gaussian

      Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

      2 2 22

      4 2

      2 2 2

      2exp

      r rh

      r x y

      determines the degrdetermines the degree of blurring that occee of blurring that occursurs

      4949

      Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

      0 0 1 0 0

      0 1 2 1 0

      1 2 16 2 1

      0 1 2 1 0

      0 0 1 0 0

      0 0 1 0 0

      0 1 2 1 0

      1 2 16 2 1

      0 1 2 1 0

      0 0 1 0 0

      The coefficient must sum to The coefficient must sum to

      zerozero

      5050

      Edge Detection and Edge Detection and SegmentationSegmentation

      Image resulting from edge detection cannot be used as a segmentation result

      Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

      5151

      75 Region-based 75 Region-based SegmentationSegmentation

      GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

      Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

      5252

      Two methods of Region Two methods of Region SegmentationSegmentation

      Region GrowingRegion Growing

      Region SplittingRegion Splitting

      Region growing is the opposite of the Region growing is the opposite of the

      split and merge approachsplit and merge approach

      5353

      Region GrowingRegion Growing

      The objective of segmentation is to The objective of segmentation is to

      partition an image into regionspartition an image into regions

      A region is a connected component with A region is a connected component with

      some uniformity (say gray-levels or some uniformity (say gray-levels or

      texture)texture)

      In region growing we start with a set In region growing we start with a set

      of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

      appending to each seedrsquos neighbor appending to each seedrsquos neighbor

      pixels if they have pixels if they have similar propertiessimilar properties

      such as specific ranges of gray level such as specific ranges of gray level

      and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

      Need initialization Need initialization similarity similarity

      criterioncriterion

      5454

      Steps of Region GrowingSteps of Region Growing

      Start by choosing an arbitrary seed Start by choosing an arbitrary seed

      pixel andpixel and compare it with neighbor compare it with neighbor

      ppixelsixels

      When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

      is grown from the seed pixel by is grown from the seed pixel by

      addingadding neighboneighborr pixel pixelss

      When the growth of one region stopsWhen the growth of one region stops

      choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

      againagain

      5555

      Region Region growing growing

      An initial set of small An initial set of small

      areas are iterativelyareas are iteratively

      merged according to merged according to

      similarity constraintssimilarity constraints

      5656

      Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

      X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

      We need initial seed points to groWe need initial seed points to grow into regionsw into regions

      On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

      Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

      SeedSeed pointspoints

      5757

      CriterionCriterion

      There is a valley at around 190 in the There is a valley at around 190 in the

      histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

      to be considered as a part of region to the to be considered as a part of region to the

      seed pointseed point

      The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

      to at least one pixel in that regionto at least one pixel in that region

      Result of region growing and boundaries of Result of region growing and boundaries of

      defectsdefects

      5858

      Region SplittingRegion Splitting

      The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

      ption that the entire image is homogeneousption that the entire image is homogeneous

      If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

      This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

      Need homogeneity criterion split ruleNeed homogeneity criterion split rule

      5959

      Region SplittingRegion Splitting

      DisadvantageDisadvantage

      they create regions that may be adjacent they create regions that may be adjacent

      and homogeneous but not mergedand homogeneous but not merged

      6060

      Region Splitting and MergingRegion Splitting and Merging

      ProcedureProcedure

      11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

      22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

      33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

      6161

      Region Splitting and Merging

      Quadtree

      (四叉树 )

      6262

      PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

      where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

      mmii is the mean gray level of that region is the mean gray level of that region

      σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

      ExampleExample

      Original Original

      imageimageThresholded imageThresholded image Result of Result of

      Splitting and Splitting and

      MergingMerging

      • Slide 1
      • Slide 2
      • Slide 3
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      • Slide 5
      • Slide 6
      • Slide 7
      • Slide 8
      • Slide 9
      • Slide 10
      • Slide 11
      • Slide 12
      • Slide 13
      • Slide 14
      • Slide 15
      • Slide 16
      • Slide 17
      • Slide 18
      • Slide 19
      • Slide 20
      • Slide 21
      • Slide 22
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      • Slide 24
      • Slide 25
      • Slide 26
      • Slide 27
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      • Slide 29
      • Slide 30
      • Slide 31
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      • Slide 33
      • Slide 34
      • Slide 35
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      • Slide 37
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        44

        Why is it difficult

        In generalIn general autonomousautonomous ( ( 自 主 的自 主 的 ))

        segmentation is one segmentation is one of of the most difficult the most difficult

        tasks in image processing It tasks in image processing It isis difficult difficult

        because of because of many reasonmany reasonss Here are some Here are some

        typical obstaclestypical obstacles Non-uniform illuminationNon-uniform illumination

        No control of the environmentNo control of the environment

        Inadequate model of the object of interestInadequate model of the object of interest

        NoiseNoise

        etcetc

        55

        Segmentation methods can be divided into three groups according to the dominant features they employ

        Segmentation based on global knowledge about an image

        ―The knowledge is usually represented by a histogram of image features

        Edge-based segmentations

        ―Utilizing edge detection processes to find a closed boundary so that an inside and an outside can be defined

        Region-based segmentations

        ―This techniques proceed by dividing the image into regions that exhibit similar properties

        Principal approachesPrincipal approaches

        66

        2 basis properties of intensity 2 basis properties of intensity valuesvalues

        Segmentation algorithms generally are baseSegmentation algorithms generally are based on one of 2 basis properties of intensity vad on one of 2 basis properties of intensity valueslues DiscontinuityDiscontinuity

        ―to partition an image based on abrupt (to partition an image based on abrupt ( 突然突然的的 ) changes in intensity (such as edges)) changes in intensity (such as edges)

        SimilaritySimilarity

        ―to partition an image into regions that are simto partition an image into regions that are similar according to a set of predefined criteriailar according to a set of predefined criteria

        77

        Detection of DiscontinuitiesDetection of Discontinuities

        detect the three basic types of gray detect the three basic types of gray

        level discontinuitieslevel discontinuities

        points lines edgespoints lines edges

        the common way is to run a mask the common way is to run a mask

        through the imagethrough the image

        88

        ContentsContents

        ThresholdingThresholding

        Point DetectionPoint Detection

        Line DetectionLine Detection

        Edge-based SegmentationEdge-based Segmentation

        Region-based SegmentationRegion-based Segmentation

        99

        71 Thresholding71 Thresholding

        Thresholding is a labeling operation Thresholding is a labeling operation

        on a gray scale image that on a gray scale image that

        distinguishes pixels of a higher distinguishes pixels of a higher

        intensity from pixels with a lower intensity from pixels with a lower

        intensity valueintensity value

        The output of thresholding usuallyThe output of thresholding usually is is a a

        binary imagebinary image

        This technique is particularly useful This technique is particularly useful

        for scenes which contain for scenes which contain solid objects solid objects

        on a uniform contrasting backgroundon a uniform contrasting background

        1010

        Classification of ThresholdingClassification of Thresholding

        Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

        where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

        T T x y p x y f x y

        1111

        Classification of ThresholdingClassification of Thresholding

        When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

        1048638 1048638 Global Global thresholdingthresholding

        both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

        Local Local thresholdingthresholding

        x and y (in addition)x and y (in addition)

        Dynamic Dynamic thresholdingthresholding

        T T x y p x y f x y

        1212

        Basic Global ThresholdingBasic Global Thresholding

        Original imageOriginal image HistogramHistogram

        SolutionSolution use T midway between the max and use T midway between the max and

        min gray levelsmin gray levels

        SolutionSolution use T midway between the max and use T midway between the max and

        min gray levelsmin gray levels

        See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

        1313

        Basic Global ThresholdingBasic Global Thresholding

        Let light objects in dark backgroundLet light objects in dark background

        To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

        e backgrounde background

        ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

        A thresholded imageA thresholded image

        1

        0

        if f x y T backgroundg x y

        if f x y T foreground

        1414

        Heuristic Global ThresholdingHeuristic Global Thresholding

        11 Select an initial estimate for TSelect an initial estimate for T

        22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

        33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

        44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

        55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

        1515

        Basic Adaptive ThresholdingBasic Adaptive Thresholding

        subdivide original image into small areassubdivide original image into small areas

        utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

        since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

        See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

        1616

        Multilevel ThresholdingMultilevel Thresholding

        a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

        another object class if f(xy) another object class if f(xy) gtgt T T22

        to background if f(xy) le Tto background if f(xy) le T11

        1717

        The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

        1 1 2 2

        1 2 1

        p z P p z P p z

        P P

        1818

        Optimal ThresholdingOptimal Thresholding

        If the form of the If the form of the

        densities is densities is

        known or known or

        assumed in assumed in

        terms of terms of

        minimum error minimum error

        determining an determining an

        optimal optimal

        threshold for threshold for

        segmenting the segmenting the

        image is image is

        possiblepossible

        1 1 2 2

        1 2 1

        p z P p z P p z

        P P

        1919

        Optimal ThresholdingOptimal Thresholding

        Probability of erroneouslyProbability of erroneously

        2020

        Optimal ThresholdingOptimal Thresholding

        Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

        ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

        find T which makesfind T which makes

        2121

        Optimal ThresholdingOptimal Thresholding

        Minimum errorMinimum error

        Specially if Specially if PP11 = P = P22 then the optimum then the optimum

        threshold is where the curve pthreshold is where the curve p11(z) and (z) and

        pp22(z) intersect(z) intersect

        2222

        Optimal ThresholdingOptimal Thresholding

        For exampleFor example

        Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

        pp22(z)(z)

        where μwhere μ11 and σ and σ1122 are the mean and are the mean and

        variance of the Gaussian density of one variance of the Gaussian density of one

        objectobject

        μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

        the Gaussian density of the other objectthe Gaussian density of the other object

        2323

        Optimal ThresholdingOptimal Thresholding

        Quadratic equation (Quadratic equation (二次方程二次方程 ))

        2424

        Problems of ThresholdingProblems of Thresholding

        Original imageOriginal image Thresholded imageThresholded image

        2525

        Problems of ThresholdingProblems of Thresholding

        (a)(a) Exact threshold Exact threshold

        segmentationsegmentation

        (b)(b) Threshold too lowThreshold too low

        (c)(c) Threshold too Threshold too

        highhigh

        2626

        72 Point Detection72 Point Detection

        a point has been detected at the a point has been detected at the

        location on which the mark is location on which the mark is

        centered ifcentered if

        |R|geT|R|geT

        where T is a nonnegative thresholdwhere T is a nonnegative threshold

        R is the sum of products of the R is the sum of products of the

        coefficients with the gray levels contained coefficients with the gray levels contained

        in the region encompassed by the markin the region encompassed by the mark

        1 1 1

        1 8 1

        1 1 1

        1 1 1

        1 8 1

        1 1 1

        2727

        72 Point Detection72 Point Detection

        Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

        The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

        1 1 1

        1 8 1

        1 1 1

        1 1 1

        1 8 1

        1 1 1

        0 1 0

        1 4 1

        0 1 0

        0 1 0

        1 4 1

        0 1 0

        2828

        ExampleExample

        2929

        73 Line Detection73 Line Detection

        Horizontal mask will result with max Horizontal mask will result with max

        response when a line passed through the response when a line passed through the

        middle row of the mask with a constant middle row of the mask with a constant

        backgroundbackground

        the similar idea is used with other masksthe similar idea is used with other masks

        Note the preferred direction of each mask Note the preferred direction of each mask

        is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

        than other possible directionsthan other possible directions

        1 1 1 1 1 2 1 2 1 2 1 1

        2 2 2 1 2 1 1 2 1 1 2 1

        1 1 1 2 1 1 1 2 1 1 1 2

        45 45Horizontal Vertical

        1 1 1 1 1 2 1 2 1 2 1 1

        2 2 2 1 2 1 1 2 1 1 2 1

        1 1 1 2 1 1 1 2 1 1 1 2

        45 45Horizontal Vertical

        3030

        Idea 1 of Line DetectionIdea 1 of Line Detection

        Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

        izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

        if at a certain point in the imageif at a certain point in the image

        |Ri||Ri|gtgt|Rj||Rj|

        for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

        3131

        Idea 2 of Line DetectionIdea 2 of Line Detection

        Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

        After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

        3232

        ExampleExample

        3333

        74 Edge-based 74 Edge-based SegmentationSegmentation

        Edge-based segmentations rely on edges found in an image by edge detecting operators

        these edges mark image locations of discontinuities in gray level

        Edge detection is the most common approach for detecting meaningful discontinuities in gray level

        There are a large group of methods based on information about edges in the image

        3434

        What is edgeWhat is edge

        Edge is where change occurs Change is measured by derivative in 1D

        ―Biggest change derivative has maximum magnitude

        Or 2nd derivative is zero we discuss approaches for implementing

        ―first-order derivative (Gradient operator)

        ―second-order derivative (Laplacian operator)

        ―we have introduced both derivatives in chapter 3

        ―Here we will talk only about their properties for edge detection

        3535

        What is edgeWhat is edge

        In other wordsIn other words an edge is a set of an edge is a set of

        connected pixelsconnected pixels

        that lie on the boundary between two that lie on the boundary between two

        regions with relatively distinct gray-level regions with relatively distinct gray-level

        propertiesproperties

        Note edge vs boundaryNote edge vs boundary

        ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

        ―whereas a region boundary owing to whereas a region boundary owing to

        the way it is defined is a more global the way it is defined is a more global

        ideaidea

        3636

        Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

        because of because of

        optics optics

        sampling sampling

        image image

        acquisition acquisition

        imperfectionimperfection

        3737

        Thick and Thin EdgeThick and Thin Edge

        The slope of the ramp is inversely The slope of the ramp is inversely

        proportional to the degree of blurring in the proportional to the degree of blurring in the

        edgeedge

        Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

        pathpath

        Instead an edge point now is any point Instead an edge point now is any point

        contained in the ramp and contained in the ramp and an edge would an edge would

        then be a set of such points that are then be a set of such points that are

        connectedconnected

        The thickness is determined by the length of the The thickness is determined by the length of the

        rampramp

        The length is determined by the slope which is in The length is determined by the slope which is in

        turn determined by the degree of blurringturn determined by the degree of blurring

        Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

        edges tend to be thinedges tend to be thin

        3838

        First and Second derivatives (First and Second derivatives ( 导数导数 ))

        the signs of the the signs of the

        derivatives would be derivatives would be

        reversed for an edge reversed for an edge

        that transitions from that transitions from

        light to darklight to dark

        First First derivatderivatee

        SeconSecond d derivatderivatee

        Gray-Gray-level level profileprofile

        3939

        Second derivativesSecond derivatives

        an undesirable featurean undesirable feature

        produces 2 values for every edge in an produces 2 values for every edge in an

        imageimage

        zero-crossing propertyzero-crossing property

        an imaginary straight line joining the an imaginary straight line joining the

        extreme positive and negative values of extreme positive and negative values of

        the second derivative would cross zero the second derivative would cross zero

        near the midpoint of the edgenear the midpoint of the edge

        quite useful for locating the centers of quite useful for locating the centers of

        thick edgesthick edges

        4040

        Basic idea of edge detectionBasic idea of edge detection

        A profile is defined perpendicularly to A profile is defined perpendicularly to

        the edge direction and the results are the edge direction and the results are

        interpretedinterpreted

        The magnitude of the first derivative is The magnitude of the first derivative is

        used to detect an edge (if a point is on a used to detect an edge (if a point is on a

        ramp)ramp)

        The sign of the second derivative can The sign of the second derivative can

        determine whether an edge pixel is on the determine whether an edge pixel is on the

        dark or light side of an edgedark or light side of an edge

        4141

        Review of First DerivateReview of First Derivate

        Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

        Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

        Sobel operators 3Sobel operators 333

        6 5 8 5x yG z z G z z

        1 2 3

        4 5 6

        7 8 9

        z z z

        z z z

        z z z

        1 2 3

        4 5 6

        7 8 9

        z z z

        z z z

        z z z

        9 5 8 6x yG z z G z z 1 0 0 1

        0 1 1 0

        1 0 0 1

        0 1 1 0

        7 8 9 1 2 3

        3 6 9 1 4 7

        2 2

        2 2

        x

        y

        G z z z z z z

        G z z z z z z

        1 2 1 1 0 1

        0 0 0 2 0 2

        1 2 1 1 0 1

        1 2 1 1 0 1

        0 0 0 2 0 2

        1 2 1 1 0 1

        x yf G G

        4242

        Edge direction and strengthEdge direction and strength

        Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

        α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

        The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

        The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

        2 2x yf G G

        4343

        Gradient MasksGradient Masks

        1 0 0 1

        0 1 1 0

        Roberts

        1 0 0 1

        0 1 1 0

        Roberts

        1 2 1 1 0 1

        0 0 0 2 0 2

        1 2 1 1 0 1

        Sobel

        1 2 1 1 0 1

        0 0 0 2 0 2

        1 2 1 1 0 1

        Sobel

        1 1 1 1 0 1

        0 0 0 1 0 1

        1 1 1 1 0 1

        Prewitt

        1 1 1 1 0 1

        0 0 0 1 0 1

        1 1 1 1 0 1

        Prewitt

        4444

        Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

        0 1 1 1 1 0

        1 0 1 1 0 1

        1 1 0 0 1 1

        Prewitt

        0 1 1 1 1 0

        1 0 1 1 0 1

        1 1 0 0 1 1

        Prewitt

        4545

        Review of Second DerivateReview of Second Derivate

        Laplacian OperatorLaplacian Operator

        21 1

        1 1 4

        f x y f x yf

        f x y f x y f x y

        0 1 0

        1 4 1

        0 1 0

        0 1 0

        1 4 1

        0 1 0

        LaplacianLaplacian

        MaskMask

        1 1 1

        1 8 1

        1 1 1

        1 1 1

        1 8 1

        1 1 1

        4646

        Example of edge detectionExample of edge detection

        See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

        Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

        derivative)derivative)

        produces double edgesproduces double edges

        unable to detect edge directionunable to detect edge direction

        4747

        Canny edge detectorCanny edge detector

        The most powerful edge-detection The most powerful edge-detection

        method method

        It differs from the other edge-It differs from the other edge-

        detection methods in that detection methods in that

        it uses two different thresholds (to detect it uses two different thresholds (to detect

        strong and weak edges) strong and weak edges)

        and includes the weak edges in the and includes the weak edges in the

        output only if they are connected to output only if they are connected to

        strong edges strong edges

        This method is therefore less likely This method is therefore less likely

        than the others to be fooled by than the others to be fooled by

        noise and more likely to detect true noise and more likely to detect true

        weak edgesweak edges

        4848

        Laplacian of GaussianLaplacian of Gaussian

        Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

        2 2 22

        4 2

        2 2 2

        2exp

        r rh

        r x y

        determines the degrdetermines the degree of blurring that occee of blurring that occursurs

        4949

        Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

        0 0 1 0 0

        0 1 2 1 0

        1 2 16 2 1

        0 1 2 1 0

        0 0 1 0 0

        0 0 1 0 0

        0 1 2 1 0

        1 2 16 2 1

        0 1 2 1 0

        0 0 1 0 0

        The coefficient must sum to The coefficient must sum to

        zerozero

        5050

        Edge Detection and Edge Detection and SegmentationSegmentation

        Image resulting from edge detection cannot be used as a segmentation result

        Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

        5151

        75 Region-based 75 Region-based SegmentationSegmentation

        GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

        Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

        PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

        For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

        PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

        For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

        5252

        Two methods of Region Two methods of Region SegmentationSegmentation

        Region GrowingRegion Growing

        Region SplittingRegion Splitting

        Region growing is the opposite of the Region growing is the opposite of the

        split and merge approachsplit and merge approach

        5353

        Region GrowingRegion Growing

        The objective of segmentation is to The objective of segmentation is to

        partition an image into regionspartition an image into regions

        A region is a connected component with A region is a connected component with

        some uniformity (say gray-levels or some uniformity (say gray-levels or

        texture)texture)

        In region growing we start with a set In region growing we start with a set

        of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

        appending to each seedrsquos neighbor appending to each seedrsquos neighbor

        pixels if they have pixels if they have similar propertiessimilar properties

        such as specific ranges of gray level such as specific ranges of gray level

        and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

        Need initialization Need initialization similarity similarity

        criterioncriterion

        5454

        Steps of Region GrowingSteps of Region Growing

        Start by choosing an arbitrary seed Start by choosing an arbitrary seed

        pixel andpixel and compare it with neighbor compare it with neighbor

        ppixelsixels

        When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

        is grown from the seed pixel by is grown from the seed pixel by

        addingadding neighboneighborr pixel pixelss

        When the growth of one region stopsWhen the growth of one region stops

        choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

        againagain

        5555

        Region Region growing growing

        An initial set of small An initial set of small

        areas are iterativelyareas are iteratively

        merged according to merged according to

        similarity constraintssimilarity constraints

        5656

        Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

        X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

        We need initial seed points to groWe need initial seed points to grow into regionsw into regions

        On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

        Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

        SeedSeed pointspoints

        5757

        CriterionCriterion

        There is a valley at around 190 in the There is a valley at around 190 in the

        histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

        to be considered as a part of region to the to be considered as a part of region to the

        seed pointseed point

        The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

        to at least one pixel in that regionto at least one pixel in that region

        Result of region growing and boundaries of Result of region growing and boundaries of

        defectsdefects

        5858

        Region SplittingRegion Splitting

        The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

        ption that the entire image is homogeneousption that the entire image is homogeneous

        If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

        This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

        Need homogeneity criterion split ruleNeed homogeneity criterion split rule

        5959

        Region SplittingRegion Splitting

        DisadvantageDisadvantage

        they create regions that may be adjacent they create regions that may be adjacent

        and homogeneous but not mergedand homogeneous but not merged

        6060

        Region Splitting and MergingRegion Splitting and Merging

        ProcedureProcedure

        11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

        22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

        33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

        6161

        Region Splitting and Merging

        Quadtree

        (四叉树 )

        6262

        PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

        where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

        mmii is the mean gray level of that region is the mean gray level of that region

        σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

        ExampleExample

        Original Original

        imageimageThresholded imageThresholded image Result of Result of

        Splitting and Splitting and

        MergingMerging

        • Slide 1
        • Slide 2
        • Slide 3
        • Slide 4
        • Slide 5
        • Slide 6
        • Slide 7
        • Slide 8
        • Slide 9
        • Slide 10
        • Slide 11
        • Slide 12
        • Slide 13
        • Slide 14
        • Slide 15
        • Slide 16
        • Slide 17
        • Slide 18
        • Slide 19
        • Slide 20
        • Slide 21
        • Slide 22
        • Slide 23
        • Slide 24
        • Slide 25
        • Slide 26
        • Slide 27
        • Slide 28
        • Slide 29
        • Slide 30
        • Slide 31
        • Slide 32
        • Slide 33
        • Slide 34
        • Slide 35
        • Slide 36
        • Slide 37
        • Slide 38
        • Slide 39
        • Slide 40
        • Slide 41
        • Slide 42
        • Slide 43
        • Slide 44
        • Slide 45
        • Slide 46
        • Slide 47
        • Slide 48
        • Slide 49
        • Slide 50
        • Slide 51
        • Slide 52
        • Slide 53
        • Slide 54
        • Slide 55
        • Slide 56
        • Slide 57
        • Slide 58
        • Slide 59
        • Slide 60
        • Slide 61
        • Slide 62

          55

          Segmentation methods can be divided into three groups according to the dominant features they employ

          Segmentation based on global knowledge about an image

          ―The knowledge is usually represented by a histogram of image features

          Edge-based segmentations

          ―Utilizing edge detection processes to find a closed boundary so that an inside and an outside can be defined

          Region-based segmentations

          ―This techniques proceed by dividing the image into regions that exhibit similar properties

          Principal approachesPrincipal approaches

          66

          2 basis properties of intensity 2 basis properties of intensity valuesvalues

          Segmentation algorithms generally are baseSegmentation algorithms generally are based on one of 2 basis properties of intensity vad on one of 2 basis properties of intensity valueslues DiscontinuityDiscontinuity

          ―to partition an image based on abrupt (to partition an image based on abrupt ( 突然突然的的 ) changes in intensity (such as edges)) changes in intensity (such as edges)

          SimilaritySimilarity

          ―to partition an image into regions that are simto partition an image into regions that are similar according to a set of predefined criteriailar according to a set of predefined criteria

          77

          Detection of DiscontinuitiesDetection of Discontinuities

          detect the three basic types of gray detect the three basic types of gray

          level discontinuitieslevel discontinuities

          points lines edgespoints lines edges

          the common way is to run a mask the common way is to run a mask

          through the imagethrough the image

          88

          ContentsContents

          ThresholdingThresholding

          Point DetectionPoint Detection

          Line DetectionLine Detection

          Edge-based SegmentationEdge-based Segmentation

          Region-based SegmentationRegion-based Segmentation

          99

          71 Thresholding71 Thresholding

          Thresholding is a labeling operation Thresholding is a labeling operation

          on a gray scale image that on a gray scale image that

          distinguishes pixels of a higher distinguishes pixels of a higher

          intensity from pixels with a lower intensity from pixels with a lower

          intensity valueintensity value

          The output of thresholding usuallyThe output of thresholding usually is is a a

          binary imagebinary image

          This technique is particularly useful This technique is particularly useful

          for scenes which contain for scenes which contain solid objects solid objects

          on a uniform contrasting backgroundon a uniform contrasting background

          1010

          Classification of ThresholdingClassification of Thresholding

          Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

          where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

          T T x y p x y f x y

          1111

          Classification of ThresholdingClassification of Thresholding

          When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

          1048638 1048638 Global Global thresholdingthresholding

          both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

          Local Local thresholdingthresholding

          x and y (in addition)x and y (in addition)

          Dynamic Dynamic thresholdingthresholding

          T T x y p x y f x y

          1212

          Basic Global ThresholdingBasic Global Thresholding

          Original imageOriginal image HistogramHistogram

          SolutionSolution use T midway between the max and use T midway between the max and

          min gray levelsmin gray levels

          SolutionSolution use T midway between the max and use T midway between the max and

          min gray levelsmin gray levels

          See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

          1313

          Basic Global ThresholdingBasic Global Thresholding

          Let light objects in dark backgroundLet light objects in dark background

          To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

          e backgrounde background

          ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

          A thresholded imageA thresholded image

          1

          0

          if f x y T backgroundg x y

          if f x y T foreground

          1414

          Heuristic Global ThresholdingHeuristic Global Thresholding

          11 Select an initial estimate for TSelect an initial estimate for T

          22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

          33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

          44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

          55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

          1515

          Basic Adaptive ThresholdingBasic Adaptive Thresholding

          subdivide original image into small areassubdivide original image into small areas

          utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

          since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

          See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

          1616

          Multilevel ThresholdingMultilevel Thresholding

          a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

          another object class if f(xy) another object class if f(xy) gtgt T T22

          to background if f(xy) le Tto background if f(xy) le T11

          1717

          The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

          1 1 2 2

          1 2 1

          p z P p z P p z

          P P

          1818

          Optimal ThresholdingOptimal Thresholding

          If the form of the If the form of the

          densities is densities is

          known or known or

          assumed in assumed in

          terms of terms of

          minimum error minimum error

          determining an determining an

          optimal optimal

          threshold for threshold for

          segmenting the segmenting the

          image is image is

          possiblepossible

          1 1 2 2

          1 2 1

          p z P p z P p z

          P P

          1919

          Optimal ThresholdingOptimal Thresholding

          Probability of erroneouslyProbability of erroneously

          2020

          Optimal ThresholdingOptimal Thresholding

          Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

          ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

          find T which makesfind T which makes

          2121

          Optimal ThresholdingOptimal Thresholding

          Minimum errorMinimum error

          Specially if Specially if PP11 = P = P22 then the optimum then the optimum

          threshold is where the curve pthreshold is where the curve p11(z) and (z) and

          pp22(z) intersect(z) intersect

          2222

          Optimal ThresholdingOptimal Thresholding

          For exampleFor example

          Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

          pp22(z)(z)

          where μwhere μ11 and σ and σ1122 are the mean and are the mean and

          variance of the Gaussian density of one variance of the Gaussian density of one

          objectobject

          μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

          the Gaussian density of the other objectthe Gaussian density of the other object

          2323

          Optimal ThresholdingOptimal Thresholding

          Quadratic equation (Quadratic equation (二次方程二次方程 ))

          2424

          Problems of ThresholdingProblems of Thresholding

          Original imageOriginal image Thresholded imageThresholded image

          2525

          Problems of ThresholdingProblems of Thresholding

          (a)(a) Exact threshold Exact threshold

          segmentationsegmentation

          (b)(b) Threshold too lowThreshold too low

          (c)(c) Threshold too Threshold too

          highhigh

          2626

          72 Point Detection72 Point Detection

          a point has been detected at the a point has been detected at the

          location on which the mark is location on which the mark is

          centered ifcentered if

          |R|geT|R|geT

          where T is a nonnegative thresholdwhere T is a nonnegative threshold

          R is the sum of products of the R is the sum of products of the

          coefficients with the gray levels contained coefficients with the gray levels contained

          in the region encompassed by the markin the region encompassed by the mark

          1 1 1

          1 8 1

          1 1 1

          1 1 1

          1 8 1

          1 1 1

          2727

          72 Point Detection72 Point Detection

          Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

          The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

          1 1 1

          1 8 1

          1 1 1

          1 1 1

          1 8 1

          1 1 1

          0 1 0

          1 4 1

          0 1 0

          0 1 0

          1 4 1

          0 1 0

          2828

          ExampleExample

          2929

          73 Line Detection73 Line Detection

          Horizontal mask will result with max Horizontal mask will result with max

          response when a line passed through the response when a line passed through the

          middle row of the mask with a constant middle row of the mask with a constant

          backgroundbackground

          the similar idea is used with other masksthe similar idea is used with other masks

          Note the preferred direction of each mask Note the preferred direction of each mask

          is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

          than other possible directionsthan other possible directions

          1 1 1 1 1 2 1 2 1 2 1 1

          2 2 2 1 2 1 1 2 1 1 2 1

          1 1 1 2 1 1 1 2 1 1 1 2

          45 45Horizontal Vertical

          1 1 1 1 1 2 1 2 1 2 1 1

          2 2 2 1 2 1 1 2 1 1 2 1

          1 1 1 2 1 1 1 2 1 1 1 2

          45 45Horizontal Vertical

          3030

          Idea 1 of Line DetectionIdea 1 of Line Detection

          Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

          izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

          if at a certain point in the imageif at a certain point in the image

          |Ri||Ri|gtgt|Rj||Rj|

          for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

          3131

          Idea 2 of Line DetectionIdea 2 of Line Detection

          Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

          After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

          3232

          ExampleExample

          3333

          74 Edge-based 74 Edge-based SegmentationSegmentation

          Edge-based segmentations rely on edges found in an image by edge detecting operators

          these edges mark image locations of discontinuities in gray level

          Edge detection is the most common approach for detecting meaningful discontinuities in gray level

          There are a large group of methods based on information about edges in the image

          3434

          What is edgeWhat is edge

          Edge is where change occurs Change is measured by derivative in 1D

          ―Biggest change derivative has maximum magnitude

          Or 2nd derivative is zero we discuss approaches for implementing

          ―first-order derivative (Gradient operator)

          ―second-order derivative (Laplacian operator)

          ―we have introduced both derivatives in chapter 3

          ―Here we will talk only about their properties for edge detection

          3535

          What is edgeWhat is edge

          In other wordsIn other words an edge is a set of an edge is a set of

          connected pixelsconnected pixels

          that lie on the boundary between two that lie on the boundary between two

          regions with relatively distinct gray-level regions with relatively distinct gray-level

          propertiesproperties

          Note edge vs boundaryNote edge vs boundary

          ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

          ―whereas a region boundary owing to whereas a region boundary owing to

          the way it is defined is a more global the way it is defined is a more global

          ideaidea

          3636

          Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

          because of because of

          optics optics

          sampling sampling

          image image

          acquisition acquisition

          imperfectionimperfection

          3737

          Thick and Thin EdgeThick and Thin Edge

          The slope of the ramp is inversely The slope of the ramp is inversely

          proportional to the degree of blurring in the proportional to the degree of blurring in the

          edgeedge

          Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

          pathpath

          Instead an edge point now is any point Instead an edge point now is any point

          contained in the ramp and contained in the ramp and an edge would an edge would

          then be a set of such points that are then be a set of such points that are

          connectedconnected

          The thickness is determined by the length of the The thickness is determined by the length of the

          rampramp

          The length is determined by the slope which is in The length is determined by the slope which is in

          turn determined by the degree of blurringturn determined by the degree of blurring

          Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

          edges tend to be thinedges tend to be thin

          3838

          First and Second derivatives (First and Second derivatives ( 导数导数 ))

          the signs of the the signs of the

          derivatives would be derivatives would be

          reversed for an edge reversed for an edge

          that transitions from that transitions from

          light to darklight to dark

          First First derivatderivatee

          SeconSecond d derivatderivatee

          Gray-Gray-level level profileprofile

          3939

          Second derivativesSecond derivatives

          an undesirable featurean undesirable feature

          produces 2 values for every edge in an produces 2 values for every edge in an

          imageimage

          zero-crossing propertyzero-crossing property

          an imaginary straight line joining the an imaginary straight line joining the

          extreme positive and negative values of extreme positive and negative values of

          the second derivative would cross zero the second derivative would cross zero

          near the midpoint of the edgenear the midpoint of the edge

          quite useful for locating the centers of quite useful for locating the centers of

          thick edgesthick edges

          4040

          Basic idea of edge detectionBasic idea of edge detection

          A profile is defined perpendicularly to A profile is defined perpendicularly to

          the edge direction and the results are the edge direction and the results are

          interpretedinterpreted

          The magnitude of the first derivative is The magnitude of the first derivative is

          used to detect an edge (if a point is on a used to detect an edge (if a point is on a

          ramp)ramp)

          The sign of the second derivative can The sign of the second derivative can

          determine whether an edge pixel is on the determine whether an edge pixel is on the

          dark or light side of an edgedark or light side of an edge

          4141

          Review of First DerivateReview of First Derivate

          Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

          Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

          Sobel operators 3Sobel operators 333

          6 5 8 5x yG z z G z z

          1 2 3

          4 5 6

          7 8 9

          z z z

          z z z

          z z z

          1 2 3

          4 5 6

          7 8 9

          z z z

          z z z

          z z z

          9 5 8 6x yG z z G z z 1 0 0 1

          0 1 1 0

          1 0 0 1

          0 1 1 0

          7 8 9 1 2 3

          3 6 9 1 4 7

          2 2

          2 2

          x

          y

          G z z z z z z

          G z z z z z z

          1 2 1 1 0 1

          0 0 0 2 0 2

          1 2 1 1 0 1

          1 2 1 1 0 1

          0 0 0 2 0 2

          1 2 1 1 0 1

          x yf G G

          4242

          Edge direction and strengthEdge direction and strength

          Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

          α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

          The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

          The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

          2 2x yf G G

          4343

          Gradient MasksGradient Masks

          1 0 0 1

          0 1 1 0

          Roberts

          1 0 0 1

          0 1 1 0

          Roberts

          1 2 1 1 0 1

          0 0 0 2 0 2

          1 2 1 1 0 1

          Sobel

          1 2 1 1 0 1

          0 0 0 2 0 2

          1 2 1 1 0 1

          Sobel

          1 1 1 1 0 1

          0 0 0 1 0 1

          1 1 1 1 0 1

          Prewitt

          1 1 1 1 0 1

          0 0 0 1 0 1

          1 1 1 1 0 1

          Prewitt

          4444

          Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

          0 1 1 1 1 0

          1 0 1 1 0 1

          1 1 0 0 1 1

          Prewitt

          0 1 1 1 1 0

          1 0 1 1 0 1

          1 1 0 0 1 1

          Prewitt

          4545

          Review of Second DerivateReview of Second Derivate

          Laplacian OperatorLaplacian Operator

          21 1

          1 1 4

          f x y f x yf

          f x y f x y f x y

          0 1 0

          1 4 1

          0 1 0

          0 1 0

          1 4 1

          0 1 0

          LaplacianLaplacian

          MaskMask

          1 1 1

          1 8 1

          1 1 1

          1 1 1

          1 8 1

          1 1 1

          4646

          Example of edge detectionExample of edge detection

          See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

          Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

          derivative)derivative)

          produces double edgesproduces double edges

          unable to detect edge directionunable to detect edge direction

          4747

          Canny edge detectorCanny edge detector

          The most powerful edge-detection The most powerful edge-detection

          method method

          It differs from the other edge-It differs from the other edge-

          detection methods in that detection methods in that

          it uses two different thresholds (to detect it uses two different thresholds (to detect

          strong and weak edges) strong and weak edges)

          and includes the weak edges in the and includes the weak edges in the

          output only if they are connected to output only if they are connected to

          strong edges strong edges

          This method is therefore less likely This method is therefore less likely

          than the others to be fooled by than the others to be fooled by

          noise and more likely to detect true noise and more likely to detect true

          weak edgesweak edges

          4848

          Laplacian of GaussianLaplacian of Gaussian

          Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

          2 2 22

          4 2

          2 2 2

          2exp

          r rh

          r x y

          determines the degrdetermines the degree of blurring that occee of blurring that occursurs

          4949

          Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

          0 0 1 0 0

          0 1 2 1 0

          1 2 16 2 1

          0 1 2 1 0

          0 0 1 0 0

          0 0 1 0 0

          0 1 2 1 0

          1 2 16 2 1

          0 1 2 1 0

          0 0 1 0 0

          The coefficient must sum to The coefficient must sum to

          zerozero

          5050

          Edge Detection and Edge Detection and SegmentationSegmentation

          Image resulting from edge detection cannot be used as a segmentation result

          Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

          5151

          75 Region-based 75 Region-based SegmentationSegmentation

          GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

          Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

          PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

          For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

          PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

          For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

          5252

          Two methods of Region Two methods of Region SegmentationSegmentation

          Region GrowingRegion Growing

          Region SplittingRegion Splitting

          Region growing is the opposite of the Region growing is the opposite of the

          split and merge approachsplit and merge approach

          5353

          Region GrowingRegion Growing

          The objective of segmentation is to The objective of segmentation is to

          partition an image into regionspartition an image into regions

          A region is a connected component with A region is a connected component with

          some uniformity (say gray-levels or some uniformity (say gray-levels or

          texture)texture)

          In region growing we start with a set In region growing we start with a set

          of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

          appending to each seedrsquos neighbor appending to each seedrsquos neighbor

          pixels if they have pixels if they have similar propertiessimilar properties

          such as specific ranges of gray level such as specific ranges of gray level

          and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

          Need initialization Need initialization similarity similarity

          criterioncriterion

          5454

          Steps of Region GrowingSteps of Region Growing

          Start by choosing an arbitrary seed Start by choosing an arbitrary seed

          pixel andpixel and compare it with neighbor compare it with neighbor

          ppixelsixels

          When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

          is grown from the seed pixel by is grown from the seed pixel by

          addingadding neighboneighborr pixel pixelss

          When the growth of one region stopsWhen the growth of one region stops

          choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

          againagain

          5555

          Region Region growing growing

          An initial set of small An initial set of small

          areas are iterativelyareas are iteratively

          merged according to merged according to

          similarity constraintssimilarity constraints

          5656

          Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

          X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

          We need initial seed points to groWe need initial seed points to grow into regionsw into regions

          On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

          Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

          SeedSeed pointspoints

          5757

          CriterionCriterion

          There is a valley at around 190 in the There is a valley at around 190 in the

          histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

          to be considered as a part of region to the to be considered as a part of region to the

          seed pointseed point

          The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

          to at least one pixel in that regionto at least one pixel in that region

          Result of region growing and boundaries of Result of region growing and boundaries of

          defectsdefects

          5858

          Region SplittingRegion Splitting

          The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

          ption that the entire image is homogeneousption that the entire image is homogeneous

          If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

          This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

          Need homogeneity criterion split ruleNeed homogeneity criterion split rule

          5959

          Region SplittingRegion Splitting

          DisadvantageDisadvantage

          they create regions that may be adjacent they create regions that may be adjacent

          and homogeneous but not mergedand homogeneous but not merged

          6060

          Region Splitting and MergingRegion Splitting and Merging

          ProcedureProcedure

          11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

          22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

          33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

          6161

          Region Splitting and Merging

          Quadtree

          (四叉树 )

          6262

          PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

          where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

          mmii is the mean gray level of that region is the mean gray level of that region

          σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

          ExampleExample

          Original Original

          imageimageThresholded imageThresholded image Result of Result of

          Splitting and Splitting and

          MergingMerging

          • Slide 1
          • Slide 2
          • Slide 3
          • Slide 4
          • Slide 5
          • Slide 6
          • Slide 7
          • Slide 8
          • Slide 9
          • Slide 10
          • Slide 11
          • Slide 12
          • Slide 13
          • Slide 14
          • Slide 15
          • Slide 16
          • Slide 17
          • Slide 18
          • Slide 19
          • Slide 20
          • Slide 21
          • Slide 22
          • Slide 23
          • Slide 24
          • Slide 25
          • Slide 26
          • Slide 27
          • Slide 28
          • Slide 29
          • Slide 30
          • Slide 31
          • Slide 32
          • Slide 33
          • Slide 34
          • Slide 35
          • Slide 36
          • Slide 37
          • Slide 38
          • Slide 39
          • Slide 40
          • Slide 41
          • Slide 42
          • Slide 43
          • Slide 44
          • Slide 45
          • Slide 46
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          • Slide 48
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          • Slide 53
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          • Slide 55
          • Slide 56
          • Slide 57
          • Slide 58
          • Slide 59
          • Slide 60
          • Slide 61
          • Slide 62

            66

            2 basis properties of intensity 2 basis properties of intensity valuesvalues

            Segmentation algorithms generally are baseSegmentation algorithms generally are based on one of 2 basis properties of intensity vad on one of 2 basis properties of intensity valueslues DiscontinuityDiscontinuity

            ―to partition an image based on abrupt (to partition an image based on abrupt ( 突然突然的的 ) changes in intensity (such as edges)) changes in intensity (such as edges)

            SimilaritySimilarity

            ―to partition an image into regions that are simto partition an image into regions that are similar according to a set of predefined criteriailar according to a set of predefined criteria

            77

            Detection of DiscontinuitiesDetection of Discontinuities

            detect the three basic types of gray detect the three basic types of gray

            level discontinuitieslevel discontinuities

            points lines edgespoints lines edges

            the common way is to run a mask the common way is to run a mask

            through the imagethrough the image

            88

            ContentsContents

            ThresholdingThresholding

            Point DetectionPoint Detection

            Line DetectionLine Detection

            Edge-based SegmentationEdge-based Segmentation

            Region-based SegmentationRegion-based Segmentation

            99

            71 Thresholding71 Thresholding

            Thresholding is a labeling operation Thresholding is a labeling operation

            on a gray scale image that on a gray scale image that

            distinguishes pixels of a higher distinguishes pixels of a higher

            intensity from pixels with a lower intensity from pixels with a lower

            intensity valueintensity value

            The output of thresholding usuallyThe output of thresholding usually is is a a

            binary imagebinary image

            This technique is particularly useful This technique is particularly useful

            for scenes which contain for scenes which contain solid objects solid objects

            on a uniform contrasting backgroundon a uniform contrasting background

            1010

            Classification of ThresholdingClassification of Thresholding

            Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

            where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

            T T x y p x y f x y

            1111

            Classification of ThresholdingClassification of Thresholding

            When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

            1048638 1048638 Global Global thresholdingthresholding

            both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

            Local Local thresholdingthresholding

            x and y (in addition)x and y (in addition)

            Dynamic Dynamic thresholdingthresholding

            T T x y p x y f x y

            1212

            Basic Global ThresholdingBasic Global Thresholding

            Original imageOriginal image HistogramHistogram

            SolutionSolution use T midway between the max and use T midway between the max and

            min gray levelsmin gray levels

            SolutionSolution use T midway between the max and use T midway between the max and

            min gray levelsmin gray levels

            See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

            1313

            Basic Global ThresholdingBasic Global Thresholding

            Let light objects in dark backgroundLet light objects in dark background

            To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

            e backgrounde background

            ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

            A thresholded imageA thresholded image

            1

            0

            if f x y T backgroundg x y

            if f x y T foreground

            1414

            Heuristic Global ThresholdingHeuristic Global Thresholding

            11 Select an initial estimate for TSelect an initial estimate for T

            22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

            33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

            44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

            55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

            1515

            Basic Adaptive ThresholdingBasic Adaptive Thresholding

            subdivide original image into small areassubdivide original image into small areas

            utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

            since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

            See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

            1616

            Multilevel ThresholdingMultilevel Thresholding

            a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

            another object class if f(xy) another object class if f(xy) gtgt T T22

            to background if f(xy) le Tto background if f(xy) le T11

            1717

            The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

            1 1 2 2

            1 2 1

            p z P p z P p z

            P P

            1818

            Optimal ThresholdingOptimal Thresholding

            If the form of the If the form of the

            densities is densities is

            known or known or

            assumed in assumed in

            terms of terms of

            minimum error minimum error

            determining an determining an

            optimal optimal

            threshold for threshold for

            segmenting the segmenting the

            image is image is

            possiblepossible

            1 1 2 2

            1 2 1

            p z P p z P p z

            P P

            1919

            Optimal ThresholdingOptimal Thresholding

            Probability of erroneouslyProbability of erroneously

            2020

            Optimal ThresholdingOptimal Thresholding

            Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

            ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

            find T which makesfind T which makes

            2121

            Optimal ThresholdingOptimal Thresholding

            Minimum errorMinimum error

            Specially if Specially if PP11 = P = P22 then the optimum then the optimum

            threshold is where the curve pthreshold is where the curve p11(z) and (z) and

            pp22(z) intersect(z) intersect

            2222

            Optimal ThresholdingOptimal Thresholding

            For exampleFor example

            Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

            pp22(z)(z)

            where μwhere μ11 and σ and σ1122 are the mean and are the mean and

            variance of the Gaussian density of one variance of the Gaussian density of one

            objectobject

            μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

            the Gaussian density of the other objectthe Gaussian density of the other object

            2323

            Optimal ThresholdingOptimal Thresholding

            Quadratic equation (Quadratic equation (二次方程二次方程 ))

            2424

            Problems of ThresholdingProblems of Thresholding

            Original imageOriginal image Thresholded imageThresholded image

            2525

            Problems of ThresholdingProblems of Thresholding

            (a)(a) Exact threshold Exact threshold

            segmentationsegmentation

            (b)(b) Threshold too lowThreshold too low

            (c)(c) Threshold too Threshold too

            highhigh

            2626

            72 Point Detection72 Point Detection

            a point has been detected at the a point has been detected at the

            location on which the mark is location on which the mark is

            centered ifcentered if

            |R|geT|R|geT

            where T is a nonnegative thresholdwhere T is a nonnegative threshold

            R is the sum of products of the R is the sum of products of the

            coefficients with the gray levels contained coefficients with the gray levels contained

            in the region encompassed by the markin the region encompassed by the mark

            1 1 1

            1 8 1

            1 1 1

            1 1 1

            1 8 1

            1 1 1

            2727

            72 Point Detection72 Point Detection

            Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

            The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

            1 1 1

            1 8 1

            1 1 1

            1 1 1

            1 8 1

            1 1 1

            0 1 0

            1 4 1

            0 1 0

            0 1 0

            1 4 1

            0 1 0

            2828

            ExampleExample

            2929

            73 Line Detection73 Line Detection

            Horizontal mask will result with max Horizontal mask will result with max

            response when a line passed through the response when a line passed through the

            middle row of the mask with a constant middle row of the mask with a constant

            backgroundbackground

            the similar idea is used with other masksthe similar idea is used with other masks

            Note the preferred direction of each mask Note the preferred direction of each mask

            is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

            than other possible directionsthan other possible directions

            1 1 1 1 1 2 1 2 1 2 1 1

            2 2 2 1 2 1 1 2 1 1 2 1

            1 1 1 2 1 1 1 2 1 1 1 2

            45 45Horizontal Vertical

            1 1 1 1 1 2 1 2 1 2 1 1

            2 2 2 1 2 1 1 2 1 1 2 1

            1 1 1 2 1 1 1 2 1 1 1 2

            45 45Horizontal Vertical

            3030

            Idea 1 of Line DetectionIdea 1 of Line Detection

            Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

            izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

            if at a certain point in the imageif at a certain point in the image

            |Ri||Ri|gtgt|Rj||Rj|

            for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

            3131

            Idea 2 of Line DetectionIdea 2 of Line Detection

            Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

            After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

            3232

            ExampleExample

            3333

            74 Edge-based 74 Edge-based SegmentationSegmentation

            Edge-based segmentations rely on edges found in an image by edge detecting operators

            these edges mark image locations of discontinuities in gray level

            Edge detection is the most common approach for detecting meaningful discontinuities in gray level

            There are a large group of methods based on information about edges in the image

            3434

            What is edgeWhat is edge

            Edge is where change occurs Change is measured by derivative in 1D

            ―Biggest change derivative has maximum magnitude

            Or 2nd derivative is zero we discuss approaches for implementing

            ―first-order derivative (Gradient operator)

            ―second-order derivative (Laplacian operator)

            ―we have introduced both derivatives in chapter 3

            ―Here we will talk only about their properties for edge detection

            3535

            What is edgeWhat is edge

            In other wordsIn other words an edge is a set of an edge is a set of

            connected pixelsconnected pixels

            that lie on the boundary between two that lie on the boundary between two

            regions with relatively distinct gray-level regions with relatively distinct gray-level

            propertiesproperties

            Note edge vs boundaryNote edge vs boundary

            ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

            ―whereas a region boundary owing to whereas a region boundary owing to

            the way it is defined is a more global the way it is defined is a more global

            ideaidea

            3636

            Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

            because of because of

            optics optics

            sampling sampling

            image image

            acquisition acquisition

            imperfectionimperfection

            3737

            Thick and Thin EdgeThick and Thin Edge

            The slope of the ramp is inversely The slope of the ramp is inversely

            proportional to the degree of blurring in the proportional to the degree of blurring in the

            edgeedge

            Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

            pathpath

            Instead an edge point now is any point Instead an edge point now is any point

            contained in the ramp and contained in the ramp and an edge would an edge would

            then be a set of such points that are then be a set of such points that are

            connectedconnected

            The thickness is determined by the length of the The thickness is determined by the length of the

            rampramp

            The length is determined by the slope which is in The length is determined by the slope which is in

            turn determined by the degree of blurringturn determined by the degree of blurring

            Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

            edges tend to be thinedges tend to be thin

            3838

            First and Second derivatives (First and Second derivatives ( 导数导数 ))

            the signs of the the signs of the

            derivatives would be derivatives would be

            reversed for an edge reversed for an edge

            that transitions from that transitions from

            light to darklight to dark

            First First derivatderivatee

            SeconSecond d derivatderivatee

            Gray-Gray-level level profileprofile

            3939

            Second derivativesSecond derivatives

            an undesirable featurean undesirable feature

            produces 2 values for every edge in an produces 2 values for every edge in an

            imageimage

            zero-crossing propertyzero-crossing property

            an imaginary straight line joining the an imaginary straight line joining the

            extreme positive and negative values of extreme positive and negative values of

            the second derivative would cross zero the second derivative would cross zero

            near the midpoint of the edgenear the midpoint of the edge

            quite useful for locating the centers of quite useful for locating the centers of

            thick edgesthick edges

            4040

            Basic idea of edge detectionBasic idea of edge detection

            A profile is defined perpendicularly to A profile is defined perpendicularly to

            the edge direction and the results are the edge direction and the results are

            interpretedinterpreted

            The magnitude of the first derivative is The magnitude of the first derivative is

            used to detect an edge (if a point is on a used to detect an edge (if a point is on a

            ramp)ramp)

            The sign of the second derivative can The sign of the second derivative can

            determine whether an edge pixel is on the determine whether an edge pixel is on the

            dark or light side of an edgedark or light side of an edge

            4141

            Review of First DerivateReview of First Derivate

            Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

            Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

            Sobel operators 3Sobel operators 333

            6 5 8 5x yG z z G z z

            1 2 3

            4 5 6

            7 8 9

            z z z

            z z z

            z z z

            1 2 3

            4 5 6

            7 8 9

            z z z

            z z z

            z z z

            9 5 8 6x yG z z G z z 1 0 0 1

            0 1 1 0

            1 0 0 1

            0 1 1 0

            7 8 9 1 2 3

            3 6 9 1 4 7

            2 2

            2 2

            x

            y

            G z z z z z z

            G z z z z z z

            1 2 1 1 0 1

            0 0 0 2 0 2

            1 2 1 1 0 1

            1 2 1 1 0 1

            0 0 0 2 0 2

            1 2 1 1 0 1

            x yf G G

            4242

            Edge direction and strengthEdge direction and strength

            Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

            α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

            The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

            The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

            2 2x yf G G

            4343

            Gradient MasksGradient Masks

            1 0 0 1

            0 1 1 0

            Roberts

            1 0 0 1

            0 1 1 0

            Roberts

            1 2 1 1 0 1

            0 0 0 2 0 2

            1 2 1 1 0 1

            Sobel

            1 2 1 1 0 1

            0 0 0 2 0 2

            1 2 1 1 0 1

            Sobel

            1 1 1 1 0 1

            0 0 0 1 0 1

            1 1 1 1 0 1

            Prewitt

            1 1 1 1 0 1

            0 0 0 1 0 1

            1 1 1 1 0 1

            Prewitt

            4444

            Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

            0 1 1 1 1 0

            1 0 1 1 0 1

            1 1 0 0 1 1

            Prewitt

            0 1 1 1 1 0

            1 0 1 1 0 1

            1 1 0 0 1 1

            Prewitt

            4545

            Review of Second DerivateReview of Second Derivate

            Laplacian OperatorLaplacian Operator

            21 1

            1 1 4

            f x y f x yf

            f x y f x y f x y

            0 1 0

            1 4 1

            0 1 0

            0 1 0

            1 4 1

            0 1 0

            LaplacianLaplacian

            MaskMask

            1 1 1

            1 8 1

            1 1 1

            1 1 1

            1 8 1

            1 1 1

            4646

            Example of edge detectionExample of edge detection

            See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

            Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

            derivative)derivative)

            produces double edgesproduces double edges

            unable to detect edge directionunable to detect edge direction

            4747

            Canny edge detectorCanny edge detector

            The most powerful edge-detection The most powerful edge-detection

            method method

            It differs from the other edge-It differs from the other edge-

            detection methods in that detection methods in that

            it uses two different thresholds (to detect it uses two different thresholds (to detect

            strong and weak edges) strong and weak edges)

            and includes the weak edges in the and includes the weak edges in the

            output only if they are connected to output only if they are connected to

            strong edges strong edges

            This method is therefore less likely This method is therefore less likely

            than the others to be fooled by than the others to be fooled by

            noise and more likely to detect true noise and more likely to detect true

            weak edgesweak edges

            4848

            Laplacian of GaussianLaplacian of Gaussian

            Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

            2 2 22

            4 2

            2 2 2

            2exp

            r rh

            r x y

            determines the degrdetermines the degree of blurring that occee of blurring that occursurs

            4949

            Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

            0 0 1 0 0

            0 1 2 1 0

            1 2 16 2 1

            0 1 2 1 0

            0 0 1 0 0

            0 0 1 0 0

            0 1 2 1 0

            1 2 16 2 1

            0 1 2 1 0

            0 0 1 0 0

            The coefficient must sum to The coefficient must sum to

            zerozero

            5050

            Edge Detection and Edge Detection and SegmentationSegmentation

            Image resulting from edge detection cannot be used as a segmentation result

            Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

            5151

            75 Region-based 75 Region-based SegmentationSegmentation

            GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

            Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

            PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

            For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

            PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

            For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

            5252

            Two methods of Region Two methods of Region SegmentationSegmentation

            Region GrowingRegion Growing

            Region SplittingRegion Splitting

            Region growing is the opposite of the Region growing is the opposite of the

            split and merge approachsplit and merge approach

            5353

            Region GrowingRegion Growing

            The objective of segmentation is to The objective of segmentation is to

            partition an image into regionspartition an image into regions

            A region is a connected component with A region is a connected component with

            some uniformity (say gray-levels or some uniformity (say gray-levels or

            texture)texture)

            In region growing we start with a set In region growing we start with a set

            of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

            appending to each seedrsquos neighbor appending to each seedrsquos neighbor

            pixels if they have pixels if they have similar propertiessimilar properties

            such as specific ranges of gray level such as specific ranges of gray level

            and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

            Need initialization Need initialization similarity similarity

            criterioncriterion

            5454

            Steps of Region GrowingSteps of Region Growing

            Start by choosing an arbitrary seed Start by choosing an arbitrary seed

            pixel andpixel and compare it with neighbor compare it with neighbor

            ppixelsixels

            When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

            is grown from the seed pixel by is grown from the seed pixel by

            addingadding neighboneighborr pixel pixelss

            When the growth of one region stopsWhen the growth of one region stops

            choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

            againagain

            5555

            Region Region growing growing

            An initial set of small An initial set of small

            areas are iterativelyareas are iteratively

            merged according to merged according to

            similarity constraintssimilarity constraints

            5656

            Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

            X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

            We need initial seed points to groWe need initial seed points to grow into regionsw into regions

            On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

            Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

            SeedSeed pointspoints

            5757

            CriterionCriterion

            There is a valley at around 190 in the There is a valley at around 190 in the

            histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

            to be considered as a part of region to the to be considered as a part of region to the

            seed pointseed point

            The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

            to at least one pixel in that regionto at least one pixel in that region

            Result of region growing and boundaries of Result of region growing and boundaries of

            defectsdefects

            5858

            Region SplittingRegion Splitting

            The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

            ption that the entire image is homogeneousption that the entire image is homogeneous

            If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

            This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

            Need homogeneity criterion split ruleNeed homogeneity criterion split rule

            5959

            Region SplittingRegion Splitting

            DisadvantageDisadvantage

            they create regions that may be adjacent they create regions that may be adjacent

            and homogeneous but not mergedand homogeneous but not merged

            6060

            Region Splitting and MergingRegion Splitting and Merging

            ProcedureProcedure

            11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

            22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

            33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

            6161

            Region Splitting and Merging

            Quadtree

            (四叉树 )

            6262

            PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

            where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

            mmii is the mean gray level of that region is the mean gray level of that region

            σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

            ExampleExample

            Original Original

            imageimageThresholded imageThresholded image Result of Result of

            Splitting and Splitting and

            MergingMerging

            • Slide 1
            • Slide 2
            • Slide 3
            • Slide 4
            • Slide 5
            • Slide 6
            • Slide 7
            • Slide 8
            • Slide 9
            • Slide 10
            • Slide 11
            • Slide 12
            • Slide 13
            • Slide 14
            • Slide 15
            • Slide 16
            • Slide 17
            • Slide 18
            • Slide 19
            • Slide 20
            • Slide 21
            • Slide 22
            • Slide 23
            • Slide 24
            • Slide 25
            • Slide 26
            • Slide 27
            • Slide 28
            • Slide 29
            • Slide 30
            • Slide 31
            • Slide 32
            • Slide 33
            • Slide 34
            • Slide 35
            • Slide 36
            • Slide 37
            • Slide 38
            • Slide 39
            • Slide 40
            • Slide 41
            • Slide 42
            • Slide 43
            • Slide 44
            • Slide 45
            • Slide 46
            • Slide 47
            • Slide 48
            • Slide 49
            • Slide 50
            • Slide 51
            • Slide 52
            • Slide 53
            • Slide 54
            • Slide 55
            • Slide 56
            • Slide 57
            • Slide 58
            • Slide 59
            • Slide 60
            • Slide 61
            • Slide 62

              77

              Detection of DiscontinuitiesDetection of Discontinuities

              detect the three basic types of gray detect the three basic types of gray

              level discontinuitieslevel discontinuities

              points lines edgespoints lines edges

              the common way is to run a mask the common way is to run a mask

              through the imagethrough the image

              88

              ContentsContents

              ThresholdingThresholding

              Point DetectionPoint Detection

              Line DetectionLine Detection

              Edge-based SegmentationEdge-based Segmentation

              Region-based SegmentationRegion-based Segmentation

              99

              71 Thresholding71 Thresholding

              Thresholding is a labeling operation Thresholding is a labeling operation

              on a gray scale image that on a gray scale image that

              distinguishes pixels of a higher distinguishes pixels of a higher

              intensity from pixels with a lower intensity from pixels with a lower

              intensity valueintensity value

              The output of thresholding usuallyThe output of thresholding usually is is a a

              binary imagebinary image

              This technique is particularly useful This technique is particularly useful

              for scenes which contain for scenes which contain solid objects solid objects

              on a uniform contrasting backgroundon a uniform contrasting background

              1010

              Classification of ThresholdingClassification of Thresholding

              Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

              where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

              T T x y p x y f x y

              1111

              Classification of ThresholdingClassification of Thresholding

              When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

              1048638 1048638 Global Global thresholdingthresholding

              both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

              Local Local thresholdingthresholding

              x and y (in addition)x and y (in addition)

              Dynamic Dynamic thresholdingthresholding

              T T x y p x y f x y

              1212

              Basic Global ThresholdingBasic Global Thresholding

              Original imageOriginal image HistogramHistogram

              SolutionSolution use T midway between the max and use T midway between the max and

              min gray levelsmin gray levels

              SolutionSolution use T midway between the max and use T midway between the max and

              min gray levelsmin gray levels

              See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

              1313

              Basic Global ThresholdingBasic Global Thresholding

              Let light objects in dark backgroundLet light objects in dark background

              To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

              e backgrounde background

              ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

              A thresholded imageA thresholded image

              1

              0

              if f x y T backgroundg x y

              if f x y T foreground

              1414

              Heuristic Global ThresholdingHeuristic Global Thresholding

              11 Select an initial estimate for TSelect an initial estimate for T

              22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

              33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

              44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

              55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

              1515

              Basic Adaptive ThresholdingBasic Adaptive Thresholding

              subdivide original image into small areassubdivide original image into small areas

              utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

              since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

              See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

              1616

              Multilevel ThresholdingMultilevel Thresholding

              a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

              another object class if f(xy) another object class if f(xy) gtgt T T22

              to background if f(xy) le Tto background if f(xy) le T11

              1717

              The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

              1 1 2 2

              1 2 1

              p z P p z P p z

              P P

              1818

              Optimal ThresholdingOptimal Thresholding

              If the form of the If the form of the

              densities is densities is

              known or known or

              assumed in assumed in

              terms of terms of

              minimum error minimum error

              determining an determining an

              optimal optimal

              threshold for threshold for

              segmenting the segmenting the

              image is image is

              possiblepossible

              1 1 2 2

              1 2 1

              p z P p z P p z

              P P

              1919

              Optimal ThresholdingOptimal Thresholding

              Probability of erroneouslyProbability of erroneously

              2020

              Optimal ThresholdingOptimal Thresholding

              Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

              ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

              find T which makesfind T which makes

              2121

              Optimal ThresholdingOptimal Thresholding

              Minimum errorMinimum error

              Specially if Specially if PP11 = P = P22 then the optimum then the optimum

              threshold is where the curve pthreshold is where the curve p11(z) and (z) and

              pp22(z) intersect(z) intersect

              2222

              Optimal ThresholdingOptimal Thresholding

              For exampleFor example

              Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

              pp22(z)(z)

              where μwhere μ11 and σ and σ1122 are the mean and are the mean and

              variance of the Gaussian density of one variance of the Gaussian density of one

              objectobject

              μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

              the Gaussian density of the other objectthe Gaussian density of the other object

              2323

              Optimal ThresholdingOptimal Thresholding

              Quadratic equation (Quadratic equation (二次方程二次方程 ))

              2424

              Problems of ThresholdingProblems of Thresholding

              Original imageOriginal image Thresholded imageThresholded image

              2525

              Problems of ThresholdingProblems of Thresholding

              (a)(a) Exact threshold Exact threshold

              segmentationsegmentation

              (b)(b) Threshold too lowThreshold too low

              (c)(c) Threshold too Threshold too

              highhigh

              2626

              72 Point Detection72 Point Detection

              a point has been detected at the a point has been detected at the

              location on which the mark is location on which the mark is

              centered ifcentered if

              |R|geT|R|geT

              where T is a nonnegative thresholdwhere T is a nonnegative threshold

              R is the sum of products of the R is the sum of products of the

              coefficients with the gray levels contained coefficients with the gray levels contained

              in the region encompassed by the markin the region encompassed by the mark

              1 1 1

              1 8 1

              1 1 1

              1 1 1

              1 8 1

              1 1 1

              2727

              72 Point Detection72 Point Detection

              Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

              The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

              1 1 1

              1 8 1

              1 1 1

              1 1 1

              1 8 1

              1 1 1

              0 1 0

              1 4 1

              0 1 0

              0 1 0

              1 4 1

              0 1 0

              2828

              ExampleExample

              2929

              73 Line Detection73 Line Detection

              Horizontal mask will result with max Horizontal mask will result with max

              response when a line passed through the response when a line passed through the

              middle row of the mask with a constant middle row of the mask with a constant

              backgroundbackground

              the similar idea is used with other masksthe similar idea is used with other masks

              Note the preferred direction of each mask Note the preferred direction of each mask

              is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

              than other possible directionsthan other possible directions

              1 1 1 1 1 2 1 2 1 2 1 1

              2 2 2 1 2 1 1 2 1 1 2 1

              1 1 1 2 1 1 1 2 1 1 1 2

              45 45Horizontal Vertical

              1 1 1 1 1 2 1 2 1 2 1 1

              2 2 2 1 2 1 1 2 1 1 2 1

              1 1 1 2 1 1 1 2 1 1 1 2

              45 45Horizontal Vertical

              3030

              Idea 1 of Line DetectionIdea 1 of Line Detection

              Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

              izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

              if at a certain point in the imageif at a certain point in the image

              |Ri||Ri|gtgt|Rj||Rj|

              for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

              3131

              Idea 2 of Line DetectionIdea 2 of Line Detection

              Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

              After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

              3232

              ExampleExample

              3333

              74 Edge-based 74 Edge-based SegmentationSegmentation

              Edge-based segmentations rely on edges found in an image by edge detecting operators

              these edges mark image locations of discontinuities in gray level

              Edge detection is the most common approach for detecting meaningful discontinuities in gray level

              There are a large group of methods based on information about edges in the image

              3434

              What is edgeWhat is edge

              Edge is where change occurs Change is measured by derivative in 1D

              ―Biggest change derivative has maximum magnitude

              Or 2nd derivative is zero we discuss approaches for implementing

              ―first-order derivative (Gradient operator)

              ―second-order derivative (Laplacian operator)

              ―we have introduced both derivatives in chapter 3

              ―Here we will talk only about their properties for edge detection

              3535

              What is edgeWhat is edge

              In other wordsIn other words an edge is a set of an edge is a set of

              connected pixelsconnected pixels

              that lie on the boundary between two that lie on the boundary between two

              regions with relatively distinct gray-level regions with relatively distinct gray-level

              propertiesproperties

              Note edge vs boundaryNote edge vs boundary

              ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

              ―whereas a region boundary owing to whereas a region boundary owing to

              the way it is defined is a more global the way it is defined is a more global

              ideaidea

              3636

              Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

              because of because of

              optics optics

              sampling sampling

              image image

              acquisition acquisition

              imperfectionimperfection

              3737

              Thick and Thin EdgeThick and Thin Edge

              The slope of the ramp is inversely The slope of the ramp is inversely

              proportional to the degree of blurring in the proportional to the degree of blurring in the

              edgeedge

              Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

              pathpath

              Instead an edge point now is any point Instead an edge point now is any point

              contained in the ramp and contained in the ramp and an edge would an edge would

              then be a set of such points that are then be a set of such points that are

              connectedconnected

              The thickness is determined by the length of the The thickness is determined by the length of the

              rampramp

              The length is determined by the slope which is in The length is determined by the slope which is in

              turn determined by the degree of blurringturn determined by the degree of blurring

              Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

              edges tend to be thinedges tend to be thin

              3838

              First and Second derivatives (First and Second derivatives ( 导数导数 ))

              the signs of the the signs of the

              derivatives would be derivatives would be

              reversed for an edge reversed for an edge

              that transitions from that transitions from

              light to darklight to dark

              First First derivatderivatee

              SeconSecond d derivatderivatee

              Gray-Gray-level level profileprofile

              3939

              Second derivativesSecond derivatives

              an undesirable featurean undesirable feature

              produces 2 values for every edge in an produces 2 values for every edge in an

              imageimage

              zero-crossing propertyzero-crossing property

              an imaginary straight line joining the an imaginary straight line joining the

              extreme positive and negative values of extreme positive and negative values of

              the second derivative would cross zero the second derivative would cross zero

              near the midpoint of the edgenear the midpoint of the edge

              quite useful for locating the centers of quite useful for locating the centers of

              thick edgesthick edges

              4040

              Basic idea of edge detectionBasic idea of edge detection

              A profile is defined perpendicularly to A profile is defined perpendicularly to

              the edge direction and the results are the edge direction and the results are

              interpretedinterpreted

              The magnitude of the first derivative is The magnitude of the first derivative is

              used to detect an edge (if a point is on a used to detect an edge (if a point is on a

              ramp)ramp)

              The sign of the second derivative can The sign of the second derivative can

              determine whether an edge pixel is on the determine whether an edge pixel is on the

              dark or light side of an edgedark or light side of an edge

              4141

              Review of First DerivateReview of First Derivate

              Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

              Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

              Sobel operators 3Sobel operators 333

              6 5 8 5x yG z z G z z

              1 2 3

              4 5 6

              7 8 9

              z z z

              z z z

              z z z

              1 2 3

              4 5 6

              7 8 9

              z z z

              z z z

              z z z

              9 5 8 6x yG z z G z z 1 0 0 1

              0 1 1 0

              1 0 0 1

              0 1 1 0

              7 8 9 1 2 3

              3 6 9 1 4 7

              2 2

              2 2

              x

              y

              G z z z z z z

              G z z z z z z

              1 2 1 1 0 1

              0 0 0 2 0 2

              1 2 1 1 0 1

              1 2 1 1 0 1

              0 0 0 2 0 2

              1 2 1 1 0 1

              x yf G G

              4242

              Edge direction and strengthEdge direction and strength

              Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

              α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

              The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

              The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

              2 2x yf G G

              4343

              Gradient MasksGradient Masks

              1 0 0 1

              0 1 1 0

              Roberts

              1 0 0 1

              0 1 1 0

              Roberts

              1 2 1 1 0 1

              0 0 0 2 0 2

              1 2 1 1 0 1

              Sobel

              1 2 1 1 0 1

              0 0 0 2 0 2

              1 2 1 1 0 1

              Sobel

              1 1 1 1 0 1

              0 0 0 1 0 1

              1 1 1 1 0 1

              Prewitt

              1 1 1 1 0 1

              0 0 0 1 0 1

              1 1 1 1 0 1

              Prewitt

              4444

              Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

              0 1 1 1 1 0

              1 0 1 1 0 1

              1 1 0 0 1 1

              Prewitt

              0 1 1 1 1 0

              1 0 1 1 0 1

              1 1 0 0 1 1

              Prewitt

              4545

              Review of Second DerivateReview of Second Derivate

              Laplacian OperatorLaplacian Operator

              21 1

              1 1 4

              f x y f x yf

              f x y f x y f x y

              0 1 0

              1 4 1

              0 1 0

              0 1 0

              1 4 1

              0 1 0

              LaplacianLaplacian

              MaskMask

              1 1 1

              1 8 1

              1 1 1

              1 1 1

              1 8 1

              1 1 1

              4646

              Example of edge detectionExample of edge detection

              See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

              Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

              derivative)derivative)

              produces double edgesproduces double edges

              unable to detect edge directionunable to detect edge direction

              4747

              Canny edge detectorCanny edge detector

              The most powerful edge-detection The most powerful edge-detection

              method method

              It differs from the other edge-It differs from the other edge-

              detection methods in that detection methods in that

              it uses two different thresholds (to detect it uses two different thresholds (to detect

              strong and weak edges) strong and weak edges)

              and includes the weak edges in the and includes the weak edges in the

              output only if they are connected to output only if they are connected to

              strong edges strong edges

              This method is therefore less likely This method is therefore less likely

              than the others to be fooled by than the others to be fooled by

              noise and more likely to detect true noise and more likely to detect true

              weak edgesweak edges

              4848

              Laplacian of GaussianLaplacian of Gaussian

              Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

              2 2 22

              4 2

              2 2 2

              2exp

              r rh

              r x y

              determines the degrdetermines the degree of blurring that occee of blurring that occursurs

              4949

              Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

              0 0 1 0 0

              0 1 2 1 0

              1 2 16 2 1

              0 1 2 1 0

              0 0 1 0 0

              0 0 1 0 0

              0 1 2 1 0

              1 2 16 2 1

              0 1 2 1 0

              0 0 1 0 0

              The coefficient must sum to The coefficient must sum to

              zerozero

              5050

              Edge Detection and Edge Detection and SegmentationSegmentation

              Image resulting from edge detection cannot be used as a segmentation result

              Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

              5151

              75 Region-based 75 Region-based SegmentationSegmentation

              GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

              Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

              PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

              For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

              PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

              For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

              5252

              Two methods of Region Two methods of Region SegmentationSegmentation

              Region GrowingRegion Growing

              Region SplittingRegion Splitting

              Region growing is the opposite of the Region growing is the opposite of the

              split and merge approachsplit and merge approach

              5353

              Region GrowingRegion Growing

              The objective of segmentation is to The objective of segmentation is to

              partition an image into regionspartition an image into regions

              A region is a connected component with A region is a connected component with

              some uniformity (say gray-levels or some uniformity (say gray-levels or

              texture)texture)

              In region growing we start with a set In region growing we start with a set

              of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

              appending to each seedrsquos neighbor appending to each seedrsquos neighbor

              pixels if they have pixels if they have similar propertiessimilar properties

              such as specific ranges of gray level such as specific ranges of gray level

              and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

              Need initialization Need initialization similarity similarity

              criterioncriterion

              5454

              Steps of Region GrowingSteps of Region Growing

              Start by choosing an arbitrary seed Start by choosing an arbitrary seed

              pixel andpixel and compare it with neighbor compare it with neighbor

              ppixelsixels

              When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

              is grown from the seed pixel by is grown from the seed pixel by

              addingadding neighboneighborr pixel pixelss

              When the growth of one region stopsWhen the growth of one region stops

              choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

              againagain

              5555

              Region Region growing growing

              An initial set of small An initial set of small

              areas are iterativelyareas are iteratively

              merged according to merged according to

              similarity constraintssimilarity constraints

              5656

              Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

              X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

              We need initial seed points to groWe need initial seed points to grow into regionsw into regions

              On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

              Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

              SeedSeed pointspoints

              5757

              CriterionCriterion

              There is a valley at around 190 in the There is a valley at around 190 in the

              histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

              to be considered as a part of region to the to be considered as a part of region to the

              seed pointseed point

              The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

              to at least one pixel in that regionto at least one pixel in that region

              Result of region growing and boundaries of Result of region growing and boundaries of

              defectsdefects

              5858

              Region SplittingRegion Splitting

              The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

              ption that the entire image is homogeneousption that the entire image is homogeneous

              If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

              This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

              Need homogeneity criterion split ruleNeed homogeneity criterion split rule

              5959

              Region SplittingRegion Splitting

              DisadvantageDisadvantage

              they create regions that may be adjacent they create regions that may be adjacent

              and homogeneous but not mergedand homogeneous but not merged

              6060

              Region Splitting and MergingRegion Splitting and Merging

              ProcedureProcedure

              11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

              22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

              33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

              6161

              Region Splitting and Merging

              Quadtree

              (四叉树 )

              6262

              PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

              where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

              mmii is the mean gray level of that region is the mean gray level of that region

              σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

              ExampleExample

              Original Original

              imageimageThresholded imageThresholded image Result of Result of

              Splitting and Splitting and

              MergingMerging

              • Slide 1
              • Slide 2
              • Slide 3
              • Slide 4
              • Slide 5
              • Slide 6
              • Slide 7
              • Slide 8
              • Slide 9
              • Slide 10
              • Slide 11
              • Slide 12
              • Slide 13
              • Slide 14
              • Slide 15
              • Slide 16
              • Slide 17
              • Slide 18
              • Slide 19
              • Slide 20
              • Slide 21
              • Slide 22
              • Slide 23
              • Slide 24
              • Slide 25
              • Slide 26
              • Slide 27
              • Slide 28
              • Slide 29
              • Slide 30
              • Slide 31
              • Slide 32
              • Slide 33
              • Slide 34
              • Slide 35
              • Slide 36
              • Slide 37
              • Slide 38
              • Slide 39
              • Slide 40
              • Slide 41
              • Slide 42
              • Slide 43
              • Slide 44
              • Slide 45
              • Slide 46
              • Slide 47
              • Slide 48
              • Slide 49
              • Slide 50
              • Slide 51
              • Slide 52
              • Slide 53
              • Slide 54
              • Slide 55
              • Slide 56
              • Slide 57
              • Slide 58
              • Slide 59
              • Slide 60
              • Slide 61
              • Slide 62

                88

                ContentsContents

                ThresholdingThresholding

                Point DetectionPoint Detection

                Line DetectionLine Detection

                Edge-based SegmentationEdge-based Segmentation

                Region-based SegmentationRegion-based Segmentation

                99

                71 Thresholding71 Thresholding

                Thresholding is a labeling operation Thresholding is a labeling operation

                on a gray scale image that on a gray scale image that

                distinguishes pixels of a higher distinguishes pixels of a higher

                intensity from pixels with a lower intensity from pixels with a lower

                intensity valueintensity value

                The output of thresholding usuallyThe output of thresholding usually is is a a

                binary imagebinary image

                This technique is particularly useful This technique is particularly useful

                for scenes which contain for scenes which contain solid objects solid objects

                on a uniform contrasting backgroundon a uniform contrasting background

                1010

                Classification of ThresholdingClassification of Thresholding

                Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

                where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

                T T x y p x y f x y

                1111

                Classification of ThresholdingClassification of Thresholding

                When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

                1048638 1048638 Global Global thresholdingthresholding

                both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

                Local Local thresholdingthresholding

                x and y (in addition)x and y (in addition)

                Dynamic Dynamic thresholdingthresholding

                T T x y p x y f x y

                1212

                Basic Global ThresholdingBasic Global Thresholding

                Original imageOriginal image HistogramHistogram

                SolutionSolution use T midway between the max and use T midway between the max and

                min gray levelsmin gray levels

                SolutionSolution use T midway between the max and use T midway between the max and

                min gray levelsmin gray levels

                See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

                1313

                Basic Global ThresholdingBasic Global Thresholding

                Let light objects in dark backgroundLet light objects in dark background

                To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

                e backgrounde background

                ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

                A thresholded imageA thresholded image

                1

                0

                if f x y T backgroundg x y

                if f x y T foreground

                1414

                Heuristic Global ThresholdingHeuristic Global Thresholding

                11 Select an initial estimate for TSelect an initial estimate for T

                22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

                33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

                44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

                55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

                1515

                Basic Adaptive ThresholdingBasic Adaptive Thresholding

                subdivide original image into small areassubdivide original image into small areas

                utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

                since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

                See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

                1616

                Multilevel ThresholdingMultilevel Thresholding

                a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

                another object class if f(xy) another object class if f(xy) gtgt T T22

                to background if f(xy) le Tto background if f(xy) le T11

                1717

                The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

                1 1 2 2

                1 2 1

                p z P p z P p z

                P P

                1818

                Optimal ThresholdingOptimal Thresholding

                If the form of the If the form of the

                densities is densities is

                known or known or

                assumed in assumed in

                terms of terms of

                minimum error minimum error

                determining an determining an

                optimal optimal

                threshold for threshold for

                segmenting the segmenting the

                image is image is

                possiblepossible

                1 1 2 2

                1 2 1

                p z P p z P p z

                P P

                1919

                Optimal ThresholdingOptimal Thresholding

                Probability of erroneouslyProbability of erroneously

                2020

                Optimal ThresholdingOptimal Thresholding

                Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                find T which makesfind T which makes

                2121

                Optimal ThresholdingOptimal Thresholding

                Minimum errorMinimum error

                Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                pp22(z) intersect(z) intersect

                2222

                Optimal ThresholdingOptimal Thresholding

                For exampleFor example

                Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                pp22(z)(z)

                where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                variance of the Gaussian density of one variance of the Gaussian density of one

                objectobject

                μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                the Gaussian density of the other objectthe Gaussian density of the other object

                2323

                Optimal ThresholdingOptimal Thresholding

                Quadratic equation (Quadratic equation (二次方程二次方程 ))

                2424

                Problems of ThresholdingProblems of Thresholding

                Original imageOriginal image Thresholded imageThresholded image

                2525

                Problems of ThresholdingProblems of Thresholding

                (a)(a) Exact threshold Exact threshold

                segmentationsegmentation

                (b)(b) Threshold too lowThreshold too low

                (c)(c) Threshold too Threshold too

                highhigh

                2626

                72 Point Detection72 Point Detection

                a point has been detected at the a point has been detected at the

                location on which the mark is location on which the mark is

                centered ifcentered if

                |R|geT|R|geT

                where T is a nonnegative thresholdwhere T is a nonnegative threshold

                R is the sum of products of the R is the sum of products of the

                coefficients with the gray levels contained coefficients with the gray levels contained

                in the region encompassed by the markin the region encompassed by the mark

                1 1 1

                1 8 1

                1 1 1

                1 1 1

                1 8 1

                1 1 1

                2727

                72 Point Detection72 Point Detection

                Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                1 1 1

                1 8 1

                1 1 1

                1 1 1

                1 8 1

                1 1 1

                0 1 0

                1 4 1

                0 1 0

                0 1 0

                1 4 1

                0 1 0

                2828

                ExampleExample

                2929

                73 Line Detection73 Line Detection

                Horizontal mask will result with max Horizontal mask will result with max

                response when a line passed through the response when a line passed through the

                middle row of the mask with a constant middle row of the mask with a constant

                backgroundbackground

                the similar idea is used with other masksthe similar idea is used with other masks

                Note the preferred direction of each mask Note the preferred direction of each mask

                is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                than other possible directionsthan other possible directions

                1 1 1 1 1 2 1 2 1 2 1 1

                2 2 2 1 2 1 1 2 1 1 2 1

                1 1 1 2 1 1 1 2 1 1 1 2

                45 45Horizontal Vertical

                1 1 1 1 1 2 1 2 1 2 1 1

                2 2 2 1 2 1 1 2 1 1 2 1

                1 1 1 2 1 1 1 2 1 1 1 2

                45 45Horizontal Vertical

                3030

                Idea 1 of Line DetectionIdea 1 of Line Detection

                Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                if at a certain point in the imageif at a certain point in the image

                |Ri||Ri|gtgt|Rj||Rj|

                for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                3131

                Idea 2 of Line DetectionIdea 2 of Line Detection

                Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                3232

                ExampleExample

                3333

                74 Edge-based 74 Edge-based SegmentationSegmentation

                Edge-based segmentations rely on edges found in an image by edge detecting operators

                these edges mark image locations of discontinuities in gray level

                Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                There are a large group of methods based on information about edges in the image

                3434

                What is edgeWhat is edge

                Edge is where change occurs Change is measured by derivative in 1D

                ―Biggest change derivative has maximum magnitude

                Or 2nd derivative is zero we discuss approaches for implementing

                ―first-order derivative (Gradient operator)

                ―second-order derivative (Laplacian operator)

                ―we have introduced both derivatives in chapter 3

                ―Here we will talk only about their properties for edge detection

                3535

                What is edgeWhat is edge

                In other wordsIn other words an edge is a set of an edge is a set of

                connected pixelsconnected pixels

                that lie on the boundary between two that lie on the boundary between two

                regions with relatively distinct gray-level regions with relatively distinct gray-level

                propertiesproperties

                Note edge vs boundaryNote edge vs boundary

                ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                ―whereas a region boundary owing to whereas a region boundary owing to

                the way it is defined is a more global the way it is defined is a more global

                ideaidea

                3636

                Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                because of because of

                optics optics

                sampling sampling

                image image

                acquisition acquisition

                imperfectionimperfection

                3737

                Thick and Thin EdgeThick and Thin Edge

                The slope of the ramp is inversely The slope of the ramp is inversely

                proportional to the degree of blurring in the proportional to the degree of blurring in the

                edgeedge

                Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                pathpath

                Instead an edge point now is any point Instead an edge point now is any point

                contained in the ramp and contained in the ramp and an edge would an edge would

                then be a set of such points that are then be a set of such points that are

                connectedconnected

                The thickness is determined by the length of the The thickness is determined by the length of the

                rampramp

                The length is determined by the slope which is in The length is determined by the slope which is in

                turn determined by the degree of blurringturn determined by the degree of blurring

                Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                edges tend to be thinedges tend to be thin

                3838

                First and Second derivatives (First and Second derivatives ( 导数导数 ))

                the signs of the the signs of the

                derivatives would be derivatives would be

                reversed for an edge reversed for an edge

                that transitions from that transitions from

                light to darklight to dark

                First First derivatderivatee

                SeconSecond d derivatderivatee

                Gray-Gray-level level profileprofile

                3939

                Second derivativesSecond derivatives

                an undesirable featurean undesirable feature

                produces 2 values for every edge in an produces 2 values for every edge in an

                imageimage

                zero-crossing propertyzero-crossing property

                an imaginary straight line joining the an imaginary straight line joining the

                extreme positive and negative values of extreme positive and negative values of

                the second derivative would cross zero the second derivative would cross zero

                near the midpoint of the edgenear the midpoint of the edge

                quite useful for locating the centers of quite useful for locating the centers of

                thick edgesthick edges

                4040

                Basic idea of edge detectionBasic idea of edge detection

                A profile is defined perpendicularly to A profile is defined perpendicularly to

                the edge direction and the results are the edge direction and the results are

                interpretedinterpreted

                The magnitude of the first derivative is The magnitude of the first derivative is

                used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                ramp)ramp)

                The sign of the second derivative can The sign of the second derivative can

                determine whether an edge pixel is on the determine whether an edge pixel is on the

                dark or light side of an edgedark or light side of an edge

                4141

                Review of First DerivateReview of First Derivate

                Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                Sobel operators 3Sobel operators 333

                6 5 8 5x yG z z G z z

                1 2 3

                4 5 6

                7 8 9

                z z z

                z z z

                z z z

                1 2 3

                4 5 6

                7 8 9

                z z z

                z z z

                z z z

                9 5 8 6x yG z z G z z 1 0 0 1

                0 1 1 0

                1 0 0 1

                0 1 1 0

                7 8 9 1 2 3

                3 6 9 1 4 7

                2 2

                2 2

                x

                y

                G z z z z z z

                G z z z z z z

                1 2 1 1 0 1

                0 0 0 2 0 2

                1 2 1 1 0 1

                1 2 1 1 0 1

                0 0 0 2 0 2

                1 2 1 1 0 1

                x yf G G

                4242

                Edge direction and strengthEdge direction and strength

                Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                2 2x yf G G

                4343

                Gradient MasksGradient Masks

                1 0 0 1

                0 1 1 0

                Roberts

                1 0 0 1

                0 1 1 0

                Roberts

                1 2 1 1 0 1

                0 0 0 2 0 2

                1 2 1 1 0 1

                Sobel

                1 2 1 1 0 1

                0 0 0 2 0 2

                1 2 1 1 0 1

                Sobel

                1 1 1 1 0 1

                0 0 0 1 0 1

                1 1 1 1 0 1

                Prewitt

                1 1 1 1 0 1

                0 0 0 1 0 1

                1 1 1 1 0 1

                Prewitt

                4444

                Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                0 1 1 1 1 0

                1 0 1 1 0 1

                1 1 0 0 1 1

                Prewitt

                0 1 1 1 1 0

                1 0 1 1 0 1

                1 1 0 0 1 1

                Prewitt

                4545

                Review of Second DerivateReview of Second Derivate

                Laplacian OperatorLaplacian Operator

                21 1

                1 1 4

                f x y f x yf

                f x y f x y f x y

                0 1 0

                1 4 1

                0 1 0

                0 1 0

                1 4 1

                0 1 0

                LaplacianLaplacian

                MaskMask

                1 1 1

                1 8 1

                1 1 1

                1 1 1

                1 8 1

                1 1 1

                4646

                Example of edge detectionExample of edge detection

                See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                derivative)derivative)

                produces double edgesproduces double edges

                unable to detect edge directionunable to detect edge direction

                4747

                Canny edge detectorCanny edge detector

                The most powerful edge-detection The most powerful edge-detection

                method method

                It differs from the other edge-It differs from the other edge-

                detection methods in that detection methods in that

                it uses two different thresholds (to detect it uses two different thresholds (to detect

                strong and weak edges) strong and weak edges)

                and includes the weak edges in the and includes the weak edges in the

                output only if they are connected to output only if they are connected to

                strong edges strong edges

                This method is therefore less likely This method is therefore less likely

                than the others to be fooled by than the others to be fooled by

                noise and more likely to detect true noise and more likely to detect true

                weak edgesweak edges

                4848

                Laplacian of GaussianLaplacian of Gaussian

                Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                2 2 22

                4 2

                2 2 2

                2exp

                r rh

                r x y

                determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                4949

                Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                0 0 1 0 0

                0 1 2 1 0

                1 2 16 2 1

                0 1 2 1 0

                0 0 1 0 0

                0 0 1 0 0

                0 1 2 1 0

                1 2 16 2 1

                0 1 2 1 0

                0 0 1 0 0

                The coefficient must sum to The coefficient must sum to

                zerozero

                5050

                Edge Detection and Edge Detection and SegmentationSegmentation

                Image resulting from edge detection cannot be used as a segmentation result

                Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                5151

                75 Region-based 75 Region-based SegmentationSegmentation

                GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                5252

                Two methods of Region Two methods of Region SegmentationSegmentation

                Region GrowingRegion Growing

                Region SplittingRegion Splitting

                Region growing is the opposite of the Region growing is the opposite of the

                split and merge approachsplit and merge approach

                5353

                Region GrowingRegion Growing

                The objective of segmentation is to The objective of segmentation is to

                partition an image into regionspartition an image into regions

                A region is a connected component with A region is a connected component with

                some uniformity (say gray-levels or some uniformity (say gray-levels or

                texture)texture)

                In region growing we start with a set In region growing we start with a set

                of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                pixels if they have pixels if they have similar propertiessimilar properties

                such as specific ranges of gray level such as specific ranges of gray level

                and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                Need initialization Need initialization similarity similarity

                criterioncriterion

                5454

                Steps of Region GrowingSteps of Region Growing

                Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                pixel andpixel and compare it with neighbor compare it with neighbor

                ppixelsixels

                When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                is grown from the seed pixel by is grown from the seed pixel by

                addingadding neighboneighborr pixel pixelss

                When the growth of one region stopsWhen the growth of one region stops

                choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                againagain

                5555

                Region Region growing growing

                An initial set of small An initial set of small

                areas are iterativelyareas are iteratively

                merged according to merged according to

                similarity constraintssimilarity constraints

                5656

                Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                SeedSeed pointspoints

                5757

                CriterionCriterion

                There is a valley at around 190 in the There is a valley at around 190 in the

                histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                to be considered as a part of region to the to be considered as a part of region to the

                seed pointseed point

                The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                to at least one pixel in that regionto at least one pixel in that region

                Result of region growing and boundaries of Result of region growing and boundaries of

                defectsdefects

                5858

                Region SplittingRegion Splitting

                The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                ption that the entire image is homogeneousption that the entire image is homogeneous

                If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                5959

                Region SplittingRegion Splitting

                DisadvantageDisadvantage

                they create regions that may be adjacent they create regions that may be adjacent

                and homogeneous but not mergedand homogeneous but not merged

                6060

                Region Splitting and MergingRegion Splitting and Merging

                ProcedureProcedure

                11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                6161

                Region Splitting and Merging

                Quadtree

                (四叉树 )

                6262

                PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                mmii is the mean gray level of that region is the mean gray level of that region

                σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                ExampleExample

                Original Original

                imageimageThresholded imageThresholded image Result of Result of

                Splitting and Splitting and

                MergingMerging

                • Slide 1
                • Slide 2
                • Slide 3
                • Slide 4
                • Slide 5
                • Slide 6
                • Slide 7
                • Slide 8
                • Slide 9
                • Slide 10
                • Slide 11
                • Slide 12
                • Slide 13
                • Slide 14
                • Slide 15
                • Slide 16
                • Slide 17
                • Slide 18
                • Slide 19
                • Slide 20
                • Slide 21
                • Slide 22
                • Slide 23
                • Slide 24
                • Slide 25
                • Slide 26
                • Slide 27
                • Slide 28
                • Slide 29
                • Slide 30
                • Slide 31
                • Slide 32
                • Slide 33
                • Slide 34
                • Slide 35
                • Slide 36
                • Slide 37
                • Slide 38
                • Slide 39
                • Slide 40
                • Slide 41
                • Slide 42
                • Slide 43
                • Slide 44
                • Slide 45
                • Slide 46
                • Slide 47
                • Slide 48
                • Slide 49
                • Slide 50
                • Slide 51
                • Slide 52
                • Slide 53
                • Slide 54
                • Slide 55
                • Slide 56
                • Slide 57
                • Slide 58
                • Slide 59
                • Slide 60
                • Slide 61
                • Slide 62

                  99

                  71 Thresholding71 Thresholding

                  Thresholding is a labeling operation Thresholding is a labeling operation

                  on a gray scale image that on a gray scale image that

                  distinguishes pixels of a higher distinguishes pixels of a higher

                  intensity from pixels with a lower intensity from pixels with a lower

                  intensity valueintensity value

                  The output of thresholding usuallyThe output of thresholding usually is is a a

                  binary imagebinary image

                  This technique is particularly useful This technique is particularly useful

                  for scenes which contain for scenes which contain solid objects solid objects

                  on a uniform contrasting backgroundon a uniform contrasting background

                  1010

                  Classification of ThresholdingClassification of Thresholding

                  Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

                  where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

                  T T x y p x y f x y

                  1111

                  Classification of ThresholdingClassification of Thresholding

                  When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

                  1048638 1048638 Global Global thresholdingthresholding

                  both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

                  Local Local thresholdingthresholding

                  x and y (in addition)x and y (in addition)

                  Dynamic Dynamic thresholdingthresholding

                  T T x y p x y f x y

                  1212

                  Basic Global ThresholdingBasic Global Thresholding

                  Original imageOriginal image HistogramHistogram

                  SolutionSolution use T midway between the max and use T midway between the max and

                  min gray levelsmin gray levels

                  SolutionSolution use T midway between the max and use T midway between the max and

                  min gray levelsmin gray levels

                  See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

                  1313

                  Basic Global ThresholdingBasic Global Thresholding

                  Let light objects in dark backgroundLet light objects in dark background

                  To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

                  e backgrounde background

                  ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

                  A thresholded imageA thresholded image

                  1

                  0

                  if f x y T backgroundg x y

                  if f x y T foreground

                  1414

                  Heuristic Global ThresholdingHeuristic Global Thresholding

                  11 Select an initial estimate for TSelect an initial estimate for T

                  22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

                  33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

                  44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

                  55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

                  1515

                  Basic Adaptive ThresholdingBasic Adaptive Thresholding

                  subdivide original image into small areassubdivide original image into small areas

                  utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

                  since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

                  See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

                  1616

                  Multilevel ThresholdingMultilevel Thresholding

                  a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

                  another object class if f(xy) another object class if f(xy) gtgt T T22

                  to background if f(xy) le Tto background if f(xy) le T11

                  1717

                  The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

                  1 1 2 2

                  1 2 1

                  p z P p z P p z

                  P P

                  1818

                  Optimal ThresholdingOptimal Thresholding

                  If the form of the If the form of the

                  densities is densities is

                  known or known or

                  assumed in assumed in

                  terms of terms of

                  minimum error minimum error

                  determining an determining an

                  optimal optimal

                  threshold for threshold for

                  segmenting the segmenting the

                  image is image is

                  possiblepossible

                  1 1 2 2

                  1 2 1

                  p z P p z P p z

                  P P

                  1919

                  Optimal ThresholdingOptimal Thresholding

                  Probability of erroneouslyProbability of erroneously

                  2020

                  Optimal ThresholdingOptimal Thresholding

                  Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                  ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                  find T which makesfind T which makes

                  2121

                  Optimal ThresholdingOptimal Thresholding

                  Minimum errorMinimum error

                  Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                  threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                  pp22(z) intersect(z) intersect

                  2222

                  Optimal ThresholdingOptimal Thresholding

                  For exampleFor example

                  Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                  pp22(z)(z)

                  where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                  variance of the Gaussian density of one variance of the Gaussian density of one

                  objectobject

                  μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                  the Gaussian density of the other objectthe Gaussian density of the other object

                  2323

                  Optimal ThresholdingOptimal Thresholding

                  Quadratic equation (Quadratic equation (二次方程二次方程 ))

                  2424

                  Problems of ThresholdingProblems of Thresholding

                  Original imageOriginal image Thresholded imageThresholded image

                  2525

                  Problems of ThresholdingProblems of Thresholding

                  (a)(a) Exact threshold Exact threshold

                  segmentationsegmentation

                  (b)(b) Threshold too lowThreshold too low

                  (c)(c) Threshold too Threshold too

                  highhigh

                  2626

                  72 Point Detection72 Point Detection

                  a point has been detected at the a point has been detected at the

                  location on which the mark is location on which the mark is

                  centered ifcentered if

                  |R|geT|R|geT

                  where T is a nonnegative thresholdwhere T is a nonnegative threshold

                  R is the sum of products of the R is the sum of products of the

                  coefficients with the gray levels contained coefficients with the gray levels contained

                  in the region encompassed by the markin the region encompassed by the mark

                  1 1 1

                  1 8 1

                  1 1 1

                  1 1 1

                  1 8 1

                  1 1 1

                  2727

                  72 Point Detection72 Point Detection

                  Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                  The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                  1 1 1

                  1 8 1

                  1 1 1

                  1 1 1

                  1 8 1

                  1 1 1

                  0 1 0

                  1 4 1

                  0 1 0

                  0 1 0

                  1 4 1

                  0 1 0

                  2828

                  ExampleExample

                  2929

                  73 Line Detection73 Line Detection

                  Horizontal mask will result with max Horizontal mask will result with max

                  response when a line passed through the response when a line passed through the

                  middle row of the mask with a constant middle row of the mask with a constant

                  backgroundbackground

                  the similar idea is used with other masksthe similar idea is used with other masks

                  Note the preferred direction of each mask Note the preferred direction of each mask

                  is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                  than other possible directionsthan other possible directions

                  1 1 1 1 1 2 1 2 1 2 1 1

                  2 2 2 1 2 1 1 2 1 1 2 1

                  1 1 1 2 1 1 1 2 1 1 1 2

                  45 45Horizontal Vertical

                  1 1 1 1 1 2 1 2 1 2 1 1

                  2 2 2 1 2 1 1 2 1 1 2 1

                  1 1 1 2 1 1 1 2 1 1 1 2

                  45 45Horizontal Vertical

                  3030

                  Idea 1 of Line DetectionIdea 1 of Line Detection

                  Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                  izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                  if at a certain point in the imageif at a certain point in the image

                  |Ri||Ri|gtgt|Rj||Rj|

                  for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                  3131

                  Idea 2 of Line DetectionIdea 2 of Line Detection

                  Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                  After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                  3232

                  ExampleExample

                  3333

                  74 Edge-based 74 Edge-based SegmentationSegmentation

                  Edge-based segmentations rely on edges found in an image by edge detecting operators

                  these edges mark image locations of discontinuities in gray level

                  Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                  There are a large group of methods based on information about edges in the image

                  3434

                  What is edgeWhat is edge

                  Edge is where change occurs Change is measured by derivative in 1D

                  ―Biggest change derivative has maximum magnitude

                  Or 2nd derivative is zero we discuss approaches for implementing

                  ―first-order derivative (Gradient operator)

                  ―second-order derivative (Laplacian operator)

                  ―we have introduced both derivatives in chapter 3

                  ―Here we will talk only about their properties for edge detection

                  3535

                  What is edgeWhat is edge

                  In other wordsIn other words an edge is a set of an edge is a set of

                  connected pixelsconnected pixels

                  that lie on the boundary between two that lie on the boundary between two

                  regions with relatively distinct gray-level regions with relatively distinct gray-level

                  propertiesproperties

                  Note edge vs boundaryNote edge vs boundary

                  ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                  ―whereas a region boundary owing to whereas a region boundary owing to

                  the way it is defined is a more global the way it is defined is a more global

                  ideaidea

                  3636

                  Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                  because of because of

                  optics optics

                  sampling sampling

                  image image

                  acquisition acquisition

                  imperfectionimperfection

                  3737

                  Thick and Thin EdgeThick and Thin Edge

                  The slope of the ramp is inversely The slope of the ramp is inversely

                  proportional to the degree of blurring in the proportional to the degree of blurring in the

                  edgeedge

                  Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                  pathpath

                  Instead an edge point now is any point Instead an edge point now is any point

                  contained in the ramp and contained in the ramp and an edge would an edge would

                  then be a set of such points that are then be a set of such points that are

                  connectedconnected

                  The thickness is determined by the length of the The thickness is determined by the length of the

                  rampramp

                  The length is determined by the slope which is in The length is determined by the slope which is in

                  turn determined by the degree of blurringturn determined by the degree of blurring

                  Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                  edges tend to be thinedges tend to be thin

                  3838

                  First and Second derivatives (First and Second derivatives ( 导数导数 ))

                  the signs of the the signs of the

                  derivatives would be derivatives would be

                  reversed for an edge reversed for an edge

                  that transitions from that transitions from

                  light to darklight to dark

                  First First derivatderivatee

                  SeconSecond d derivatderivatee

                  Gray-Gray-level level profileprofile

                  3939

                  Second derivativesSecond derivatives

                  an undesirable featurean undesirable feature

                  produces 2 values for every edge in an produces 2 values for every edge in an

                  imageimage

                  zero-crossing propertyzero-crossing property

                  an imaginary straight line joining the an imaginary straight line joining the

                  extreme positive and negative values of extreme positive and negative values of

                  the second derivative would cross zero the second derivative would cross zero

                  near the midpoint of the edgenear the midpoint of the edge

                  quite useful for locating the centers of quite useful for locating the centers of

                  thick edgesthick edges

                  4040

                  Basic idea of edge detectionBasic idea of edge detection

                  A profile is defined perpendicularly to A profile is defined perpendicularly to

                  the edge direction and the results are the edge direction and the results are

                  interpretedinterpreted

                  The magnitude of the first derivative is The magnitude of the first derivative is

                  used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                  ramp)ramp)

                  The sign of the second derivative can The sign of the second derivative can

                  determine whether an edge pixel is on the determine whether an edge pixel is on the

                  dark or light side of an edgedark or light side of an edge

                  4141

                  Review of First DerivateReview of First Derivate

                  Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                  Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                  Sobel operators 3Sobel operators 333

                  6 5 8 5x yG z z G z z

                  1 2 3

                  4 5 6

                  7 8 9

                  z z z

                  z z z

                  z z z

                  1 2 3

                  4 5 6

                  7 8 9

                  z z z

                  z z z

                  z z z

                  9 5 8 6x yG z z G z z 1 0 0 1

                  0 1 1 0

                  1 0 0 1

                  0 1 1 0

                  7 8 9 1 2 3

                  3 6 9 1 4 7

                  2 2

                  2 2

                  x

                  y

                  G z z z z z z

                  G z z z z z z

                  1 2 1 1 0 1

                  0 0 0 2 0 2

                  1 2 1 1 0 1

                  1 2 1 1 0 1

                  0 0 0 2 0 2

                  1 2 1 1 0 1

                  x yf G G

                  4242

                  Edge direction and strengthEdge direction and strength

                  Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                  α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                  The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                  The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                  2 2x yf G G

                  4343

                  Gradient MasksGradient Masks

                  1 0 0 1

                  0 1 1 0

                  Roberts

                  1 0 0 1

                  0 1 1 0

                  Roberts

                  1 2 1 1 0 1

                  0 0 0 2 0 2

                  1 2 1 1 0 1

                  Sobel

                  1 2 1 1 0 1

                  0 0 0 2 0 2

                  1 2 1 1 0 1

                  Sobel

                  1 1 1 1 0 1

                  0 0 0 1 0 1

                  1 1 1 1 0 1

                  Prewitt

                  1 1 1 1 0 1

                  0 0 0 1 0 1

                  1 1 1 1 0 1

                  Prewitt

                  4444

                  Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                  0 1 1 1 1 0

                  1 0 1 1 0 1

                  1 1 0 0 1 1

                  Prewitt

                  0 1 1 1 1 0

                  1 0 1 1 0 1

                  1 1 0 0 1 1

                  Prewitt

                  4545

                  Review of Second DerivateReview of Second Derivate

                  Laplacian OperatorLaplacian Operator

                  21 1

                  1 1 4

                  f x y f x yf

                  f x y f x y f x y

                  0 1 0

                  1 4 1

                  0 1 0

                  0 1 0

                  1 4 1

                  0 1 0

                  LaplacianLaplacian

                  MaskMask

                  1 1 1

                  1 8 1

                  1 1 1

                  1 1 1

                  1 8 1

                  1 1 1

                  4646

                  Example of edge detectionExample of edge detection

                  See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                  Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                  derivative)derivative)

                  produces double edgesproduces double edges

                  unable to detect edge directionunable to detect edge direction

                  4747

                  Canny edge detectorCanny edge detector

                  The most powerful edge-detection The most powerful edge-detection

                  method method

                  It differs from the other edge-It differs from the other edge-

                  detection methods in that detection methods in that

                  it uses two different thresholds (to detect it uses two different thresholds (to detect

                  strong and weak edges) strong and weak edges)

                  and includes the weak edges in the and includes the weak edges in the

                  output only if they are connected to output only if they are connected to

                  strong edges strong edges

                  This method is therefore less likely This method is therefore less likely

                  than the others to be fooled by than the others to be fooled by

                  noise and more likely to detect true noise and more likely to detect true

                  weak edgesweak edges

                  4848

                  Laplacian of GaussianLaplacian of Gaussian

                  Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                  2 2 22

                  4 2

                  2 2 2

                  2exp

                  r rh

                  r x y

                  determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                  4949

                  Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                  0 0 1 0 0

                  0 1 2 1 0

                  1 2 16 2 1

                  0 1 2 1 0

                  0 0 1 0 0

                  0 0 1 0 0

                  0 1 2 1 0

                  1 2 16 2 1

                  0 1 2 1 0

                  0 0 1 0 0

                  The coefficient must sum to The coefficient must sum to

                  zerozero

                  5050

                  Edge Detection and Edge Detection and SegmentationSegmentation

                  Image resulting from edge detection cannot be used as a segmentation result

                  Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                  5151

                  75 Region-based 75 Region-based SegmentationSegmentation

                  GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                  Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                  PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                  For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                  PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                  For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                  5252

                  Two methods of Region Two methods of Region SegmentationSegmentation

                  Region GrowingRegion Growing

                  Region SplittingRegion Splitting

                  Region growing is the opposite of the Region growing is the opposite of the

                  split and merge approachsplit and merge approach

                  5353

                  Region GrowingRegion Growing

                  The objective of segmentation is to The objective of segmentation is to

                  partition an image into regionspartition an image into regions

                  A region is a connected component with A region is a connected component with

                  some uniformity (say gray-levels or some uniformity (say gray-levels or

                  texture)texture)

                  In region growing we start with a set In region growing we start with a set

                  of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                  appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                  pixels if they have pixels if they have similar propertiessimilar properties

                  such as specific ranges of gray level such as specific ranges of gray level

                  and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                  Need initialization Need initialization similarity similarity

                  criterioncriterion

                  5454

                  Steps of Region GrowingSteps of Region Growing

                  Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                  pixel andpixel and compare it with neighbor compare it with neighbor

                  ppixelsixels

                  When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                  is grown from the seed pixel by is grown from the seed pixel by

                  addingadding neighboneighborr pixel pixelss

                  When the growth of one region stopsWhen the growth of one region stops

                  choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                  againagain

                  5555

                  Region Region growing growing

                  An initial set of small An initial set of small

                  areas are iterativelyareas are iteratively

                  merged according to merged according to

                  similarity constraintssimilarity constraints

                  5656

                  Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                  X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                  We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                  On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                  Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                  SeedSeed pointspoints

                  5757

                  CriterionCriterion

                  There is a valley at around 190 in the There is a valley at around 190 in the

                  histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                  to be considered as a part of region to the to be considered as a part of region to the

                  seed pointseed point

                  The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                  to at least one pixel in that regionto at least one pixel in that region

                  Result of region growing and boundaries of Result of region growing and boundaries of

                  defectsdefects

                  5858

                  Region SplittingRegion Splitting

                  The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                  ption that the entire image is homogeneousption that the entire image is homogeneous

                  If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                  This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                  Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                  5959

                  Region SplittingRegion Splitting

                  DisadvantageDisadvantage

                  they create regions that may be adjacent they create regions that may be adjacent

                  and homogeneous but not mergedand homogeneous but not merged

                  6060

                  Region Splitting and MergingRegion Splitting and Merging

                  ProcedureProcedure

                  11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                  22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                  33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                  6161

                  Region Splitting and Merging

                  Quadtree

                  (四叉树 )

                  6262

                  PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                  where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                  mmii is the mean gray level of that region is the mean gray level of that region

                  σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                  ExampleExample

                  Original Original

                  imageimageThresholded imageThresholded image Result of Result of

                  Splitting and Splitting and

                  MergingMerging

                  • Slide 1
                  • Slide 2
                  • Slide 3
                  • Slide 4
                  • Slide 5
                  • Slide 6
                  • Slide 7
                  • Slide 8
                  • Slide 9
                  • Slide 10
                  • Slide 11
                  • Slide 12
                  • Slide 13
                  • Slide 14
                  • Slide 15
                  • Slide 16
                  • Slide 17
                  • Slide 18
                  • Slide 19
                  • Slide 20
                  • Slide 21
                  • Slide 22
                  • Slide 23
                  • Slide 24
                  • Slide 25
                  • Slide 26
                  • Slide 27
                  • Slide 28
                  • Slide 29
                  • Slide 30
                  • Slide 31
                  • Slide 32
                  • Slide 33
                  • Slide 34
                  • Slide 35
                  • Slide 36
                  • Slide 37
                  • Slide 38
                  • Slide 39
                  • Slide 40
                  • Slide 41
                  • Slide 42
                  • Slide 43
                  • Slide 44
                  • Slide 45
                  • Slide 46
                  • Slide 47
                  • Slide 48
                  • Slide 49
                  • Slide 50
                  • Slide 51
                  • Slide 52
                  • Slide 53
                  • Slide 54
                  • Slide 55
                  • Slide 56
                  • Slide 57
                  • Slide 58
                  • Slide 59
                  • Slide 60
                  • Slide 61
                  • Slide 62

                    1010

                    Classification of ThresholdingClassification of Thresholding

                    Thresholding can be viewed as an operation Thresholding can be viewed as an operation that involves tests against a function T of ththat involves tests against a function T of the forme form

                    where p(xy) denotes some local property of this where p(xy) denotes some local property of this pointpoint

                    T T x y p x y f x y

                    1111

                    Classification of ThresholdingClassification of Thresholding

                    When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

                    1048638 1048638 Global Global thresholdingthresholding

                    both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

                    Local Local thresholdingthresholding

                    x and y (in addition)x and y (in addition)

                    Dynamic Dynamic thresholdingthresholding

                    T T x y p x y f x y

                    1212

                    Basic Global ThresholdingBasic Global Thresholding

                    Original imageOriginal image HistogramHistogram

                    SolutionSolution use T midway between the max and use T midway between the max and

                    min gray levelsmin gray levels

                    SolutionSolution use T midway between the max and use T midway between the max and

                    min gray levelsmin gray levels

                    See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

                    1313

                    Basic Global ThresholdingBasic Global Thresholding

                    Let light objects in dark backgroundLet light objects in dark background

                    To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

                    e backgrounde background

                    ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

                    A thresholded imageA thresholded image

                    1

                    0

                    if f x y T backgroundg x y

                    if f x y T foreground

                    1414

                    Heuristic Global ThresholdingHeuristic Global Thresholding

                    11 Select an initial estimate for TSelect an initial estimate for T

                    22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

                    33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

                    44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

                    55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

                    1515

                    Basic Adaptive ThresholdingBasic Adaptive Thresholding

                    subdivide original image into small areassubdivide original image into small areas

                    utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

                    since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

                    See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

                    1616

                    Multilevel ThresholdingMultilevel Thresholding

                    a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

                    another object class if f(xy) another object class if f(xy) gtgt T T22

                    to background if f(xy) le Tto background if f(xy) le T11

                    1717

                    The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

                    1 1 2 2

                    1 2 1

                    p z P p z P p z

                    P P

                    1818

                    Optimal ThresholdingOptimal Thresholding

                    If the form of the If the form of the

                    densities is densities is

                    known or known or

                    assumed in assumed in

                    terms of terms of

                    minimum error minimum error

                    determining an determining an

                    optimal optimal

                    threshold for threshold for

                    segmenting the segmenting the

                    image is image is

                    possiblepossible

                    1 1 2 2

                    1 2 1

                    p z P p z P p z

                    P P

                    1919

                    Optimal ThresholdingOptimal Thresholding

                    Probability of erroneouslyProbability of erroneously

                    2020

                    Optimal ThresholdingOptimal Thresholding

                    Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                    ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                    find T which makesfind T which makes

                    2121

                    Optimal ThresholdingOptimal Thresholding

                    Minimum errorMinimum error

                    Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                    threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                    pp22(z) intersect(z) intersect

                    2222

                    Optimal ThresholdingOptimal Thresholding

                    For exampleFor example

                    Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                    pp22(z)(z)

                    where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                    variance of the Gaussian density of one variance of the Gaussian density of one

                    objectobject

                    μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                    the Gaussian density of the other objectthe Gaussian density of the other object

                    2323

                    Optimal ThresholdingOptimal Thresholding

                    Quadratic equation (Quadratic equation (二次方程二次方程 ))

                    2424

                    Problems of ThresholdingProblems of Thresholding

                    Original imageOriginal image Thresholded imageThresholded image

                    2525

                    Problems of ThresholdingProblems of Thresholding

                    (a)(a) Exact threshold Exact threshold

                    segmentationsegmentation

                    (b)(b) Threshold too lowThreshold too low

                    (c)(c) Threshold too Threshold too

                    highhigh

                    2626

                    72 Point Detection72 Point Detection

                    a point has been detected at the a point has been detected at the

                    location on which the mark is location on which the mark is

                    centered ifcentered if

                    |R|geT|R|geT

                    where T is a nonnegative thresholdwhere T is a nonnegative threshold

                    R is the sum of products of the R is the sum of products of the

                    coefficients with the gray levels contained coefficients with the gray levels contained

                    in the region encompassed by the markin the region encompassed by the mark

                    1 1 1

                    1 8 1

                    1 1 1

                    1 1 1

                    1 8 1

                    1 1 1

                    2727

                    72 Point Detection72 Point Detection

                    Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                    The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                    1 1 1

                    1 8 1

                    1 1 1

                    1 1 1

                    1 8 1

                    1 1 1

                    0 1 0

                    1 4 1

                    0 1 0

                    0 1 0

                    1 4 1

                    0 1 0

                    2828

                    ExampleExample

                    2929

                    73 Line Detection73 Line Detection

                    Horizontal mask will result with max Horizontal mask will result with max

                    response when a line passed through the response when a line passed through the

                    middle row of the mask with a constant middle row of the mask with a constant

                    backgroundbackground

                    the similar idea is used with other masksthe similar idea is used with other masks

                    Note the preferred direction of each mask Note the preferred direction of each mask

                    is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                    than other possible directionsthan other possible directions

                    1 1 1 1 1 2 1 2 1 2 1 1

                    2 2 2 1 2 1 1 2 1 1 2 1

                    1 1 1 2 1 1 1 2 1 1 1 2

                    45 45Horizontal Vertical

                    1 1 1 1 1 2 1 2 1 2 1 1

                    2 2 2 1 2 1 1 2 1 1 2 1

                    1 1 1 2 1 1 1 2 1 1 1 2

                    45 45Horizontal Vertical

                    3030

                    Idea 1 of Line DetectionIdea 1 of Line Detection

                    Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                    izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                    if at a certain point in the imageif at a certain point in the image

                    |Ri||Ri|gtgt|Rj||Rj|

                    for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                    3131

                    Idea 2 of Line DetectionIdea 2 of Line Detection

                    Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                    After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                    3232

                    ExampleExample

                    3333

                    74 Edge-based 74 Edge-based SegmentationSegmentation

                    Edge-based segmentations rely on edges found in an image by edge detecting operators

                    these edges mark image locations of discontinuities in gray level

                    Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                    There are a large group of methods based on information about edges in the image

                    3434

                    What is edgeWhat is edge

                    Edge is where change occurs Change is measured by derivative in 1D

                    ―Biggest change derivative has maximum magnitude

                    Or 2nd derivative is zero we discuss approaches for implementing

                    ―first-order derivative (Gradient operator)

                    ―second-order derivative (Laplacian operator)

                    ―we have introduced both derivatives in chapter 3

                    ―Here we will talk only about their properties for edge detection

                    3535

                    What is edgeWhat is edge

                    In other wordsIn other words an edge is a set of an edge is a set of

                    connected pixelsconnected pixels

                    that lie on the boundary between two that lie on the boundary between two

                    regions with relatively distinct gray-level regions with relatively distinct gray-level

                    propertiesproperties

                    Note edge vs boundaryNote edge vs boundary

                    ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                    ―whereas a region boundary owing to whereas a region boundary owing to

                    the way it is defined is a more global the way it is defined is a more global

                    ideaidea

                    3636

                    Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                    because of because of

                    optics optics

                    sampling sampling

                    image image

                    acquisition acquisition

                    imperfectionimperfection

                    3737

                    Thick and Thin EdgeThick and Thin Edge

                    The slope of the ramp is inversely The slope of the ramp is inversely

                    proportional to the degree of blurring in the proportional to the degree of blurring in the

                    edgeedge

                    Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                    pathpath

                    Instead an edge point now is any point Instead an edge point now is any point

                    contained in the ramp and contained in the ramp and an edge would an edge would

                    then be a set of such points that are then be a set of such points that are

                    connectedconnected

                    The thickness is determined by the length of the The thickness is determined by the length of the

                    rampramp

                    The length is determined by the slope which is in The length is determined by the slope which is in

                    turn determined by the degree of blurringturn determined by the degree of blurring

                    Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                    edges tend to be thinedges tend to be thin

                    3838

                    First and Second derivatives (First and Second derivatives ( 导数导数 ))

                    the signs of the the signs of the

                    derivatives would be derivatives would be

                    reversed for an edge reversed for an edge

                    that transitions from that transitions from

                    light to darklight to dark

                    First First derivatderivatee

                    SeconSecond d derivatderivatee

                    Gray-Gray-level level profileprofile

                    3939

                    Second derivativesSecond derivatives

                    an undesirable featurean undesirable feature

                    produces 2 values for every edge in an produces 2 values for every edge in an

                    imageimage

                    zero-crossing propertyzero-crossing property

                    an imaginary straight line joining the an imaginary straight line joining the

                    extreme positive and negative values of extreme positive and negative values of

                    the second derivative would cross zero the second derivative would cross zero

                    near the midpoint of the edgenear the midpoint of the edge

                    quite useful for locating the centers of quite useful for locating the centers of

                    thick edgesthick edges

                    4040

                    Basic idea of edge detectionBasic idea of edge detection

                    A profile is defined perpendicularly to A profile is defined perpendicularly to

                    the edge direction and the results are the edge direction and the results are

                    interpretedinterpreted

                    The magnitude of the first derivative is The magnitude of the first derivative is

                    used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                    ramp)ramp)

                    The sign of the second derivative can The sign of the second derivative can

                    determine whether an edge pixel is on the determine whether an edge pixel is on the

                    dark or light side of an edgedark or light side of an edge

                    4141

                    Review of First DerivateReview of First Derivate

                    Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                    Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                    Sobel operators 3Sobel operators 333

                    6 5 8 5x yG z z G z z

                    1 2 3

                    4 5 6

                    7 8 9

                    z z z

                    z z z

                    z z z

                    1 2 3

                    4 5 6

                    7 8 9

                    z z z

                    z z z

                    z z z

                    9 5 8 6x yG z z G z z 1 0 0 1

                    0 1 1 0

                    1 0 0 1

                    0 1 1 0

                    7 8 9 1 2 3

                    3 6 9 1 4 7

                    2 2

                    2 2

                    x

                    y

                    G z z z z z z

                    G z z z z z z

                    1 2 1 1 0 1

                    0 0 0 2 0 2

                    1 2 1 1 0 1

                    1 2 1 1 0 1

                    0 0 0 2 0 2

                    1 2 1 1 0 1

                    x yf G G

                    4242

                    Edge direction and strengthEdge direction and strength

                    Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                    α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                    The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                    The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                    2 2x yf G G

                    4343

                    Gradient MasksGradient Masks

                    1 0 0 1

                    0 1 1 0

                    Roberts

                    1 0 0 1

                    0 1 1 0

                    Roberts

                    1 2 1 1 0 1

                    0 0 0 2 0 2

                    1 2 1 1 0 1

                    Sobel

                    1 2 1 1 0 1

                    0 0 0 2 0 2

                    1 2 1 1 0 1

                    Sobel

                    1 1 1 1 0 1

                    0 0 0 1 0 1

                    1 1 1 1 0 1

                    Prewitt

                    1 1 1 1 0 1

                    0 0 0 1 0 1

                    1 1 1 1 0 1

                    Prewitt

                    4444

                    Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                    0 1 1 1 1 0

                    1 0 1 1 0 1

                    1 1 0 0 1 1

                    Prewitt

                    0 1 1 1 1 0

                    1 0 1 1 0 1

                    1 1 0 0 1 1

                    Prewitt

                    4545

                    Review of Second DerivateReview of Second Derivate

                    Laplacian OperatorLaplacian Operator

                    21 1

                    1 1 4

                    f x y f x yf

                    f x y f x y f x y

                    0 1 0

                    1 4 1

                    0 1 0

                    0 1 0

                    1 4 1

                    0 1 0

                    LaplacianLaplacian

                    MaskMask

                    1 1 1

                    1 8 1

                    1 1 1

                    1 1 1

                    1 8 1

                    1 1 1

                    4646

                    Example of edge detectionExample of edge detection

                    See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                    Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                    derivative)derivative)

                    produces double edgesproduces double edges

                    unable to detect edge directionunable to detect edge direction

                    4747

                    Canny edge detectorCanny edge detector

                    The most powerful edge-detection The most powerful edge-detection

                    method method

                    It differs from the other edge-It differs from the other edge-

                    detection methods in that detection methods in that

                    it uses two different thresholds (to detect it uses two different thresholds (to detect

                    strong and weak edges) strong and weak edges)

                    and includes the weak edges in the and includes the weak edges in the

                    output only if they are connected to output only if they are connected to

                    strong edges strong edges

                    This method is therefore less likely This method is therefore less likely

                    than the others to be fooled by than the others to be fooled by

                    noise and more likely to detect true noise and more likely to detect true

                    weak edgesweak edges

                    4848

                    Laplacian of GaussianLaplacian of Gaussian

                    Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                    2 2 22

                    4 2

                    2 2 2

                    2exp

                    r rh

                    r x y

                    determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                    4949

                    Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                    0 0 1 0 0

                    0 1 2 1 0

                    1 2 16 2 1

                    0 1 2 1 0

                    0 0 1 0 0

                    0 0 1 0 0

                    0 1 2 1 0

                    1 2 16 2 1

                    0 1 2 1 0

                    0 0 1 0 0

                    The coefficient must sum to The coefficient must sum to

                    zerozero

                    5050

                    Edge Detection and Edge Detection and SegmentationSegmentation

                    Image resulting from edge detection cannot be used as a segmentation result

                    Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                    5151

                    75 Region-based 75 Region-based SegmentationSegmentation

                    GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                    Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                    5252

                    Two methods of Region Two methods of Region SegmentationSegmentation

                    Region GrowingRegion Growing

                    Region SplittingRegion Splitting

                    Region growing is the opposite of the Region growing is the opposite of the

                    split and merge approachsplit and merge approach

                    5353

                    Region GrowingRegion Growing

                    The objective of segmentation is to The objective of segmentation is to

                    partition an image into regionspartition an image into regions

                    A region is a connected component with A region is a connected component with

                    some uniformity (say gray-levels or some uniformity (say gray-levels or

                    texture)texture)

                    In region growing we start with a set In region growing we start with a set

                    of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                    appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                    pixels if they have pixels if they have similar propertiessimilar properties

                    such as specific ranges of gray level such as specific ranges of gray level

                    and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                    Need initialization Need initialization similarity similarity

                    criterioncriterion

                    5454

                    Steps of Region GrowingSteps of Region Growing

                    Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                    pixel andpixel and compare it with neighbor compare it with neighbor

                    ppixelsixels

                    When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                    is grown from the seed pixel by is grown from the seed pixel by

                    addingadding neighboneighborr pixel pixelss

                    When the growth of one region stopsWhen the growth of one region stops

                    choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                    againagain

                    5555

                    Region Region growing growing

                    An initial set of small An initial set of small

                    areas are iterativelyareas are iteratively

                    merged according to merged according to

                    similarity constraintssimilarity constraints

                    5656

                    Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                    X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                    We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                    On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                    Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                    SeedSeed pointspoints

                    5757

                    CriterionCriterion

                    There is a valley at around 190 in the There is a valley at around 190 in the

                    histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                    to be considered as a part of region to the to be considered as a part of region to the

                    seed pointseed point

                    The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                    to at least one pixel in that regionto at least one pixel in that region

                    Result of region growing and boundaries of Result of region growing and boundaries of

                    defectsdefects

                    5858

                    Region SplittingRegion Splitting

                    The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                    ption that the entire image is homogeneousption that the entire image is homogeneous

                    If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                    This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                    Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                    5959

                    Region SplittingRegion Splitting

                    DisadvantageDisadvantage

                    they create regions that may be adjacent they create regions that may be adjacent

                    and homogeneous but not mergedand homogeneous but not merged

                    6060

                    Region Splitting and MergingRegion Splitting and Merging

                    ProcedureProcedure

                    11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                    22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                    33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                    6161

                    Region Splitting and Merging

                    Quadtree

                    (四叉树 )

                    6262

                    PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                    where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                    mmii is the mean gray level of that region is the mean gray level of that region

                    σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                    ExampleExample

                    Original Original

                    imageimageThresholded imageThresholded image Result of Result of

                    Splitting and Splitting and

                    MergingMerging

                    • Slide 1
                    • Slide 2
                    • Slide 3
                    • Slide 4
                    • Slide 5
                    • Slide 6
                    • Slide 7
                    • Slide 8
                    • Slide 9
                    • Slide 10
                    • Slide 11
                    • Slide 12
                    • Slide 13
                    • Slide 14
                    • Slide 15
                    • Slide 16
                    • Slide 17
                    • Slide 18
                    • Slide 19
                    • Slide 20
                    • Slide 21
                    • Slide 22
                    • Slide 23
                    • Slide 24
                    • Slide 25
                    • Slide 26
                    • Slide 27
                    • Slide 28
                    • Slide 29
                    • Slide 30
                    • Slide 31
                    • Slide 32
                    • Slide 33
                    • Slide 34
                    • Slide 35
                    • Slide 36
                    • Slide 37
                    • Slide 38
                    • Slide 39
                    • Slide 40
                    • Slide 41
                    • Slide 42
                    • Slide 43
                    • Slide 44
                    • Slide 45
                    • Slide 46
                    • Slide 47
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                    • Slide 49
                    • Slide 50
                    • Slide 51
                    • Slide 52
                    • Slide 53
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                    • Slide 55
                    • Slide 56
                    • Slide 57
                    • Slide 58
                    • Slide 59
                    • Slide 60
                    • Slide 61
                    • Slide 62

                      1111

                      Classification of ThresholdingClassification of Thresholding

                      When T depends onWhen T depends on only f(xy) only on gray-level valuesonly f(xy) only on gray-level values

                      1048638 1048638 Global Global thresholdingthresholding

                      both f(xy) and p(xy) on gray-level values and itboth f(xy) and p(xy) on gray-level values and its neighborss neighbors

                      Local Local thresholdingthresholding

                      x and y (in addition)x and y (in addition)

                      Dynamic Dynamic thresholdingthresholding

                      T T x y p x y f x y

                      1212

                      Basic Global ThresholdingBasic Global Thresholding

                      Original imageOriginal image HistogramHistogram

                      SolutionSolution use T midway between the max and use T midway between the max and

                      min gray levelsmin gray levels

                      SolutionSolution use T midway between the max and use T midway between the max and

                      min gray levelsmin gray levels

                      See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

                      1313

                      Basic Global ThresholdingBasic Global Thresholding

                      Let light objects in dark backgroundLet light objects in dark background

                      To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

                      e backgrounde background

                      ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

                      A thresholded imageA thresholded image

                      1

                      0

                      if f x y T backgroundg x y

                      if f x y T foreground

                      1414

                      Heuristic Global ThresholdingHeuristic Global Thresholding

                      11 Select an initial estimate for TSelect an initial estimate for T

                      22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

                      33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

                      44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

                      55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

                      1515

                      Basic Adaptive ThresholdingBasic Adaptive Thresholding

                      subdivide original image into small areassubdivide original image into small areas

                      utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

                      since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

                      See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

                      1616

                      Multilevel ThresholdingMultilevel Thresholding

                      a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

                      another object class if f(xy) another object class if f(xy) gtgt T T22

                      to background if f(xy) le Tto background if f(xy) le T11

                      1717

                      The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

                      1 1 2 2

                      1 2 1

                      p z P p z P p z

                      P P

                      1818

                      Optimal ThresholdingOptimal Thresholding

                      If the form of the If the form of the

                      densities is densities is

                      known or known or

                      assumed in assumed in

                      terms of terms of

                      minimum error minimum error

                      determining an determining an

                      optimal optimal

                      threshold for threshold for

                      segmenting the segmenting the

                      image is image is

                      possiblepossible

                      1 1 2 2

                      1 2 1

                      p z P p z P p z

                      P P

                      1919

                      Optimal ThresholdingOptimal Thresholding

                      Probability of erroneouslyProbability of erroneously

                      2020

                      Optimal ThresholdingOptimal Thresholding

                      Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                      ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                      find T which makesfind T which makes

                      2121

                      Optimal ThresholdingOptimal Thresholding

                      Minimum errorMinimum error

                      Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                      threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                      pp22(z) intersect(z) intersect

                      2222

                      Optimal ThresholdingOptimal Thresholding

                      For exampleFor example

                      Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                      pp22(z)(z)

                      where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                      variance of the Gaussian density of one variance of the Gaussian density of one

                      objectobject

                      μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                      the Gaussian density of the other objectthe Gaussian density of the other object

                      2323

                      Optimal ThresholdingOptimal Thresholding

                      Quadratic equation (Quadratic equation (二次方程二次方程 ))

                      2424

                      Problems of ThresholdingProblems of Thresholding

                      Original imageOriginal image Thresholded imageThresholded image

                      2525

                      Problems of ThresholdingProblems of Thresholding

                      (a)(a) Exact threshold Exact threshold

                      segmentationsegmentation

                      (b)(b) Threshold too lowThreshold too low

                      (c)(c) Threshold too Threshold too

                      highhigh

                      2626

                      72 Point Detection72 Point Detection

                      a point has been detected at the a point has been detected at the

                      location on which the mark is location on which the mark is

                      centered ifcentered if

                      |R|geT|R|geT

                      where T is a nonnegative thresholdwhere T is a nonnegative threshold

                      R is the sum of products of the R is the sum of products of the

                      coefficients with the gray levels contained coefficients with the gray levels contained

                      in the region encompassed by the markin the region encompassed by the mark

                      1 1 1

                      1 8 1

                      1 1 1

                      1 1 1

                      1 8 1

                      1 1 1

                      2727

                      72 Point Detection72 Point Detection

                      Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                      The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                      1 1 1

                      1 8 1

                      1 1 1

                      1 1 1

                      1 8 1

                      1 1 1

                      0 1 0

                      1 4 1

                      0 1 0

                      0 1 0

                      1 4 1

                      0 1 0

                      2828

                      ExampleExample

                      2929

                      73 Line Detection73 Line Detection

                      Horizontal mask will result with max Horizontal mask will result with max

                      response when a line passed through the response when a line passed through the

                      middle row of the mask with a constant middle row of the mask with a constant

                      backgroundbackground

                      the similar idea is used with other masksthe similar idea is used with other masks

                      Note the preferred direction of each mask Note the preferred direction of each mask

                      is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                      than other possible directionsthan other possible directions

                      1 1 1 1 1 2 1 2 1 2 1 1

                      2 2 2 1 2 1 1 2 1 1 2 1

                      1 1 1 2 1 1 1 2 1 1 1 2

                      45 45Horizontal Vertical

                      1 1 1 1 1 2 1 2 1 2 1 1

                      2 2 2 1 2 1 1 2 1 1 2 1

                      1 1 1 2 1 1 1 2 1 1 1 2

                      45 45Horizontal Vertical

                      3030

                      Idea 1 of Line DetectionIdea 1 of Line Detection

                      Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                      izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                      if at a certain point in the imageif at a certain point in the image

                      |Ri||Ri|gtgt|Rj||Rj|

                      for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                      3131

                      Idea 2 of Line DetectionIdea 2 of Line Detection

                      Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                      After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                      3232

                      ExampleExample

                      3333

                      74 Edge-based 74 Edge-based SegmentationSegmentation

                      Edge-based segmentations rely on edges found in an image by edge detecting operators

                      these edges mark image locations of discontinuities in gray level

                      Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                      There are a large group of methods based on information about edges in the image

                      3434

                      What is edgeWhat is edge

                      Edge is where change occurs Change is measured by derivative in 1D

                      ―Biggest change derivative has maximum magnitude

                      Or 2nd derivative is zero we discuss approaches for implementing

                      ―first-order derivative (Gradient operator)

                      ―second-order derivative (Laplacian operator)

                      ―we have introduced both derivatives in chapter 3

                      ―Here we will talk only about their properties for edge detection

                      3535

                      What is edgeWhat is edge

                      In other wordsIn other words an edge is a set of an edge is a set of

                      connected pixelsconnected pixels

                      that lie on the boundary between two that lie on the boundary between two

                      regions with relatively distinct gray-level regions with relatively distinct gray-level

                      propertiesproperties

                      Note edge vs boundaryNote edge vs boundary

                      ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                      ―whereas a region boundary owing to whereas a region boundary owing to

                      the way it is defined is a more global the way it is defined is a more global

                      ideaidea

                      3636

                      Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                      because of because of

                      optics optics

                      sampling sampling

                      image image

                      acquisition acquisition

                      imperfectionimperfection

                      3737

                      Thick and Thin EdgeThick and Thin Edge

                      The slope of the ramp is inversely The slope of the ramp is inversely

                      proportional to the degree of blurring in the proportional to the degree of blurring in the

                      edgeedge

                      Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                      pathpath

                      Instead an edge point now is any point Instead an edge point now is any point

                      contained in the ramp and contained in the ramp and an edge would an edge would

                      then be a set of such points that are then be a set of such points that are

                      connectedconnected

                      The thickness is determined by the length of the The thickness is determined by the length of the

                      rampramp

                      The length is determined by the slope which is in The length is determined by the slope which is in

                      turn determined by the degree of blurringturn determined by the degree of blurring

                      Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                      edges tend to be thinedges tend to be thin

                      3838

                      First and Second derivatives (First and Second derivatives ( 导数导数 ))

                      the signs of the the signs of the

                      derivatives would be derivatives would be

                      reversed for an edge reversed for an edge

                      that transitions from that transitions from

                      light to darklight to dark

                      First First derivatderivatee

                      SeconSecond d derivatderivatee

                      Gray-Gray-level level profileprofile

                      3939

                      Second derivativesSecond derivatives

                      an undesirable featurean undesirable feature

                      produces 2 values for every edge in an produces 2 values for every edge in an

                      imageimage

                      zero-crossing propertyzero-crossing property

                      an imaginary straight line joining the an imaginary straight line joining the

                      extreme positive and negative values of extreme positive and negative values of

                      the second derivative would cross zero the second derivative would cross zero

                      near the midpoint of the edgenear the midpoint of the edge

                      quite useful for locating the centers of quite useful for locating the centers of

                      thick edgesthick edges

                      4040

                      Basic idea of edge detectionBasic idea of edge detection

                      A profile is defined perpendicularly to A profile is defined perpendicularly to

                      the edge direction and the results are the edge direction and the results are

                      interpretedinterpreted

                      The magnitude of the first derivative is The magnitude of the first derivative is

                      used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                      ramp)ramp)

                      The sign of the second derivative can The sign of the second derivative can

                      determine whether an edge pixel is on the determine whether an edge pixel is on the

                      dark or light side of an edgedark or light side of an edge

                      4141

                      Review of First DerivateReview of First Derivate

                      Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                      Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                      Sobel operators 3Sobel operators 333

                      6 5 8 5x yG z z G z z

                      1 2 3

                      4 5 6

                      7 8 9

                      z z z

                      z z z

                      z z z

                      1 2 3

                      4 5 6

                      7 8 9

                      z z z

                      z z z

                      z z z

                      9 5 8 6x yG z z G z z 1 0 0 1

                      0 1 1 0

                      1 0 0 1

                      0 1 1 0

                      7 8 9 1 2 3

                      3 6 9 1 4 7

                      2 2

                      2 2

                      x

                      y

                      G z z z z z z

                      G z z z z z z

                      1 2 1 1 0 1

                      0 0 0 2 0 2

                      1 2 1 1 0 1

                      1 2 1 1 0 1

                      0 0 0 2 0 2

                      1 2 1 1 0 1

                      x yf G G

                      4242

                      Edge direction and strengthEdge direction and strength

                      Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                      α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                      The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                      The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                      2 2x yf G G

                      4343

                      Gradient MasksGradient Masks

                      1 0 0 1

                      0 1 1 0

                      Roberts

                      1 0 0 1

                      0 1 1 0

                      Roberts

                      1 2 1 1 0 1

                      0 0 0 2 0 2

                      1 2 1 1 0 1

                      Sobel

                      1 2 1 1 0 1

                      0 0 0 2 0 2

                      1 2 1 1 0 1

                      Sobel

                      1 1 1 1 0 1

                      0 0 0 1 0 1

                      1 1 1 1 0 1

                      Prewitt

                      1 1 1 1 0 1

                      0 0 0 1 0 1

                      1 1 1 1 0 1

                      Prewitt

                      4444

                      Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                      0 1 1 1 1 0

                      1 0 1 1 0 1

                      1 1 0 0 1 1

                      Prewitt

                      0 1 1 1 1 0

                      1 0 1 1 0 1

                      1 1 0 0 1 1

                      Prewitt

                      4545

                      Review of Second DerivateReview of Second Derivate

                      Laplacian OperatorLaplacian Operator

                      21 1

                      1 1 4

                      f x y f x yf

                      f x y f x y f x y

                      0 1 0

                      1 4 1

                      0 1 0

                      0 1 0

                      1 4 1

                      0 1 0

                      LaplacianLaplacian

                      MaskMask

                      1 1 1

                      1 8 1

                      1 1 1

                      1 1 1

                      1 8 1

                      1 1 1

                      4646

                      Example of edge detectionExample of edge detection

                      See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                      Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                      derivative)derivative)

                      produces double edgesproduces double edges

                      unable to detect edge directionunable to detect edge direction

                      4747

                      Canny edge detectorCanny edge detector

                      The most powerful edge-detection The most powerful edge-detection

                      method method

                      It differs from the other edge-It differs from the other edge-

                      detection methods in that detection methods in that

                      it uses two different thresholds (to detect it uses two different thresholds (to detect

                      strong and weak edges) strong and weak edges)

                      and includes the weak edges in the and includes the weak edges in the

                      output only if they are connected to output only if they are connected to

                      strong edges strong edges

                      This method is therefore less likely This method is therefore less likely

                      than the others to be fooled by than the others to be fooled by

                      noise and more likely to detect true noise and more likely to detect true

                      weak edgesweak edges

                      4848

                      Laplacian of GaussianLaplacian of Gaussian

                      Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                      2 2 22

                      4 2

                      2 2 2

                      2exp

                      r rh

                      r x y

                      determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                      4949

                      Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                      0 0 1 0 0

                      0 1 2 1 0

                      1 2 16 2 1

                      0 1 2 1 0

                      0 0 1 0 0

                      0 0 1 0 0

                      0 1 2 1 0

                      1 2 16 2 1

                      0 1 2 1 0

                      0 0 1 0 0

                      The coefficient must sum to The coefficient must sum to

                      zerozero

                      5050

                      Edge Detection and Edge Detection and SegmentationSegmentation

                      Image resulting from edge detection cannot be used as a segmentation result

                      Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                      5151

                      75 Region-based 75 Region-based SegmentationSegmentation

                      GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                      Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                      5252

                      Two methods of Region Two methods of Region SegmentationSegmentation

                      Region GrowingRegion Growing

                      Region SplittingRegion Splitting

                      Region growing is the opposite of the Region growing is the opposite of the

                      split and merge approachsplit and merge approach

                      5353

                      Region GrowingRegion Growing

                      The objective of segmentation is to The objective of segmentation is to

                      partition an image into regionspartition an image into regions

                      A region is a connected component with A region is a connected component with

                      some uniformity (say gray-levels or some uniformity (say gray-levels or

                      texture)texture)

                      In region growing we start with a set In region growing we start with a set

                      of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                      appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                      pixels if they have pixels if they have similar propertiessimilar properties

                      such as specific ranges of gray level such as specific ranges of gray level

                      and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                      Need initialization Need initialization similarity similarity

                      criterioncriterion

                      5454

                      Steps of Region GrowingSteps of Region Growing

                      Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                      pixel andpixel and compare it with neighbor compare it with neighbor

                      ppixelsixels

                      When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                      is grown from the seed pixel by is grown from the seed pixel by

                      addingadding neighboneighborr pixel pixelss

                      When the growth of one region stopsWhen the growth of one region stops

                      choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                      againagain

                      5555

                      Region Region growing growing

                      An initial set of small An initial set of small

                      areas are iterativelyareas are iteratively

                      merged according to merged according to

                      similarity constraintssimilarity constraints

                      5656

                      Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                      X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                      We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                      On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                      Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                      SeedSeed pointspoints

                      5757

                      CriterionCriterion

                      There is a valley at around 190 in the There is a valley at around 190 in the

                      histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                      to be considered as a part of region to the to be considered as a part of region to the

                      seed pointseed point

                      The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                      to at least one pixel in that regionto at least one pixel in that region

                      Result of region growing and boundaries of Result of region growing and boundaries of

                      defectsdefects

                      5858

                      Region SplittingRegion Splitting

                      The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                      ption that the entire image is homogeneousption that the entire image is homogeneous

                      If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                      This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                      Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                      5959

                      Region SplittingRegion Splitting

                      DisadvantageDisadvantage

                      they create regions that may be adjacent they create regions that may be adjacent

                      and homogeneous but not mergedand homogeneous but not merged

                      6060

                      Region Splitting and MergingRegion Splitting and Merging

                      ProcedureProcedure

                      11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                      22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                      33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                      6161

                      Region Splitting and Merging

                      Quadtree

                      (四叉树 )

                      6262

                      PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                      where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                      mmii is the mean gray level of that region is the mean gray level of that region

                      σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                      ExampleExample

                      Original Original

                      imageimageThresholded imageThresholded image Result of Result of

                      Splitting and Splitting and

                      MergingMerging

                      • Slide 1
                      • Slide 2
                      • Slide 3
                      • Slide 4
                      • Slide 5
                      • Slide 6
                      • Slide 7
                      • Slide 8
                      • Slide 9
                      • Slide 10
                      • Slide 11
                      • Slide 12
                      • Slide 13
                      • Slide 14
                      • Slide 15
                      • Slide 16
                      • Slide 17
                      • Slide 18
                      • Slide 19
                      • Slide 20
                      • Slide 21
                      • Slide 22
                      • Slide 23
                      • Slide 24
                      • Slide 25
                      • Slide 26
                      • Slide 27
                      • Slide 28
                      • Slide 29
                      • Slide 30
                      • Slide 31
                      • Slide 32
                      • Slide 33
                      • Slide 34
                      • Slide 35
                      • Slide 36
                      • Slide 37
                      • Slide 38
                      • Slide 39
                      • Slide 40
                      • Slide 41
                      • Slide 42
                      • Slide 43
                      • Slide 44
                      • Slide 45
                      • Slide 46
                      • Slide 47
                      • Slide 48
                      • Slide 49
                      • Slide 50
                      • Slide 51
                      • Slide 52
                      • Slide 53
                      • Slide 54
                      • Slide 55
                      • Slide 56
                      • Slide 57
                      • Slide 58
                      • Slide 59
                      • Slide 60
                      • Slide 61
                      • Slide 62

                        1212

                        Basic Global ThresholdingBasic Global Thresholding

                        Original imageOriginal image HistogramHistogram

                        SolutionSolution use T midway between the max and use T midway between the max and

                        min gray levelsmin gray levels

                        SolutionSolution use T midway between the max and use T midway between the max and

                        min gray levelsmin gray levels

                        See ldquobasic_global_threSee ldquobasic_global_thremrdquomrdquo

                        1313

                        Basic Global ThresholdingBasic Global Thresholding

                        Let light objects in dark backgroundLet light objects in dark background

                        To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

                        e backgrounde background

                        ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

                        A thresholded imageA thresholded image

                        1

                        0

                        if f x y T backgroundg x y

                        if f x y T foreground

                        1414

                        Heuristic Global ThresholdingHeuristic Global Thresholding

                        11 Select an initial estimate for TSelect an initial estimate for T

                        22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

                        33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

                        44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

                        55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

                        1515

                        Basic Adaptive ThresholdingBasic Adaptive Thresholding

                        subdivide original image into small areassubdivide original image into small areas

                        utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

                        since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

                        See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

                        1616

                        Multilevel ThresholdingMultilevel Thresholding

                        a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

                        another object class if f(xy) another object class if f(xy) gtgt T T22

                        to background if f(xy) le Tto background if f(xy) le T11

                        1717

                        The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

                        1 1 2 2

                        1 2 1

                        p z P p z P p z

                        P P

                        1818

                        Optimal ThresholdingOptimal Thresholding

                        If the form of the If the form of the

                        densities is densities is

                        known or known or

                        assumed in assumed in

                        terms of terms of

                        minimum error minimum error

                        determining an determining an

                        optimal optimal

                        threshold for threshold for

                        segmenting the segmenting the

                        image is image is

                        possiblepossible

                        1 1 2 2

                        1 2 1

                        p z P p z P p z

                        P P

                        1919

                        Optimal ThresholdingOptimal Thresholding

                        Probability of erroneouslyProbability of erroneously

                        2020

                        Optimal ThresholdingOptimal Thresholding

                        Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                        ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                        find T which makesfind T which makes

                        2121

                        Optimal ThresholdingOptimal Thresholding

                        Minimum errorMinimum error

                        Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                        threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                        pp22(z) intersect(z) intersect

                        2222

                        Optimal ThresholdingOptimal Thresholding

                        For exampleFor example

                        Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                        pp22(z)(z)

                        where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                        variance of the Gaussian density of one variance of the Gaussian density of one

                        objectobject

                        μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                        the Gaussian density of the other objectthe Gaussian density of the other object

                        2323

                        Optimal ThresholdingOptimal Thresholding

                        Quadratic equation (Quadratic equation (二次方程二次方程 ))

                        2424

                        Problems of ThresholdingProblems of Thresholding

                        Original imageOriginal image Thresholded imageThresholded image

                        2525

                        Problems of ThresholdingProblems of Thresholding

                        (a)(a) Exact threshold Exact threshold

                        segmentationsegmentation

                        (b)(b) Threshold too lowThreshold too low

                        (c)(c) Threshold too Threshold too

                        highhigh

                        2626

                        72 Point Detection72 Point Detection

                        a point has been detected at the a point has been detected at the

                        location on which the mark is location on which the mark is

                        centered ifcentered if

                        |R|geT|R|geT

                        where T is a nonnegative thresholdwhere T is a nonnegative threshold

                        R is the sum of products of the R is the sum of products of the

                        coefficients with the gray levels contained coefficients with the gray levels contained

                        in the region encompassed by the markin the region encompassed by the mark

                        1 1 1

                        1 8 1

                        1 1 1

                        1 1 1

                        1 8 1

                        1 1 1

                        2727

                        72 Point Detection72 Point Detection

                        Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                        The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                        1 1 1

                        1 8 1

                        1 1 1

                        1 1 1

                        1 8 1

                        1 1 1

                        0 1 0

                        1 4 1

                        0 1 0

                        0 1 0

                        1 4 1

                        0 1 0

                        2828

                        ExampleExample

                        2929

                        73 Line Detection73 Line Detection

                        Horizontal mask will result with max Horizontal mask will result with max

                        response when a line passed through the response when a line passed through the

                        middle row of the mask with a constant middle row of the mask with a constant

                        backgroundbackground

                        the similar idea is used with other masksthe similar idea is used with other masks

                        Note the preferred direction of each mask Note the preferred direction of each mask

                        is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                        than other possible directionsthan other possible directions

                        1 1 1 1 1 2 1 2 1 2 1 1

                        2 2 2 1 2 1 1 2 1 1 2 1

                        1 1 1 2 1 1 1 2 1 1 1 2

                        45 45Horizontal Vertical

                        1 1 1 1 1 2 1 2 1 2 1 1

                        2 2 2 1 2 1 1 2 1 1 2 1

                        1 1 1 2 1 1 1 2 1 1 1 2

                        45 45Horizontal Vertical

                        3030

                        Idea 1 of Line DetectionIdea 1 of Line Detection

                        Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                        izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                        if at a certain point in the imageif at a certain point in the image

                        |Ri||Ri|gtgt|Rj||Rj|

                        for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                        3131

                        Idea 2 of Line DetectionIdea 2 of Line Detection

                        Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                        After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                        3232

                        ExampleExample

                        3333

                        74 Edge-based 74 Edge-based SegmentationSegmentation

                        Edge-based segmentations rely on edges found in an image by edge detecting operators

                        these edges mark image locations of discontinuities in gray level

                        Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                        There are a large group of methods based on information about edges in the image

                        3434

                        What is edgeWhat is edge

                        Edge is where change occurs Change is measured by derivative in 1D

                        ―Biggest change derivative has maximum magnitude

                        Or 2nd derivative is zero we discuss approaches for implementing

                        ―first-order derivative (Gradient operator)

                        ―second-order derivative (Laplacian operator)

                        ―we have introduced both derivatives in chapter 3

                        ―Here we will talk only about their properties for edge detection

                        3535

                        What is edgeWhat is edge

                        In other wordsIn other words an edge is a set of an edge is a set of

                        connected pixelsconnected pixels

                        that lie on the boundary between two that lie on the boundary between two

                        regions with relatively distinct gray-level regions with relatively distinct gray-level

                        propertiesproperties

                        Note edge vs boundaryNote edge vs boundary

                        ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                        ―whereas a region boundary owing to whereas a region boundary owing to

                        the way it is defined is a more global the way it is defined is a more global

                        ideaidea

                        3636

                        Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                        because of because of

                        optics optics

                        sampling sampling

                        image image

                        acquisition acquisition

                        imperfectionimperfection

                        3737

                        Thick and Thin EdgeThick and Thin Edge

                        The slope of the ramp is inversely The slope of the ramp is inversely

                        proportional to the degree of blurring in the proportional to the degree of blurring in the

                        edgeedge

                        Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                        pathpath

                        Instead an edge point now is any point Instead an edge point now is any point

                        contained in the ramp and contained in the ramp and an edge would an edge would

                        then be a set of such points that are then be a set of such points that are

                        connectedconnected

                        The thickness is determined by the length of the The thickness is determined by the length of the

                        rampramp

                        The length is determined by the slope which is in The length is determined by the slope which is in

                        turn determined by the degree of blurringturn determined by the degree of blurring

                        Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                        edges tend to be thinedges tend to be thin

                        3838

                        First and Second derivatives (First and Second derivatives ( 导数导数 ))

                        the signs of the the signs of the

                        derivatives would be derivatives would be

                        reversed for an edge reversed for an edge

                        that transitions from that transitions from

                        light to darklight to dark

                        First First derivatderivatee

                        SeconSecond d derivatderivatee

                        Gray-Gray-level level profileprofile

                        3939

                        Second derivativesSecond derivatives

                        an undesirable featurean undesirable feature

                        produces 2 values for every edge in an produces 2 values for every edge in an

                        imageimage

                        zero-crossing propertyzero-crossing property

                        an imaginary straight line joining the an imaginary straight line joining the

                        extreme positive and negative values of extreme positive and negative values of

                        the second derivative would cross zero the second derivative would cross zero

                        near the midpoint of the edgenear the midpoint of the edge

                        quite useful for locating the centers of quite useful for locating the centers of

                        thick edgesthick edges

                        4040

                        Basic idea of edge detectionBasic idea of edge detection

                        A profile is defined perpendicularly to A profile is defined perpendicularly to

                        the edge direction and the results are the edge direction and the results are

                        interpretedinterpreted

                        The magnitude of the first derivative is The magnitude of the first derivative is

                        used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                        ramp)ramp)

                        The sign of the second derivative can The sign of the second derivative can

                        determine whether an edge pixel is on the determine whether an edge pixel is on the

                        dark or light side of an edgedark or light side of an edge

                        4141

                        Review of First DerivateReview of First Derivate

                        Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                        Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                        Sobel operators 3Sobel operators 333

                        6 5 8 5x yG z z G z z

                        1 2 3

                        4 5 6

                        7 8 9

                        z z z

                        z z z

                        z z z

                        1 2 3

                        4 5 6

                        7 8 9

                        z z z

                        z z z

                        z z z

                        9 5 8 6x yG z z G z z 1 0 0 1

                        0 1 1 0

                        1 0 0 1

                        0 1 1 0

                        7 8 9 1 2 3

                        3 6 9 1 4 7

                        2 2

                        2 2

                        x

                        y

                        G z z z z z z

                        G z z z z z z

                        1 2 1 1 0 1

                        0 0 0 2 0 2

                        1 2 1 1 0 1

                        1 2 1 1 0 1

                        0 0 0 2 0 2

                        1 2 1 1 0 1

                        x yf G G

                        4242

                        Edge direction and strengthEdge direction and strength

                        Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                        α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                        The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                        The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                        2 2x yf G G

                        4343

                        Gradient MasksGradient Masks

                        1 0 0 1

                        0 1 1 0

                        Roberts

                        1 0 0 1

                        0 1 1 0

                        Roberts

                        1 2 1 1 0 1

                        0 0 0 2 0 2

                        1 2 1 1 0 1

                        Sobel

                        1 2 1 1 0 1

                        0 0 0 2 0 2

                        1 2 1 1 0 1

                        Sobel

                        1 1 1 1 0 1

                        0 0 0 1 0 1

                        1 1 1 1 0 1

                        Prewitt

                        1 1 1 1 0 1

                        0 0 0 1 0 1

                        1 1 1 1 0 1

                        Prewitt

                        4444

                        Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                        0 1 1 1 1 0

                        1 0 1 1 0 1

                        1 1 0 0 1 1

                        Prewitt

                        0 1 1 1 1 0

                        1 0 1 1 0 1

                        1 1 0 0 1 1

                        Prewitt

                        4545

                        Review of Second DerivateReview of Second Derivate

                        Laplacian OperatorLaplacian Operator

                        21 1

                        1 1 4

                        f x y f x yf

                        f x y f x y f x y

                        0 1 0

                        1 4 1

                        0 1 0

                        0 1 0

                        1 4 1

                        0 1 0

                        LaplacianLaplacian

                        MaskMask

                        1 1 1

                        1 8 1

                        1 1 1

                        1 1 1

                        1 8 1

                        1 1 1

                        4646

                        Example of edge detectionExample of edge detection

                        See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                        Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                        derivative)derivative)

                        produces double edgesproduces double edges

                        unable to detect edge directionunable to detect edge direction

                        4747

                        Canny edge detectorCanny edge detector

                        The most powerful edge-detection The most powerful edge-detection

                        method method

                        It differs from the other edge-It differs from the other edge-

                        detection methods in that detection methods in that

                        it uses two different thresholds (to detect it uses two different thresholds (to detect

                        strong and weak edges) strong and weak edges)

                        and includes the weak edges in the and includes the weak edges in the

                        output only if they are connected to output only if they are connected to

                        strong edges strong edges

                        This method is therefore less likely This method is therefore less likely

                        than the others to be fooled by than the others to be fooled by

                        noise and more likely to detect true noise and more likely to detect true

                        weak edgesweak edges

                        4848

                        Laplacian of GaussianLaplacian of Gaussian

                        Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                        2 2 22

                        4 2

                        2 2 2

                        2exp

                        r rh

                        r x y

                        determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                        4949

                        Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                        0 0 1 0 0

                        0 1 2 1 0

                        1 2 16 2 1

                        0 1 2 1 0

                        0 0 1 0 0

                        0 0 1 0 0

                        0 1 2 1 0

                        1 2 16 2 1

                        0 1 2 1 0

                        0 0 1 0 0

                        The coefficient must sum to The coefficient must sum to

                        zerozero

                        5050

                        Edge Detection and Edge Detection and SegmentationSegmentation

                        Image resulting from edge detection cannot be used as a segmentation result

                        Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                        5151

                        75 Region-based 75 Region-based SegmentationSegmentation

                        GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                        Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                        PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                        For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                        PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                        For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                        5252

                        Two methods of Region Two methods of Region SegmentationSegmentation

                        Region GrowingRegion Growing

                        Region SplittingRegion Splitting

                        Region growing is the opposite of the Region growing is the opposite of the

                        split and merge approachsplit and merge approach

                        5353

                        Region GrowingRegion Growing

                        The objective of segmentation is to The objective of segmentation is to

                        partition an image into regionspartition an image into regions

                        A region is a connected component with A region is a connected component with

                        some uniformity (say gray-levels or some uniformity (say gray-levels or

                        texture)texture)

                        In region growing we start with a set In region growing we start with a set

                        of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                        appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                        pixels if they have pixels if they have similar propertiessimilar properties

                        such as specific ranges of gray level such as specific ranges of gray level

                        and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                        Need initialization Need initialization similarity similarity

                        criterioncriterion

                        5454

                        Steps of Region GrowingSteps of Region Growing

                        Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                        pixel andpixel and compare it with neighbor compare it with neighbor

                        ppixelsixels

                        When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                        is grown from the seed pixel by is grown from the seed pixel by

                        addingadding neighboneighborr pixel pixelss

                        When the growth of one region stopsWhen the growth of one region stops

                        choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                        againagain

                        5555

                        Region Region growing growing

                        An initial set of small An initial set of small

                        areas are iterativelyareas are iteratively

                        merged according to merged according to

                        similarity constraintssimilarity constraints

                        5656

                        Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                        X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                        We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                        On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                        Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                        SeedSeed pointspoints

                        5757

                        CriterionCriterion

                        There is a valley at around 190 in the There is a valley at around 190 in the

                        histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                        to be considered as a part of region to the to be considered as a part of region to the

                        seed pointseed point

                        The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                        to at least one pixel in that regionto at least one pixel in that region

                        Result of region growing and boundaries of Result of region growing and boundaries of

                        defectsdefects

                        5858

                        Region SplittingRegion Splitting

                        The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                        ption that the entire image is homogeneousption that the entire image is homogeneous

                        If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                        This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                        Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                        5959

                        Region SplittingRegion Splitting

                        DisadvantageDisadvantage

                        they create regions that may be adjacent they create regions that may be adjacent

                        and homogeneous but not mergedand homogeneous but not merged

                        6060

                        Region Splitting and MergingRegion Splitting and Merging

                        ProcedureProcedure

                        11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                        22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                        33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                        6161

                        Region Splitting and Merging

                        Quadtree

                        (四叉树 )

                        6262

                        PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                        where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                        mmii is the mean gray level of that region is the mean gray level of that region

                        σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                        ExampleExample

                        Original Original

                        imageimageThresholded imageThresholded image Result of Result of

                        Splitting and Splitting and

                        MergingMerging

                        • Slide 1
                        • Slide 2
                        • Slide 3
                        • Slide 4
                        • Slide 5
                        • Slide 6
                        • Slide 7
                        • Slide 8
                        • Slide 9
                        • Slide 10
                        • Slide 11
                        • Slide 12
                        • Slide 13
                        • Slide 14
                        • Slide 15
                        • Slide 16
                        • Slide 17
                        • Slide 18
                        • Slide 19
                        • Slide 20
                        • Slide 21
                        • Slide 22
                        • Slide 23
                        • Slide 24
                        • Slide 25
                        • Slide 26
                        • Slide 27
                        • Slide 28
                        • Slide 29
                        • Slide 30
                        • Slide 31
                        • Slide 32
                        • Slide 33
                        • Slide 34
                        • Slide 35
                        • Slide 36
                        • Slide 37
                        • Slide 38
                        • Slide 39
                        • Slide 40
                        • Slide 41
                        • Slide 42
                        • Slide 43
                        • Slide 44
                        • Slide 45
                        • Slide 46
                        • Slide 47
                        • Slide 48
                        • Slide 49
                        • Slide 50
                        • Slide 51
                        • Slide 52
                        • Slide 53
                        • Slide 54
                        • Slide 55
                        • Slide 56
                        • Slide 57
                        • Slide 58
                        • Slide 59
                        • Slide 60
                        • Slide 61
                        • Slide 62

                          1313

                          Basic Global ThresholdingBasic Global Thresholding

                          Let light objects in dark backgroundLet light objects in dark background

                          To extract the objectsTo extract the objects Select a ldquoTrdquo that separates the objects from thSelect a ldquoTrdquo that separates the objects from th

                          e backgrounde background

                          ie any (xy) for which f(xy)gtT is an object point ie any (xy) for which f(xy)gtT is an object point

                          A thresholded imageA thresholded image

                          1

                          0

                          if f x y T backgroundg x y

                          if f x y T foreground

                          1414

                          Heuristic Global ThresholdingHeuristic Global Thresholding

                          11 Select an initial estimate for TSelect an initial estimate for T

                          22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

                          33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

                          44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

                          55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

                          1515

                          Basic Adaptive ThresholdingBasic Adaptive Thresholding

                          subdivide original image into small areassubdivide original image into small areas

                          utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

                          since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

                          See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

                          1616

                          Multilevel ThresholdingMultilevel Thresholding

                          a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

                          another object class if f(xy) another object class if f(xy) gtgt T T22

                          to background if f(xy) le Tto background if f(xy) le T11

                          1717

                          The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

                          1 1 2 2

                          1 2 1

                          p z P p z P p z

                          P P

                          1818

                          Optimal ThresholdingOptimal Thresholding

                          If the form of the If the form of the

                          densities is densities is

                          known or known or

                          assumed in assumed in

                          terms of terms of

                          minimum error minimum error

                          determining an determining an

                          optimal optimal

                          threshold for threshold for

                          segmenting the segmenting the

                          image is image is

                          possiblepossible

                          1 1 2 2

                          1 2 1

                          p z P p z P p z

                          P P

                          1919

                          Optimal ThresholdingOptimal Thresholding

                          Probability of erroneouslyProbability of erroneously

                          2020

                          Optimal ThresholdingOptimal Thresholding

                          Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                          ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                          find T which makesfind T which makes

                          2121

                          Optimal ThresholdingOptimal Thresholding

                          Minimum errorMinimum error

                          Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                          threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                          pp22(z) intersect(z) intersect

                          2222

                          Optimal ThresholdingOptimal Thresholding

                          For exampleFor example

                          Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                          pp22(z)(z)

                          where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                          variance of the Gaussian density of one variance of the Gaussian density of one

                          objectobject

                          μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                          the Gaussian density of the other objectthe Gaussian density of the other object

                          2323

                          Optimal ThresholdingOptimal Thresholding

                          Quadratic equation (Quadratic equation (二次方程二次方程 ))

                          2424

                          Problems of ThresholdingProblems of Thresholding

                          Original imageOriginal image Thresholded imageThresholded image

                          2525

                          Problems of ThresholdingProblems of Thresholding

                          (a)(a) Exact threshold Exact threshold

                          segmentationsegmentation

                          (b)(b) Threshold too lowThreshold too low

                          (c)(c) Threshold too Threshold too

                          highhigh

                          2626

                          72 Point Detection72 Point Detection

                          a point has been detected at the a point has been detected at the

                          location on which the mark is location on which the mark is

                          centered ifcentered if

                          |R|geT|R|geT

                          where T is a nonnegative thresholdwhere T is a nonnegative threshold

                          R is the sum of products of the R is the sum of products of the

                          coefficients with the gray levels contained coefficients with the gray levels contained

                          in the region encompassed by the markin the region encompassed by the mark

                          1 1 1

                          1 8 1

                          1 1 1

                          1 1 1

                          1 8 1

                          1 1 1

                          2727

                          72 Point Detection72 Point Detection

                          Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                          The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                          1 1 1

                          1 8 1

                          1 1 1

                          1 1 1

                          1 8 1

                          1 1 1

                          0 1 0

                          1 4 1

                          0 1 0

                          0 1 0

                          1 4 1

                          0 1 0

                          2828

                          ExampleExample

                          2929

                          73 Line Detection73 Line Detection

                          Horizontal mask will result with max Horizontal mask will result with max

                          response when a line passed through the response when a line passed through the

                          middle row of the mask with a constant middle row of the mask with a constant

                          backgroundbackground

                          the similar idea is used with other masksthe similar idea is used with other masks

                          Note the preferred direction of each mask Note the preferred direction of each mask

                          is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                          than other possible directionsthan other possible directions

                          1 1 1 1 1 2 1 2 1 2 1 1

                          2 2 2 1 2 1 1 2 1 1 2 1

                          1 1 1 2 1 1 1 2 1 1 1 2

                          45 45Horizontal Vertical

                          1 1 1 1 1 2 1 2 1 2 1 1

                          2 2 2 1 2 1 1 2 1 1 2 1

                          1 1 1 2 1 1 1 2 1 1 1 2

                          45 45Horizontal Vertical

                          3030

                          Idea 1 of Line DetectionIdea 1 of Line Detection

                          Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                          izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                          if at a certain point in the imageif at a certain point in the image

                          |Ri||Ri|gtgt|Rj||Rj|

                          for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                          3131

                          Idea 2 of Line DetectionIdea 2 of Line Detection

                          Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                          After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                          3232

                          ExampleExample

                          3333

                          74 Edge-based 74 Edge-based SegmentationSegmentation

                          Edge-based segmentations rely on edges found in an image by edge detecting operators

                          these edges mark image locations of discontinuities in gray level

                          Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                          There are a large group of methods based on information about edges in the image

                          3434

                          What is edgeWhat is edge

                          Edge is where change occurs Change is measured by derivative in 1D

                          ―Biggest change derivative has maximum magnitude

                          Or 2nd derivative is zero we discuss approaches for implementing

                          ―first-order derivative (Gradient operator)

                          ―second-order derivative (Laplacian operator)

                          ―we have introduced both derivatives in chapter 3

                          ―Here we will talk only about their properties for edge detection

                          3535

                          What is edgeWhat is edge

                          In other wordsIn other words an edge is a set of an edge is a set of

                          connected pixelsconnected pixels

                          that lie on the boundary between two that lie on the boundary between two

                          regions with relatively distinct gray-level regions with relatively distinct gray-level

                          propertiesproperties

                          Note edge vs boundaryNote edge vs boundary

                          ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                          ―whereas a region boundary owing to whereas a region boundary owing to

                          the way it is defined is a more global the way it is defined is a more global

                          ideaidea

                          3636

                          Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                          because of because of

                          optics optics

                          sampling sampling

                          image image

                          acquisition acquisition

                          imperfectionimperfection

                          3737

                          Thick and Thin EdgeThick and Thin Edge

                          The slope of the ramp is inversely The slope of the ramp is inversely

                          proportional to the degree of blurring in the proportional to the degree of blurring in the

                          edgeedge

                          Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                          pathpath

                          Instead an edge point now is any point Instead an edge point now is any point

                          contained in the ramp and contained in the ramp and an edge would an edge would

                          then be a set of such points that are then be a set of such points that are

                          connectedconnected

                          The thickness is determined by the length of the The thickness is determined by the length of the

                          rampramp

                          The length is determined by the slope which is in The length is determined by the slope which is in

                          turn determined by the degree of blurringturn determined by the degree of blurring

                          Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                          edges tend to be thinedges tend to be thin

                          3838

                          First and Second derivatives (First and Second derivatives ( 导数导数 ))

                          the signs of the the signs of the

                          derivatives would be derivatives would be

                          reversed for an edge reversed for an edge

                          that transitions from that transitions from

                          light to darklight to dark

                          First First derivatderivatee

                          SeconSecond d derivatderivatee

                          Gray-Gray-level level profileprofile

                          3939

                          Second derivativesSecond derivatives

                          an undesirable featurean undesirable feature

                          produces 2 values for every edge in an produces 2 values for every edge in an

                          imageimage

                          zero-crossing propertyzero-crossing property

                          an imaginary straight line joining the an imaginary straight line joining the

                          extreme positive and negative values of extreme positive and negative values of

                          the second derivative would cross zero the second derivative would cross zero

                          near the midpoint of the edgenear the midpoint of the edge

                          quite useful for locating the centers of quite useful for locating the centers of

                          thick edgesthick edges

                          4040

                          Basic idea of edge detectionBasic idea of edge detection

                          A profile is defined perpendicularly to A profile is defined perpendicularly to

                          the edge direction and the results are the edge direction and the results are

                          interpretedinterpreted

                          The magnitude of the first derivative is The magnitude of the first derivative is

                          used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                          ramp)ramp)

                          The sign of the second derivative can The sign of the second derivative can

                          determine whether an edge pixel is on the determine whether an edge pixel is on the

                          dark or light side of an edgedark or light side of an edge

                          4141

                          Review of First DerivateReview of First Derivate

                          Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                          Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                          Sobel operators 3Sobel operators 333

                          6 5 8 5x yG z z G z z

                          1 2 3

                          4 5 6

                          7 8 9

                          z z z

                          z z z

                          z z z

                          1 2 3

                          4 5 6

                          7 8 9

                          z z z

                          z z z

                          z z z

                          9 5 8 6x yG z z G z z 1 0 0 1

                          0 1 1 0

                          1 0 0 1

                          0 1 1 0

                          7 8 9 1 2 3

                          3 6 9 1 4 7

                          2 2

                          2 2

                          x

                          y

                          G z z z z z z

                          G z z z z z z

                          1 2 1 1 0 1

                          0 0 0 2 0 2

                          1 2 1 1 0 1

                          1 2 1 1 0 1

                          0 0 0 2 0 2

                          1 2 1 1 0 1

                          x yf G G

                          4242

                          Edge direction and strengthEdge direction and strength

                          Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                          α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                          The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                          The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                          2 2x yf G G

                          4343

                          Gradient MasksGradient Masks

                          1 0 0 1

                          0 1 1 0

                          Roberts

                          1 0 0 1

                          0 1 1 0

                          Roberts

                          1 2 1 1 0 1

                          0 0 0 2 0 2

                          1 2 1 1 0 1

                          Sobel

                          1 2 1 1 0 1

                          0 0 0 2 0 2

                          1 2 1 1 0 1

                          Sobel

                          1 1 1 1 0 1

                          0 0 0 1 0 1

                          1 1 1 1 0 1

                          Prewitt

                          1 1 1 1 0 1

                          0 0 0 1 0 1

                          1 1 1 1 0 1

                          Prewitt

                          4444

                          Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                          0 1 1 1 1 0

                          1 0 1 1 0 1

                          1 1 0 0 1 1

                          Prewitt

                          0 1 1 1 1 0

                          1 0 1 1 0 1

                          1 1 0 0 1 1

                          Prewitt

                          4545

                          Review of Second DerivateReview of Second Derivate

                          Laplacian OperatorLaplacian Operator

                          21 1

                          1 1 4

                          f x y f x yf

                          f x y f x y f x y

                          0 1 0

                          1 4 1

                          0 1 0

                          0 1 0

                          1 4 1

                          0 1 0

                          LaplacianLaplacian

                          MaskMask

                          1 1 1

                          1 8 1

                          1 1 1

                          1 1 1

                          1 8 1

                          1 1 1

                          4646

                          Example of edge detectionExample of edge detection

                          See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                          Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                          derivative)derivative)

                          produces double edgesproduces double edges

                          unable to detect edge directionunable to detect edge direction

                          4747

                          Canny edge detectorCanny edge detector

                          The most powerful edge-detection The most powerful edge-detection

                          method method

                          It differs from the other edge-It differs from the other edge-

                          detection methods in that detection methods in that

                          it uses two different thresholds (to detect it uses two different thresholds (to detect

                          strong and weak edges) strong and weak edges)

                          and includes the weak edges in the and includes the weak edges in the

                          output only if they are connected to output only if they are connected to

                          strong edges strong edges

                          This method is therefore less likely This method is therefore less likely

                          than the others to be fooled by than the others to be fooled by

                          noise and more likely to detect true noise and more likely to detect true

                          weak edgesweak edges

                          4848

                          Laplacian of GaussianLaplacian of Gaussian

                          Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                          2 2 22

                          4 2

                          2 2 2

                          2exp

                          r rh

                          r x y

                          determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                          4949

                          Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                          0 0 1 0 0

                          0 1 2 1 0

                          1 2 16 2 1

                          0 1 2 1 0

                          0 0 1 0 0

                          0 0 1 0 0

                          0 1 2 1 0

                          1 2 16 2 1

                          0 1 2 1 0

                          0 0 1 0 0

                          The coefficient must sum to The coefficient must sum to

                          zerozero

                          5050

                          Edge Detection and Edge Detection and SegmentationSegmentation

                          Image resulting from edge detection cannot be used as a segmentation result

                          Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                          5151

                          75 Region-based 75 Region-based SegmentationSegmentation

                          GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                          Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                          PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                          For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                          PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                          For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                          5252

                          Two methods of Region Two methods of Region SegmentationSegmentation

                          Region GrowingRegion Growing

                          Region SplittingRegion Splitting

                          Region growing is the opposite of the Region growing is the opposite of the

                          split and merge approachsplit and merge approach

                          5353

                          Region GrowingRegion Growing

                          The objective of segmentation is to The objective of segmentation is to

                          partition an image into regionspartition an image into regions

                          A region is a connected component with A region is a connected component with

                          some uniformity (say gray-levels or some uniformity (say gray-levels or

                          texture)texture)

                          In region growing we start with a set In region growing we start with a set

                          of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                          appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                          pixels if they have pixels if they have similar propertiessimilar properties

                          such as specific ranges of gray level such as specific ranges of gray level

                          and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                          Need initialization Need initialization similarity similarity

                          criterioncriterion

                          5454

                          Steps of Region GrowingSteps of Region Growing

                          Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                          pixel andpixel and compare it with neighbor compare it with neighbor

                          ppixelsixels

                          When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                          is grown from the seed pixel by is grown from the seed pixel by

                          addingadding neighboneighborr pixel pixelss

                          When the growth of one region stopsWhen the growth of one region stops

                          choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                          againagain

                          5555

                          Region Region growing growing

                          An initial set of small An initial set of small

                          areas are iterativelyareas are iteratively

                          merged according to merged according to

                          similarity constraintssimilarity constraints

                          5656

                          Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                          X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                          We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                          On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                          Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                          SeedSeed pointspoints

                          5757

                          CriterionCriterion

                          There is a valley at around 190 in the There is a valley at around 190 in the

                          histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                          to be considered as a part of region to the to be considered as a part of region to the

                          seed pointseed point

                          The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                          to at least one pixel in that regionto at least one pixel in that region

                          Result of region growing and boundaries of Result of region growing and boundaries of

                          defectsdefects

                          5858

                          Region SplittingRegion Splitting

                          The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                          ption that the entire image is homogeneousption that the entire image is homogeneous

                          If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                          This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                          Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                          5959

                          Region SplittingRegion Splitting

                          DisadvantageDisadvantage

                          they create regions that may be adjacent they create regions that may be adjacent

                          and homogeneous but not mergedand homogeneous but not merged

                          6060

                          Region Splitting and MergingRegion Splitting and Merging

                          ProcedureProcedure

                          11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                          22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                          33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                          6161

                          Region Splitting and Merging

                          Quadtree

                          (四叉树 )

                          6262

                          PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                          where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                          mmii is the mean gray level of that region is the mean gray level of that region

                          σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                          ExampleExample

                          Original Original

                          imageimageThresholded imageThresholded image Result of Result of

                          Splitting and Splitting and

                          MergingMerging

                          • Slide 1
                          • Slide 2
                          • Slide 3
                          • Slide 4
                          • Slide 5
                          • Slide 6
                          • Slide 7
                          • Slide 8
                          • Slide 9
                          • Slide 10
                          • Slide 11
                          • Slide 12
                          • Slide 13
                          • Slide 14
                          • Slide 15
                          • Slide 16
                          • Slide 17
                          • Slide 18
                          • Slide 19
                          • Slide 20
                          • Slide 21
                          • Slide 22
                          • Slide 23
                          • Slide 24
                          • Slide 25
                          • Slide 26
                          • Slide 27
                          • Slide 28
                          • Slide 29
                          • Slide 30
                          • Slide 31
                          • Slide 32
                          • Slide 33
                          • Slide 34
                          • Slide 35
                          • Slide 36
                          • Slide 37
                          • Slide 38
                          • Slide 39
                          • Slide 40
                          • Slide 41
                          • Slide 42
                          • Slide 43
                          • Slide 44
                          • Slide 45
                          • Slide 46
                          • Slide 47
                          • Slide 48
                          • Slide 49
                          • Slide 50
                          • Slide 51
                          • Slide 52
                          • Slide 53
                          • Slide 54
                          • Slide 55
                          • Slide 56
                          • Slide 57
                          • Slide 58
                          • Slide 59
                          • Slide 60
                          • Slide 61
                          • Slide 62

                            1414

                            Heuristic Global ThresholdingHeuristic Global Thresholding

                            11 Select an initial estimate for TSelect an initial estimate for T

                            22 Segment the image using T This will produce two Segment the image using T This will produce two groups of pixels Ggroups of pixels G11 consisting of all pixels with gra consisting of all pixels with gray level values lt T and Gy level values lt T and G22 consisting of pixels with g consisting of pixels with gray level values ray level values T T

                            33 Compute the average gray level values μCompute the average gray level values μ11 and μ and μ22 fo for the pixels in regions Gr the pixels in regions G11 and G and G22

                            44 Compute a new threshold value T=05(μCompute a new threshold value T=05(μ11+μ+μ22))

                            55 Repeat steps 2 through 4 until the difference in T iRepeat steps 2 through 4 until the difference in T in successive iterations is smaller than a predefinen successive iterations is smaller than a predefined parameter Td parameter T00 seerdquohhellipmrdquo seerdquohhellipmrdquo

                            1515

                            Basic Adaptive ThresholdingBasic Adaptive Thresholding

                            subdivide original image into small areassubdivide original image into small areas

                            utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

                            since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

                            See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

                            1616

                            Multilevel ThresholdingMultilevel Thresholding

                            a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

                            another object class if f(xy) another object class if f(xy) gtgt T T22

                            to background if f(xy) le Tto background if f(xy) le T11

                            1717

                            The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

                            1 1 2 2

                            1 2 1

                            p z P p z P p z

                            P P

                            1818

                            Optimal ThresholdingOptimal Thresholding

                            If the form of the If the form of the

                            densities is densities is

                            known or known or

                            assumed in assumed in

                            terms of terms of

                            minimum error minimum error

                            determining an determining an

                            optimal optimal

                            threshold for threshold for

                            segmenting the segmenting the

                            image is image is

                            possiblepossible

                            1 1 2 2

                            1 2 1

                            p z P p z P p z

                            P P

                            1919

                            Optimal ThresholdingOptimal Thresholding

                            Probability of erroneouslyProbability of erroneously

                            2020

                            Optimal ThresholdingOptimal Thresholding

                            Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                            ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                            find T which makesfind T which makes

                            2121

                            Optimal ThresholdingOptimal Thresholding

                            Minimum errorMinimum error

                            Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                            threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                            pp22(z) intersect(z) intersect

                            2222

                            Optimal ThresholdingOptimal Thresholding

                            For exampleFor example

                            Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                            pp22(z)(z)

                            where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                            variance of the Gaussian density of one variance of the Gaussian density of one

                            objectobject

                            μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                            the Gaussian density of the other objectthe Gaussian density of the other object

                            2323

                            Optimal ThresholdingOptimal Thresholding

                            Quadratic equation (Quadratic equation (二次方程二次方程 ))

                            2424

                            Problems of ThresholdingProblems of Thresholding

                            Original imageOriginal image Thresholded imageThresholded image

                            2525

                            Problems of ThresholdingProblems of Thresholding

                            (a)(a) Exact threshold Exact threshold

                            segmentationsegmentation

                            (b)(b) Threshold too lowThreshold too low

                            (c)(c) Threshold too Threshold too

                            highhigh

                            2626

                            72 Point Detection72 Point Detection

                            a point has been detected at the a point has been detected at the

                            location on which the mark is location on which the mark is

                            centered ifcentered if

                            |R|geT|R|geT

                            where T is a nonnegative thresholdwhere T is a nonnegative threshold

                            R is the sum of products of the R is the sum of products of the

                            coefficients with the gray levels contained coefficients with the gray levels contained

                            in the region encompassed by the markin the region encompassed by the mark

                            1 1 1

                            1 8 1

                            1 1 1

                            1 1 1

                            1 8 1

                            1 1 1

                            2727

                            72 Point Detection72 Point Detection

                            Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                            The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                            1 1 1

                            1 8 1

                            1 1 1

                            1 1 1

                            1 8 1

                            1 1 1

                            0 1 0

                            1 4 1

                            0 1 0

                            0 1 0

                            1 4 1

                            0 1 0

                            2828

                            ExampleExample

                            2929

                            73 Line Detection73 Line Detection

                            Horizontal mask will result with max Horizontal mask will result with max

                            response when a line passed through the response when a line passed through the

                            middle row of the mask with a constant middle row of the mask with a constant

                            backgroundbackground

                            the similar idea is used with other masksthe similar idea is used with other masks

                            Note the preferred direction of each mask Note the preferred direction of each mask

                            is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                            than other possible directionsthan other possible directions

                            1 1 1 1 1 2 1 2 1 2 1 1

                            2 2 2 1 2 1 1 2 1 1 2 1

                            1 1 1 2 1 1 1 2 1 1 1 2

                            45 45Horizontal Vertical

                            1 1 1 1 1 2 1 2 1 2 1 1

                            2 2 2 1 2 1 1 2 1 1 2 1

                            1 1 1 2 1 1 1 2 1 1 1 2

                            45 45Horizontal Vertical

                            3030

                            Idea 1 of Line DetectionIdea 1 of Line Detection

                            Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                            izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                            if at a certain point in the imageif at a certain point in the image

                            |Ri||Ri|gtgt|Rj||Rj|

                            for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                            3131

                            Idea 2 of Line DetectionIdea 2 of Line Detection

                            Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                            After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                            3232

                            ExampleExample

                            3333

                            74 Edge-based 74 Edge-based SegmentationSegmentation

                            Edge-based segmentations rely on edges found in an image by edge detecting operators

                            these edges mark image locations of discontinuities in gray level

                            Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                            There are a large group of methods based on information about edges in the image

                            3434

                            What is edgeWhat is edge

                            Edge is where change occurs Change is measured by derivative in 1D

                            ―Biggest change derivative has maximum magnitude

                            Or 2nd derivative is zero we discuss approaches for implementing

                            ―first-order derivative (Gradient operator)

                            ―second-order derivative (Laplacian operator)

                            ―we have introduced both derivatives in chapter 3

                            ―Here we will talk only about their properties for edge detection

                            3535

                            What is edgeWhat is edge

                            In other wordsIn other words an edge is a set of an edge is a set of

                            connected pixelsconnected pixels

                            that lie on the boundary between two that lie on the boundary between two

                            regions with relatively distinct gray-level regions with relatively distinct gray-level

                            propertiesproperties

                            Note edge vs boundaryNote edge vs boundary

                            ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                            ―whereas a region boundary owing to whereas a region boundary owing to

                            the way it is defined is a more global the way it is defined is a more global

                            ideaidea

                            3636

                            Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                            because of because of

                            optics optics

                            sampling sampling

                            image image

                            acquisition acquisition

                            imperfectionimperfection

                            3737

                            Thick and Thin EdgeThick and Thin Edge

                            The slope of the ramp is inversely The slope of the ramp is inversely

                            proportional to the degree of blurring in the proportional to the degree of blurring in the

                            edgeedge

                            Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                            pathpath

                            Instead an edge point now is any point Instead an edge point now is any point

                            contained in the ramp and contained in the ramp and an edge would an edge would

                            then be a set of such points that are then be a set of such points that are

                            connectedconnected

                            The thickness is determined by the length of the The thickness is determined by the length of the

                            rampramp

                            The length is determined by the slope which is in The length is determined by the slope which is in

                            turn determined by the degree of blurringturn determined by the degree of blurring

                            Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                            edges tend to be thinedges tend to be thin

                            3838

                            First and Second derivatives (First and Second derivatives ( 导数导数 ))

                            the signs of the the signs of the

                            derivatives would be derivatives would be

                            reversed for an edge reversed for an edge

                            that transitions from that transitions from

                            light to darklight to dark

                            First First derivatderivatee

                            SeconSecond d derivatderivatee

                            Gray-Gray-level level profileprofile

                            3939

                            Second derivativesSecond derivatives

                            an undesirable featurean undesirable feature

                            produces 2 values for every edge in an produces 2 values for every edge in an

                            imageimage

                            zero-crossing propertyzero-crossing property

                            an imaginary straight line joining the an imaginary straight line joining the

                            extreme positive and negative values of extreme positive and negative values of

                            the second derivative would cross zero the second derivative would cross zero

                            near the midpoint of the edgenear the midpoint of the edge

                            quite useful for locating the centers of quite useful for locating the centers of

                            thick edgesthick edges

                            4040

                            Basic idea of edge detectionBasic idea of edge detection

                            A profile is defined perpendicularly to A profile is defined perpendicularly to

                            the edge direction and the results are the edge direction and the results are

                            interpretedinterpreted

                            The magnitude of the first derivative is The magnitude of the first derivative is

                            used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                            ramp)ramp)

                            The sign of the second derivative can The sign of the second derivative can

                            determine whether an edge pixel is on the determine whether an edge pixel is on the

                            dark or light side of an edgedark or light side of an edge

                            4141

                            Review of First DerivateReview of First Derivate

                            Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                            Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                            Sobel operators 3Sobel operators 333

                            6 5 8 5x yG z z G z z

                            1 2 3

                            4 5 6

                            7 8 9

                            z z z

                            z z z

                            z z z

                            1 2 3

                            4 5 6

                            7 8 9

                            z z z

                            z z z

                            z z z

                            9 5 8 6x yG z z G z z 1 0 0 1

                            0 1 1 0

                            1 0 0 1

                            0 1 1 0

                            7 8 9 1 2 3

                            3 6 9 1 4 7

                            2 2

                            2 2

                            x

                            y

                            G z z z z z z

                            G z z z z z z

                            1 2 1 1 0 1

                            0 0 0 2 0 2

                            1 2 1 1 0 1

                            1 2 1 1 0 1

                            0 0 0 2 0 2

                            1 2 1 1 0 1

                            x yf G G

                            4242

                            Edge direction and strengthEdge direction and strength

                            Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                            α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                            The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                            The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                            2 2x yf G G

                            4343

                            Gradient MasksGradient Masks

                            1 0 0 1

                            0 1 1 0

                            Roberts

                            1 0 0 1

                            0 1 1 0

                            Roberts

                            1 2 1 1 0 1

                            0 0 0 2 0 2

                            1 2 1 1 0 1

                            Sobel

                            1 2 1 1 0 1

                            0 0 0 2 0 2

                            1 2 1 1 0 1

                            Sobel

                            1 1 1 1 0 1

                            0 0 0 1 0 1

                            1 1 1 1 0 1

                            Prewitt

                            1 1 1 1 0 1

                            0 0 0 1 0 1

                            1 1 1 1 0 1

                            Prewitt

                            4444

                            Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                            0 1 1 1 1 0

                            1 0 1 1 0 1

                            1 1 0 0 1 1

                            Prewitt

                            0 1 1 1 1 0

                            1 0 1 1 0 1

                            1 1 0 0 1 1

                            Prewitt

                            4545

                            Review of Second DerivateReview of Second Derivate

                            Laplacian OperatorLaplacian Operator

                            21 1

                            1 1 4

                            f x y f x yf

                            f x y f x y f x y

                            0 1 0

                            1 4 1

                            0 1 0

                            0 1 0

                            1 4 1

                            0 1 0

                            LaplacianLaplacian

                            MaskMask

                            1 1 1

                            1 8 1

                            1 1 1

                            1 1 1

                            1 8 1

                            1 1 1

                            4646

                            Example of edge detectionExample of edge detection

                            See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                            Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                            derivative)derivative)

                            produces double edgesproduces double edges

                            unable to detect edge directionunable to detect edge direction

                            4747

                            Canny edge detectorCanny edge detector

                            The most powerful edge-detection The most powerful edge-detection

                            method method

                            It differs from the other edge-It differs from the other edge-

                            detection methods in that detection methods in that

                            it uses two different thresholds (to detect it uses two different thresholds (to detect

                            strong and weak edges) strong and weak edges)

                            and includes the weak edges in the and includes the weak edges in the

                            output only if they are connected to output only if they are connected to

                            strong edges strong edges

                            This method is therefore less likely This method is therefore less likely

                            than the others to be fooled by than the others to be fooled by

                            noise and more likely to detect true noise and more likely to detect true

                            weak edgesweak edges

                            4848

                            Laplacian of GaussianLaplacian of Gaussian

                            Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                            2 2 22

                            4 2

                            2 2 2

                            2exp

                            r rh

                            r x y

                            determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                            4949

                            Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                            0 0 1 0 0

                            0 1 2 1 0

                            1 2 16 2 1

                            0 1 2 1 0

                            0 0 1 0 0

                            0 0 1 0 0

                            0 1 2 1 0

                            1 2 16 2 1

                            0 1 2 1 0

                            0 0 1 0 0

                            The coefficient must sum to The coefficient must sum to

                            zerozero

                            5050

                            Edge Detection and Edge Detection and SegmentationSegmentation

                            Image resulting from edge detection cannot be used as a segmentation result

                            Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                            5151

                            75 Region-based 75 Region-based SegmentationSegmentation

                            GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                            Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                            PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                            For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                            PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                            For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                            5252

                            Two methods of Region Two methods of Region SegmentationSegmentation

                            Region GrowingRegion Growing

                            Region SplittingRegion Splitting

                            Region growing is the opposite of the Region growing is the opposite of the

                            split and merge approachsplit and merge approach

                            5353

                            Region GrowingRegion Growing

                            The objective of segmentation is to The objective of segmentation is to

                            partition an image into regionspartition an image into regions

                            A region is a connected component with A region is a connected component with

                            some uniformity (say gray-levels or some uniformity (say gray-levels or

                            texture)texture)

                            In region growing we start with a set In region growing we start with a set

                            of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                            appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                            pixels if they have pixels if they have similar propertiessimilar properties

                            such as specific ranges of gray level such as specific ranges of gray level

                            and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                            Need initialization Need initialization similarity similarity

                            criterioncriterion

                            5454

                            Steps of Region GrowingSteps of Region Growing

                            Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                            pixel andpixel and compare it with neighbor compare it with neighbor

                            ppixelsixels

                            When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                            is grown from the seed pixel by is grown from the seed pixel by

                            addingadding neighboneighborr pixel pixelss

                            When the growth of one region stopsWhen the growth of one region stops

                            choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                            againagain

                            5555

                            Region Region growing growing

                            An initial set of small An initial set of small

                            areas are iterativelyareas are iteratively

                            merged according to merged according to

                            similarity constraintssimilarity constraints

                            5656

                            Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                            X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                            We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                            On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                            Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                            SeedSeed pointspoints

                            5757

                            CriterionCriterion

                            There is a valley at around 190 in the There is a valley at around 190 in the

                            histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                            to be considered as a part of region to the to be considered as a part of region to the

                            seed pointseed point

                            The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                            to at least one pixel in that regionto at least one pixel in that region

                            Result of region growing and boundaries of Result of region growing and boundaries of

                            defectsdefects

                            5858

                            Region SplittingRegion Splitting

                            The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                            ption that the entire image is homogeneousption that the entire image is homogeneous

                            If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                            This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                            Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                            5959

                            Region SplittingRegion Splitting

                            DisadvantageDisadvantage

                            they create regions that may be adjacent they create regions that may be adjacent

                            and homogeneous but not mergedand homogeneous but not merged

                            6060

                            Region Splitting and MergingRegion Splitting and Merging

                            ProcedureProcedure

                            11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                            22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                            33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                            6161

                            Region Splitting and Merging

                            Quadtree

                            (四叉树 )

                            6262

                            PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                            where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                            mmii is the mean gray level of that region is the mean gray level of that region

                            σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                            ExampleExample

                            Original Original

                            imageimageThresholded imageThresholded image Result of Result of

                            Splitting and Splitting and

                            MergingMerging

                            • Slide 1
                            • Slide 2
                            • Slide 3
                            • Slide 4
                            • Slide 5
                            • Slide 6
                            • Slide 7
                            • Slide 8
                            • Slide 9
                            • Slide 10
                            • Slide 11
                            • Slide 12
                            • Slide 13
                            • Slide 14
                            • Slide 15
                            • Slide 16
                            • Slide 17
                            • Slide 18
                            • Slide 19
                            • Slide 20
                            • Slide 21
                            • Slide 22
                            • Slide 23
                            • Slide 24
                            • Slide 25
                            • Slide 26
                            • Slide 27
                            • Slide 28
                            • Slide 29
                            • Slide 30
                            • Slide 31
                            • Slide 32
                            • Slide 33
                            • Slide 34
                            • Slide 35
                            • Slide 36
                            • Slide 37
                            • Slide 38
                            • Slide 39
                            • Slide 40
                            • Slide 41
                            • Slide 42
                            • Slide 43
                            • Slide 44
                            • Slide 45
                            • Slide 46
                            • Slide 47
                            • Slide 48
                            • Slide 49
                            • Slide 50
                            • Slide 51
                            • Slide 52
                            • Slide 53
                            • Slide 54
                            • Slide 55
                            • Slide 56
                            • Slide 57
                            • Slide 58
                            • Slide 59
                            • Slide 60
                            • Slide 61
                            • Slide 62

                              1515

                              Basic Adaptive ThresholdingBasic Adaptive Thresholding

                              subdivide original image into small areassubdivide original image into small areas

                              utilize a different threshold to segment each utilize a different threshold to segment each subimagessubimages

                              since the threshold used for each pixel depesince the threshold used for each pixel depends on the location of the pixel in terms of tnds on the location of the pixel in terms of the subimages this type of thresholding is ahe subimages this type of thresholding is adaptivedaptive

                              See ldquoAdaptive_thresholdmrdquoSee ldquoAdaptive_thresholdmrdquo

                              1616

                              Multilevel ThresholdingMultilevel Thresholding

                              a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

                              another object class if f(xy) another object class if f(xy) gtgt T T22

                              to background if f(xy) le Tto background if f(xy) le T11

                              1717

                              The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

                              1 1 2 2

                              1 2 1

                              p z P p z P p z

                              P P

                              1818

                              Optimal ThresholdingOptimal Thresholding

                              If the form of the If the form of the

                              densities is densities is

                              known or known or

                              assumed in assumed in

                              terms of terms of

                              minimum error minimum error

                              determining an determining an

                              optimal optimal

                              threshold for threshold for

                              segmenting the segmenting the

                              image is image is

                              possiblepossible

                              1 1 2 2

                              1 2 1

                              p z P p z P p z

                              P P

                              1919

                              Optimal ThresholdingOptimal Thresholding

                              Probability of erroneouslyProbability of erroneously

                              2020

                              Optimal ThresholdingOptimal Thresholding

                              Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                              ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                              find T which makesfind T which makes

                              2121

                              Optimal ThresholdingOptimal Thresholding

                              Minimum errorMinimum error

                              Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                              threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                              pp22(z) intersect(z) intersect

                              2222

                              Optimal ThresholdingOptimal Thresholding

                              For exampleFor example

                              Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                              pp22(z)(z)

                              where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                              variance of the Gaussian density of one variance of the Gaussian density of one

                              objectobject

                              μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                              the Gaussian density of the other objectthe Gaussian density of the other object

                              2323

                              Optimal ThresholdingOptimal Thresholding

                              Quadratic equation (Quadratic equation (二次方程二次方程 ))

                              2424

                              Problems of ThresholdingProblems of Thresholding

                              Original imageOriginal image Thresholded imageThresholded image

                              2525

                              Problems of ThresholdingProblems of Thresholding

                              (a)(a) Exact threshold Exact threshold

                              segmentationsegmentation

                              (b)(b) Threshold too lowThreshold too low

                              (c)(c) Threshold too Threshold too

                              highhigh

                              2626

                              72 Point Detection72 Point Detection

                              a point has been detected at the a point has been detected at the

                              location on which the mark is location on which the mark is

                              centered ifcentered if

                              |R|geT|R|geT

                              where T is a nonnegative thresholdwhere T is a nonnegative threshold

                              R is the sum of products of the R is the sum of products of the

                              coefficients with the gray levels contained coefficients with the gray levels contained

                              in the region encompassed by the markin the region encompassed by the mark

                              1 1 1

                              1 8 1

                              1 1 1

                              1 1 1

                              1 8 1

                              1 1 1

                              2727

                              72 Point Detection72 Point Detection

                              Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                              The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                              1 1 1

                              1 8 1

                              1 1 1

                              1 1 1

                              1 8 1

                              1 1 1

                              0 1 0

                              1 4 1

                              0 1 0

                              0 1 0

                              1 4 1

                              0 1 0

                              2828

                              ExampleExample

                              2929

                              73 Line Detection73 Line Detection

                              Horizontal mask will result with max Horizontal mask will result with max

                              response when a line passed through the response when a line passed through the

                              middle row of the mask with a constant middle row of the mask with a constant

                              backgroundbackground

                              the similar idea is used with other masksthe similar idea is used with other masks

                              Note the preferred direction of each mask Note the preferred direction of each mask

                              is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                              than other possible directionsthan other possible directions

                              1 1 1 1 1 2 1 2 1 2 1 1

                              2 2 2 1 2 1 1 2 1 1 2 1

                              1 1 1 2 1 1 1 2 1 1 1 2

                              45 45Horizontal Vertical

                              1 1 1 1 1 2 1 2 1 2 1 1

                              2 2 2 1 2 1 1 2 1 1 2 1

                              1 1 1 2 1 1 1 2 1 1 1 2

                              45 45Horizontal Vertical

                              3030

                              Idea 1 of Line DetectionIdea 1 of Line Detection

                              Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                              izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                              if at a certain point in the imageif at a certain point in the image

                              |Ri||Ri|gtgt|Rj||Rj|

                              for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                              3131

                              Idea 2 of Line DetectionIdea 2 of Line Detection

                              Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                              After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                              3232

                              ExampleExample

                              3333

                              74 Edge-based 74 Edge-based SegmentationSegmentation

                              Edge-based segmentations rely on edges found in an image by edge detecting operators

                              these edges mark image locations of discontinuities in gray level

                              Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                              There are a large group of methods based on information about edges in the image

                              3434

                              What is edgeWhat is edge

                              Edge is where change occurs Change is measured by derivative in 1D

                              ―Biggest change derivative has maximum magnitude

                              Or 2nd derivative is zero we discuss approaches for implementing

                              ―first-order derivative (Gradient operator)

                              ―second-order derivative (Laplacian operator)

                              ―we have introduced both derivatives in chapter 3

                              ―Here we will talk only about their properties for edge detection

                              3535

                              What is edgeWhat is edge

                              In other wordsIn other words an edge is a set of an edge is a set of

                              connected pixelsconnected pixels

                              that lie on the boundary between two that lie on the boundary between two

                              regions with relatively distinct gray-level regions with relatively distinct gray-level

                              propertiesproperties

                              Note edge vs boundaryNote edge vs boundary

                              ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                              ―whereas a region boundary owing to whereas a region boundary owing to

                              the way it is defined is a more global the way it is defined is a more global

                              ideaidea

                              3636

                              Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                              because of because of

                              optics optics

                              sampling sampling

                              image image

                              acquisition acquisition

                              imperfectionimperfection

                              3737

                              Thick and Thin EdgeThick and Thin Edge

                              The slope of the ramp is inversely The slope of the ramp is inversely

                              proportional to the degree of blurring in the proportional to the degree of blurring in the

                              edgeedge

                              Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                              pathpath

                              Instead an edge point now is any point Instead an edge point now is any point

                              contained in the ramp and contained in the ramp and an edge would an edge would

                              then be a set of such points that are then be a set of such points that are

                              connectedconnected

                              The thickness is determined by the length of the The thickness is determined by the length of the

                              rampramp

                              The length is determined by the slope which is in The length is determined by the slope which is in

                              turn determined by the degree of blurringturn determined by the degree of blurring

                              Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                              edges tend to be thinedges tend to be thin

                              3838

                              First and Second derivatives (First and Second derivatives ( 导数导数 ))

                              the signs of the the signs of the

                              derivatives would be derivatives would be

                              reversed for an edge reversed for an edge

                              that transitions from that transitions from

                              light to darklight to dark

                              First First derivatderivatee

                              SeconSecond d derivatderivatee

                              Gray-Gray-level level profileprofile

                              3939

                              Second derivativesSecond derivatives

                              an undesirable featurean undesirable feature

                              produces 2 values for every edge in an produces 2 values for every edge in an

                              imageimage

                              zero-crossing propertyzero-crossing property

                              an imaginary straight line joining the an imaginary straight line joining the

                              extreme positive and negative values of extreme positive and negative values of

                              the second derivative would cross zero the second derivative would cross zero

                              near the midpoint of the edgenear the midpoint of the edge

                              quite useful for locating the centers of quite useful for locating the centers of

                              thick edgesthick edges

                              4040

                              Basic idea of edge detectionBasic idea of edge detection

                              A profile is defined perpendicularly to A profile is defined perpendicularly to

                              the edge direction and the results are the edge direction and the results are

                              interpretedinterpreted

                              The magnitude of the first derivative is The magnitude of the first derivative is

                              used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                              ramp)ramp)

                              The sign of the second derivative can The sign of the second derivative can

                              determine whether an edge pixel is on the determine whether an edge pixel is on the

                              dark or light side of an edgedark or light side of an edge

                              4141

                              Review of First DerivateReview of First Derivate

                              Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                              Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                              Sobel operators 3Sobel operators 333

                              6 5 8 5x yG z z G z z

                              1 2 3

                              4 5 6

                              7 8 9

                              z z z

                              z z z

                              z z z

                              1 2 3

                              4 5 6

                              7 8 9

                              z z z

                              z z z

                              z z z

                              9 5 8 6x yG z z G z z 1 0 0 1

                              0 1 1 0

                              1 0 0 1

                              0 1 1 0

                              7 8 9 1 2 3

                              3 6 9 1 4 7

                              2 2

                              2 2

                              x

                              y

                              G z z z z z z

                              G z z z z z z

                              1 2 1 1 0 1

                              0 0 0 2 0 2

                              1 2 1 1 0 1

                              1 2 1 1 0 1

                              0 0 0 2 0 2

                              1 2 1 1 0 1

                              x yf G G

                              4242

                              Edge direction and strengthEdge direction and strength

                              Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                              α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                              The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                              The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                              2 2x yf G G

                              4343

                              Gradient MasksGradient Masks

                              1 0 0 1

                              0 1 1 0

                              Roberts

                              1 0 0 1

                              0 1 1 0

                              Roberts

                              1 2 1 1 0 1

                              0 0 0 2 0 2

                              1 2 1 1 0 1

                              Sobel

                              1 2 1 1 0 1

                              0 0 0 2 0 2

                              1 2 1 1 0 1

                              Sobel

                              1 1 1 1 0 1

                              0 0 0 1 0 1

                              1 1 1 1 0 1

                              Prewitt

                              1 1 1 1 0 1

                              0 0 0 1 0 1

                              1 1 1 1 0 1

                              Prewitt

                              4444

                              Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                              0 1 1 1 1 0

                              1 0 1 1 0 1

                              1 1 0 0 1 1

                              Prewitt

                              0 1 1 1 1 0

                              1 0 1 1 0 1

                              1 1 0 0 1 1

                              Prewitt

                              4545

                              Review of Second DerivateReview of Second Derivate

                              Laplacian OperatorLaplacian Operator

                              21 1

                              1 1 4

                              f x y f x yf

                              f x y f x y f x y

                              0 1 0

                              1 4 1

                              0 1 0

                              0 1 0

                              1 4 1

                              0 1 0

                              LaplacianLaplacian

                              MaskMask

                              1 1 1

                              1 8 1

                              1 1 1

                              1 1 1

                              1 8 1

                              1 1 1

                              4646

                              Example of edge detectionExample of edge detection

                              See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                              Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                              derivative)derivative)

                              produces double edgesproduces double edges

                              unable to detect edge directionunable to detect edge direction

                              4747

                              Canny edge detectorCanny edge detector

                              The most powerful edge-detection The most powerful edge-detection

                              method method

                              It differs from the other edge-It differs from the other edge-

                              detection methods in that detection methods in that

                              it uses two different thresholds (to detect it uses two different thresholds (to detect

                              strong and weak edges) strong and weak edges)

                              and includes the weak edges in the and includes the weak edges in the

                              output only if they are connected to output only if they are connected to

                              strong edges strong edges

                              This method is therefore less likely This method is therefore less likely

                              than the others to be fooled by than the others to be fooled by

                              noise and more likely to detect true noise and more likely to detect true

                              weak edgesweak edges

                              4848

                              Laplacian of GaussianLaplacian of Gaussian

                              Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                              2 2 22

                              4 2

                              2 2 2

                              2exp

                              r rh

                              r x y

                              determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                              4949

                              Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                              0 0 1 0 0

                              0 1 2 1 0

                              1 2 16 2 1

                              0 1 2 1 0

                              0 0 1 0 0

                              0 0 1 0 0

                              0 1 2 1 0

                              1 2 16 2 1

                              0 1 2 1 0

                              0 0 1 0 0

                              The coefficient must sum to The coefficient must sum to

                              zerozero

                              5050

                              Edge Detection and Edge Detection and SegmentationSegmentation

                              Image resulting from edge detection cannot be used as a segmentation result

                              Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                              5151

                              75 Region-based 75 Region-based SegmentationSegmentation

                              GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                              Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                              PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                              For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                              PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                              For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                              5252

                              Two methods of Region Two methods of Region SegmentationSegmentation

                              Region GrowingRegion Growing

                              Region SplittingRegion Splitting

                              Region growing is the opposite of the Region growing is the opposite of the

                              split and merge approachsplit and merge approach

                              5353

                              Region GrowingRegion Growing

                              The objective of segmentation is to The objective of segmentation is to

                              partition an image into regionspartition an image into regions

                              A region is a connected component with A region is a connected component with

                              some uniformity (say gray-levels or some uniformity (say gray-levels or

                              texture)texture)

                              In region growing we start with a set In region growing we start with a set

                              of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                              appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                              pixels if they have pixels if they have similar propertiessimilar properties

                              such as specific ranges of gray level such as specific ranges of gray level

                              and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                              Need initialization Need initialization similarity similarity

                              criterioncriterion

                              5454

                              Steps of Region GrowingSteps of Region Growing

                              Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                              pixel andpixel and compare it with neighbor compare it with neighbor

                              ppixelsixels

                              When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                              is grown from the seed pixel by is grown from the seed pixel by

                              addingadding neighboneighborr pixel pixelss

                              When the growth of one region stopsWhen the growth of one region stops

                              choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                              againagain

                              5555

                              Region Region growing growing

                              An initial set of small An initial set of small

                              areas are iterativelyareas are iteratively

                              merged according to merged according to

                              similarity constraintssimilarity constraints

                              5656

                              Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                              X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                              We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                              On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                              Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                              SeedSeed pointspoints

                              5757

                              CriterionCriterion

                              There is a valley at around 190 in the There is a valley at around 190 in the

                              histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                              to be considered as a part of region to the to be considered as a part of region to the

                              seed pointseed point

                              The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                              to at least one pixel in that regionto at least one pixel in that region

                              Result of region growing and boundaries of Result of region growing and boundaries of

                              defectsdefects

                              5858

                              Region SplittingRegion Splitting

                              The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                              ption that the entire image is homogeneousption that the entire image is homogeneous

                              If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                              This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                              Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                              5959

                              Region SplittingRegion Splitting

                              DisadvantageDisadvantage

                              they create regions that may be adjacent they create regions that may be adjacent

                              and homogeneous but not mergedand homogeneous but not merged

                              6060

                              Region Splitting and MergingRegion Splitting and Merging

                              ProcedureProcedure

                              11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                              22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                              33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                              6161

                              Region Splitting and Merging

                              Quadtree

                              (四叉树 )

                              6262

                              PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                              where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                              mmii is the mean gray level of that region is the mean gray level of that region

                              σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                              ExampleExample

                              Original Original

                              imageimageThresholded imageThresholded image Result of Result of

                              Splitting and Splitting and

                              MergingMerging

                              • Slide 1
                              • Slide 2
                              • Slide 3
                              • Slide 4
                              • Slide 5
                              • Slide 6
                              • Slide 7
                              • Slide 8
                              • Slide 9
                              • Slide 10
                              • Slide 11
                              • Slide 12
                              • Slide 13
                              • Slide 14
                              • Slide 15
                              • Slide 16
                              • Slide 17
                              • Slide 18
                              • Slide 19
                              • Slide 20
                              • Slide 21
                              • Slide 22
                              • Slide 23
                              • Slide 24
                              • Slide 25
                              • Slide 26
                              • Slide 27
                              • Slide 28
                              • Slide 29
                              • Slide 30
                              • Slide 31
                              • Slide 32
                              • Slide 33
                              • Slide 34
                              • Slide 35
                              • Slide 36
                              • Slide 37
                              • Slide 38
                              • Slide 39
                              • Slide 40
                              • Slide 41
                              • Slide 42
                              • Slide 43
                              • Slide 44
                              • Slide 45
                              • Slide 46
                              • Slide 47
                              • Slide 48
                              • Slide 49
                              • Slide 50
                              • Slide 51
                              • Slide 52
                              • Slide 53
                              • Slide 54
                              • Slide 55
                              • Slide 56
                              • Slide 57
                              • Slide 58
                              • Slide 59
                              • Slide 60
                              • Slide 61
                              • Slide 62

                                1616

                                Multilevel ThresholdingMultilevel Thresholding

                                a point (xy) belongs toa point (xy) belongs to an object class if Tan object class if T11 ltlt f(xy) le T f(xy) le T22

                                another object class if f(xy) another object class if f(xy) gtgt T T22

                                to background if f(xy) le Tto background if f(xy) le T11

                                1717

                                The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

                                1 1 2 2

                                1 2 1

                                p z P p z P p z

                                P P

                                1818

                                Optimal ThresholdingOptimal Thresholding

                                If the form of the If the form of the

                                densities is densities is

                                known or known or

                                assumed in assumed in

                                terms of terms of

                                minimum error minimum error

                                determining an determining an

                                optimal optimal

                                threshold for threshold for

                                segmenting the segmenting the

                                image is image is

                                possiblepossible

                                1 1 2 2

                                1 2 1

                                p z P p z P p z

                                P P

                                1919

                                Optimal ThresholdingOptimal Thresholding

                                Probability of erroneouslyProbability of erroneously

                                2020

                                Optimal ThresholdingOptimal Thresholding

                                Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                                ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                                find T which makesfind T which makes

                                2121

                                Optimal ThresholdingOptimal Thresholding

                                Minimum errorMinimum error

                                Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                                threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                                pp22(z) intersect(z) intersect

                                2222

                                Optimal ThresholdingOptimal Thresholding

                                For exampleFor example

                                Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                                pp22(z)(z)

                                where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                                variance of the Gaussian density of one variance of the Gaussian density of one

                                objectobject

                                μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                                the Gaussian density of the other objectthe Gaussian density of the other object

                                2323

                                Optimal ThresholdingOptimal Thresholding

                                Quadratic equation (Quadratic equation (二次方程二次方程 ))

                                2424

                                Problems of ThresholdingProblems of Thresholding

                                Original imageOriginal image Thresholded imageThresholded image

                                2525

                                Problems of ThresholdingProblems of Thresholding

                                (a)(a) Exact threshold Exact threshold

                                segmentationsegmentation

                                (b)(b) Threshold too lowThreshold too low

                                (c)(c) Threshold too Threshold too

                                highhigh

                                2626

                                72 Point Detection72 Point Detection

                                a point has been detected at the a point has been detected at the

                                location on which the mark is location on which the mark is

                                centered ifcentered if

                                |R|geT|R|geT

                                where T is a nonnegative thresholdwhere T is a nonnegative threshold

                                R is the sum of products of the R is the sum of products of the

                                coefficients with the gray levels contained coefficients with the gray levels contained

                                in the region encompassed by the markin the region encompassed by the mark

                                1 1 1

                                1 8 1

                                1 1 1

                                1 1 1

                                1 8 1

                                1 1 1

                                2727

                                72 Point Detection72 Point Detection

                                Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                                The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                                1 1 1

                                1 8 1

                                1 1 1

                                1 1 1

                                1 8 1

                                1 1 1

                                0 1 0

                                1 4 1

                                0 1 0

                                0 1 0

                                1 4 1

                                0 1 0

                                2828

                                ExampleExample

                                2929

                                73 Line Detection73 Line Detection

                                Horizontal mask will result with max Horizontal mask will result with max

                                response when a line passed through the response when a line passed through the

                                middle row of the mask with a constant middle row of the mask with a constant

                                backgroundbackground

                                the similar idea is used with other masksthe similar idea is used with other masks

                                Note the preferred direction of each mask Note the preferred direction of each mask

                                is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                than other possible directionsthan other possible directions

                                1 1 1 1 1 2 1 2 1 2 1 1

                                2 2 2 1 2 1 1 2 1 1 2 1

                                1 1 1 2 1 1 1 2 1 1 1 2

                                45 45Horizontal Vertical

                                1 1 1 1 1 2 1 2 1 2 1 1

                                2 2 2 1 2 1 1 2 1 1 2 1

                                1 1 1 2 1 1 1 2 1 1 1 2

                                45 45Horizontal Vertical

                                3030

                                Idea 1 of Line DetectionIdea 1 of Line Detection

                                Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                if at a certain point in the imageif at a certain point in the image

                                |Ri||Ri|gtgt|Rj||Rj|

                                for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                3131

                                Idea 2 of Line DetectionIdea 2 of Line Detection

                                Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                3232

                                ExampleExample

                                3333

                                74 Edge-based 74 Edge-based SegmentationSegmentation

                                Edge-based segmentations rely on edges found in an image by edge detecting operators

                                these edges mark image locations of discontinuities in gray level

                                Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                There are a large group of methods based on information about edges in the image

                                3434

                                What is edgeWhat is edge

                                Edge is where change occurs Change is measured by derivative in 1D

                                ―Biggest change derivative has maximum magnitude

                                Or 2nd derivative is zero we discuss approaches for implementing

                                ―first-order derivative (Gradient operator)

                                ―second-order derivative (Laplacian operator)

                                ―we have introduced both derivatives in chapter 3

                                ―Here we will talk only about their properties for edge detection

                                3535

                                What is edgeWhat is edge

                                In other wordsIn other words an edge is a set of an edge is a set of

                                connected pixelsconnected pixels

                                that lie on the boundary between two that lie on the boundary between two

                                regions with relatively distinct gray-level regions with relatively distinct gray-level

                                propertiesproperties

                                Note edge vs boundaryNote edge vs boundary

                                ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                ―whereas a region boundary owing to whereas a region boundary owing to

                                the way it is defined is a more global the way it is defined is a more global

                                ideaidea

                                3636

                                Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                because of because of

                                optics optics

                                sampling sampling

                                image image

                                acquisition acquisition

                                imperfectionimperfection

                                3737

                                Thick and Thin EdgeThick and Thin Edge

                                The slope of the ramp is inversely The slope of the ramp is inversely

                                proportional to the degree of blurring in the proportional to the degree of blurring in the

                                edgeedge

                                Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                pathpath

                                Instead an edge point now is any point Instead an edge point now is any point

                                contained in the ramp and contained in the ramp and an edge would an edge would

                                then be a set of such points that are then be a set of such points that are

                                connectedconnected

                                The thickness is determined by the length of the The thickness is determined by the length of the

                                rampramp

                                The length is determined by the slope which is in The length is determined by the slope which is in

                                turn determined by the degree of blurringturn determined by the degree of blurring

                                Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                edges tend to be thinedges tend to be thin

                                3838

                                First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                the signs of the the signs of the

                                derivatives would be derivatives would be

                                reversed for an edge reversed for an edge

                                that transitions from that transitions from

                                light to darklight to dark

                                First First derivatderivatee

                                SeconSecond d derivatderivatee

                                Gray-Gray-level level profileprofile

                                3939

                                Second derivativesSecond derivatives

                                an undesirable featurean undesirable feature

                                produces 2 values for every edge in an produces 2 values for every edge in an

                                imageimage

                                zero-crossing propertyzero-crossing property

                                an imaginary straight line joining the an imaginary straight line joining the

                                extreme positive and negative values of extreme positive and negative values of

                                the second derivative would cross zero the second derivative would cross zero

                                near the midpoint of the edgenear the midpoint of the edge

                                quite useful for locating the centers of quite useful for locating the centers of

                                thick edgesthick edges

                                4040

                                Basic idea of edge detectionBasic idea of edge detection

                                A profile is defined perpendicularly to A profile is defined perpendicularly to

                                the edge direction and the results are the edge direction and the results are

                                interpretedinterpreted

                                The magnitude of the first derivative is The magnitude of the first derivative is

                                used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                ramp)ramp)

                                The sign of the second derivative can The sign of the second derivative can

                                determine whether an edge pixel is on the determine whether an edge pixel is on the

                                dark or light side of an edgedark or light side of an edge

                                4141

                                Review of First DerivateReview of First Derivate

                                Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                Sobel operators 3Sobel operators 333

                                6 5 8 5x yG z z G z z

                                1 2 3

                                4 5 6

                                7 8 9

                                z z z

                                z z z

                                z z z

                                1 2 3

                                4 5 6

                                7 8 9

                                z z z

                                z z z

                                z z z

                                9 5 8 6x yG z z G z z 1 0 0 1

                                0 1 1 0

                                1 0 0 1

                                0 1 1 0

                                7 8 9 1 2 3

                                3 6 9 1 4 7

                                2 2

                                2 2

                                x

                                y

                                G z z z z z z

                                G z z z z z z

                                1 2 1 1 0 1

                                0 0 0 2 0 2

                                1 2 1 1 0 1

                                1 2 1 1 0 1

                                0 0 0 2 0 2

                                1 2 1 1 0 1

                                x yf G G

                                4242

                                Edge direction and strengthEdge direction and strength

                                Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                2 2x yf G G

                                4343

                                Gradient MasksGradient Masks

                                1 0 0 1

                                0 1 1 0

                                Roberts

                                1 0 0 1

                                0 1 1 0

                                Roberts

                                1 2 1 1 0 1

                                0 0 0 2 0 2

                                1 2 1 1 0 1

                                Sobel

                                1 2 1 1 0 1

                                0 0 0 2 0 2

                                1 2 1 1 0 1

                                Sobel

                                1 1 1 1 0 1

                                0 0 0 1 0 1

                                1 1 1 1 0 1

                                Prewitt

                                1 1 1 1 0 1

                                0 0 0 1 0 1

                                1 1 1 1 0 1

                                Prewitt

                                4444

                                Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                0 1 1 1 1 0

                                1 0 1 1 0 1

                                1 1 0 0 1 1

                                Prewitt

                                0 1 1 1 1 0

                                1 0 1 1 0 1

                                1 1 0 0 1 1

                                Prewitt

                                4545

                                Review of Second DerivateReview of Second Derivate

                                Laplacian OperatorLaplacian Operator

                                21 1

                                1 1 4

                                f x y f x yf

                                f x y f x y f x y

                                0 1 0

                                1 4 1

                                0 1 0

                                0 1 0

                                1 4 1

                                0 1 0

                                LaplacianLaplacian

                                MaskMask

                                1 1 1

                                1 8 1

                                1 1 1

                                1 1 1

                                1 8 1

                                1 1 1

                                4646

                                Example of edge detectionExample of edge detection

                                See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                derivative)derivative)

                                produces double edgesproduces double edges

                                unable to detect edge directionunable to detect edge direction

                                4747

                                Canny edge detectorCanny edge detector

                                The most powerful edge-detection The most powerful edge-detection

                                method method

                                It differs from the other edge-It differs from the other edge-

                                detection methods in that detection methods in that

                                it uses two different thresholds (to detect it uses two different thresholds (to detect

                                strong and weak edges) strong and weak edges)

                                and includes the weak edges in the and includes the weak edges in the

                                output only if they are connected to output only if they are connected to

                                strong edges strong edges

                                This method is therefore less likely This method is therefore less likely

                                than the others to be fooled by than the others to be fooled by

                                noise and more likely to detect true noise and more likely to detect true

                                weak edgesweak edges

                                4848

                                Laplacian of GaussianLaplacian of Gaussian

                                Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                2 2 22

                                4 2

                                2 2 2

                                2exp

                                r rh

                                r x y

                                determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                4949

                                Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                0 0 1 0 0

                                0 1 2 1 0

                                1 2 16 2 1

                                0 1 2 1 0

                                0 0 1 0 0

                                0 0 1 0 0

                                0 1 2 1 0

                                1 2 16 2 1

                                0 1 2 1 0

                                0 0 1 0 0

                                The coefficient must sum to The coefficient must sum to

                                zerozero

                                5050

                                Edge Detection and Edge Detection and SegmentationSegmentation

                                Image resulting from edge detection cannot be used as a segmentation result

                                Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                5151

                                75 Region-based 75 Region-based SegmentationSegmentation

                                GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                5252

                                Two methods of Region Two methods of Region SegmentationSegmentation

                                Region GrowingRegion Growing

                                Region SplittingRegion Splitting

                                Region growing is the opposite of the Region growing is the opposite of the

                                split and merge approachsplit and merge approach

                                5353

                                Region GrowingRegion Growing

                                The objective of segmentation is to The objective of segmentation is to

                                partition an image into regionspartition an image into regions

                                A region is a connected component with A region is a connected component with

                                some uniformity (say gray-levels or some uniformity (say gray-levels or

                                texture)texture)

                                In region growing we start with a set In region growing we start with a set

                                of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                pixels if they have pixels if they have similar propertiessimilar properties

                                such as specific ranges of gray level such as specific ranges of gray level

                                and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                Need initialization Need initialization similarity similarity

                                criterioncriterion

                                5454

                                Steps of Region GrowingSteps of Region Growing

                                Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                pixel andpixel and compare it with neighbor compare it with neighbor

                                ppixelsixels

                                When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                is grown from the seed pixel by is grown from the seed pixel by

                                addingadding neighboneighborr pixel pixelss

                                When the growth of one region stopsWhen the growth of one region stops

                                choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                againagain

                                5555

                                Region Region growing growing

                                An initial set of small An initial set of small

                                areas are iterativelyareas are iteratively

                                merged according to merged according to

                                similarity constraintssimilarity constraints

                                5656

                                Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                SeedSeed pointspoints

                                5757

                                CriterionCriterion

                                There is a valley at around 190 in the There is a valley at around 190 in the

                                histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                to be considered as a part of region to the to be considered as a part of region to the

                                seed pointseed point

                                The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                to at least one pixel in that regionto at least one pixel in that region

                                Result of region growing and boundaries of Result of region growing and boundaries of

                                defectsdefects

                                5858

                                Region SplittingRegion Splitting

                                The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                ption that the entire image is homogeneousption that the entire image is homogeneous

                                If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                5959

                                Region SplittingRegion Splitting

                                DisadvantageDisadvantage

                                they create regions that may be adjacent they create regions that may be adjacent

                                and homogeneous but not mergedand homogeneous but not merged

                                6060

                                Region Splitting and MergingRegion Splitting and Merging

                                ProcedureProcedure

                                11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                6161

                                Region Splitting and Merging

                                Quadtree

                                (四叉树 )

                                6262

                                PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                mmii is the mean gray level of that region is the mean gray level of that region

                                σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                ExampleExample

                                Original Original

                                imageimageThresholded imageThresholded image Result of Result of

                                Splitting and Splitting and

                                MergingMerging

                                • Slide 1
                                • Slide 2
                                • Slide 3
                                • Slide 4
                                • Slide 5
                                • Slide 6
                                • Slide 7
                                • Slide 8
                                • Slide 9
                                • Slide 10
                                • Slide 11
                                • Slide 12
                                • Slide 13
                                • Slide 14
                                • Slide 15
                                • Slide 16
                                • Slide 17
                                • Slide 18
                                • Slide 19
                                • Slide 20
                                • Slide 21
                                • Slide 22
                                • Slide 23
                                • Slide 24
                                • Slide 25
                                • Slide 26
                                • Slide 27
                                • Slide 28
                                • Slide 29
                                • Slide 30
                                • Slide 31
                                • Slide 32
                                • Slide 33
                                • Slide 34
                                • Slide 35
                                • Slide 36
                                • Slide 37
                                • Slide 38
                                • Slide 39
                                • Slide 40
                                • Slide 41
                                • Slide 42
                                • Slide 43
                                • Slide 44
                                • Slide 45
                                • Slide 46
                                • Slide 47
                                • Slide 48
                                • Slide 49
                                • Slide 50
                                • Slide 51
                                • Slide 52
                                • Slide 53
                                • Slide 54
                                • Slide 55
                                • Slide 56
                                • Slide 57
                                • Slide 58
                                • Slide 59
                                • Slide 60
                                • Slide 61
                                • Slide 62

                                  1717

                                  The histogram of an image containing two pThe histogram of an image containing two principal brightness regions can be considererincipal brightness regions can be considered an estimate of the brightness probability d an estimate of the brightness probability density function p(z)density function p(z) p(z) is the sum (or mixture) of two unimodal (p(z) is the sum (or mixture) of two unimodal ( 单单峰的峰的 ) densities (one for light one for dark region) densities (one for light one for dark regions)s)

                                  1 1 2 2

                                  1 2 1

                                  p z P p z P p z

                                  P P

                                  1818

                                  Optimal ThresholdingOptimal Thresholding

                                  If the form of the If the form of the

                                  densities is densities is

                                  known or known or

                                  assumed in assumed in

                                  terms of terms of

                                  minimum error minimum error

                                  determining an determining an

                                  optimal optimal

                                  threshold for threshold for

                                  segmenting the segmenting the

                                  image is image is

                                  possiblepossible

                                  1 1 2 2

                                  1 2 1

                                  p z P p z P p z

                                  P P

                                  1919

                                  Optimal ThresholdingOptimal Thresholding

                                  Probability of erroneouslyProbability of erroneously

                                  2020

                                  Optimal ThresholdingOptimal Thresholding

                                  Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                                  ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                                  find T which makesfind T which makes

                                  2121

                                  Optimal ThresholdingOptimal Thresholding

                                  Minimum errorMinimum error

                                  Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                                  threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                                  pp22(z) intersect(z) intersect

                                  2222

                                  Optimal ThresholdingOptimal Thresholding

                                  For exampleFor example

                                  Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                                  pp22(z)(z)

                                  where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                                  variance of the Gaussian density of one variance of the Gaussian density of one

                                  objectobject

                                  μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                                  the Gaussian density of the other objectthe Gaussian density of the other object

                                  2323

                                  Optimal ThresholdingOptimal Thresholding

                                  Quadratic equation (Quadratic equation (二次方程二次方程 ))

                                  2424

                                  Problems of ThresholdingProblems of Thresholding

                                  Original imageOriginal image Thresholded imageThresholded image

                                  2525

                                  Problems of ThresholdingProblems of Thresholding

                                  (a)(a) Exact threshold Exact threshold

                                  segmentationsegmentation

                                  (b)(b) Threshold too lowThreshold too low

                                  (c)(c) Threshold too Threshold too

                                  highhigh

                                  2626

                                  72 Point Detection72 Point Detection

                                  a point has been detected at the a point has been detected at the

                                  location on which the mark is location on which the mark is

                                  centered ifcentered if

                                  |R|geT|R|geT

                                  where T is a nonnegative thresholdwhere T is a nonnegative threshold

                                  R is the sum of products of the R is the sum of products of the

                                  coefficients with the gray levels contained coefficients with the gray levels contained

                                  in the region encompassed by the markin the region encompassed by the mark

                                  1 1 1

                                  1 8 1

                                  1 1 1

                                  1 1 1

                                  1 8 1

                                  1 1 1

                                  2727

                                  72 Point Detection72 Point Detection

                                  Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                                  The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                                  1 1 1

                                  1 8 1

                                  1 1 1

                                  1 1 1

                                  1 8 1

                                  1 1 1

                                  0 1 0

                                  1 4 1

                                  0 1 0

                                  0 1 0

                                  1 4 1

                                  0 1 0

                                  2828

                                  ExampleExample

                                  2929

                                  73 Line Detection73 Line Detection

                                  Horizontal mask will result with max Horizontal mask will result with max

                                  response when a line passed through the response when a line passed through the

                                  middle row of the mask with a constant middle row of the mask with a constant

                                  backgroundbackground

                                  the similar idea is used with other masksthe similar idea is used with other masks

                                  Note the preferred direction of each mask Note the preferred direction of each mask

                                  is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                  than other possible directionsthan other possible directions

                                  1 1 1 1 1 2 1 2 1 2 1 1

                                  2 2 2 1 2 1 1 2 1 1 2 1

                                  1 1 1 2 1 1 1 2 1 1 1 2

                                  45 45Horizontal Vertical

                                  1 1 1 1 1 2 1 2 1 2 1 1

                                  2 2 2 1 2 1 1 2 1 1 2 1

                                  1 1 1 2 1 1 1 2 1 1 1 2

                                  45 45Horizontal Vertical

                                  3030

                                  Idea 1 of Line DetectionIdea 1 of Line Detection

                                  Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                  izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                  if at a certain point in the imageif at a certain point in the image

                                  |Ri||Ri|gtgt|Rj||Rj|

                                  for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                  3131

                                  Idea 2 of Line DetectionIdea 2 of Line Detection

                                  Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                  After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                  3232

                                  ExampleExample

                                  3333

                                  74 Edge-based 74 Edge-based SegmentationSegmentation

                                  Edge-based segmentations rely on edges found in an image by edge detecting operators

                                  these edges mark image locations of discontinuities in gray level

                                  Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                  There are a large group of methods based on information about edges in the image

                                  3434

                                  What is edgeWhat is edge

                                  Edge is where change occurs Change is measured by derivative in 1D

                                  ―Biggest change derivative has maximum magnitude

                                  Or 2nd derivative is zero we discuss approaches for implementing

                                  ―first-order derivative (Gradient operator)

                                  ―second-order derivative (Laplacian operator)

                                  ―we have introduced both derivatives in chapter 3

                                  ―Here we will talk only about their properties for edge detection

                                  3535

                                  What is edgeWhat is edge

                                  In other wordsIn other words an edge is a set of an edge is a set of

                                  connected pixelsconnected pixels

                                  that lie on the boundary between two that lie on the boundary between two

                                  regions with relatively distinct gray-level regions with relatively distinct gray-level

                                  propertiesproperties

                                  Note edge vs boundaryNote edge vs boundary

                                  ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                  ―whereas a region boundary owing to whereas a region boundary owing to

                                  the way it is defined is a more global the way it is defined is a more global

                                  ideaidea

                                  3636

                                  Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                  because of because of

                                  optics optics

                                  sampling sampling

                                  image image

                                  acquisition acquisition

                                  imperfectionimperfection

                                  3737

                                  Thick and Thin EdgeThick and Thin Edge

                                  The slope of the ramp is inversely The slope of the ramp is inversely

                                  proportional to the degree of blurring in the proportional to the degree of blurring in the

                                  edgeedge

                                  Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                  pathpath

                                  Instead an edge point now is any point Instead an edge point now is any point

                                  contained in the ramp and contained in the ramp and an edge would an edge would

                                  then be a set of such points that are then be a set of such points that are

                                  connectedconnected

                                  The thickness is determined by the length of the The thickness is determined by the length of the

                                  rampramp

                                  The length is determined by the slope which is in The length is determined by the slope which is in

                                  turn determined by the degree of blurringturn determined by the degree of blurring

                                  Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                  edges tend to be thinedges tend to be thin

                                  3838

                                  First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                  the signs of the the signs of the

                                  derivatives would be derivatives would be

                                  reversed for an edge reversed for an edge

                                  that transitions from that transitions from

                                  light to darklight to dark

                                  First First derivatderivatee

                                  SeconSecond d derivatderivatee

                                  Gray-Gray-level level profileprofile

                                  3939

                                  Second derivativesSecond derivatives

                                  an undesirable featurean undesirable feature

                                  produces 2 values for every edge in an produces 2 values for every edge in an

                                  imageimage

                                  zero-crossing propertyzero-crossing property

                                  an imaginary straight line joining the an imaginary straight line joining the

                                  extreme positive and negative values of extreme positive and negative values of

                                  the second derivative would cross zero the second derivative would cross zero

                                  near the midpoint of the edgenear the midpoint of the edge

                                  quite useful for locating the centers of quite useful for locating the centers of

                                  thick edgesthick edges

                                  4040

                                  Basic idea of edge detectionBasic idea of edge detection

                                  A profile is defined perpendicularly to A profile is defined perpendicularly to

                                  the edge direction and the results are the edge direction and the results are

                                  interpretedinterpreted

                                  The magnitude of the first derivative is The magnitude of the first derivative is

                                  used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                  ramp)ramp)

                                  The sign of the second derivative can The sign of the second derivative can

                                  determine whether an edge pixel is on the determine whether an edge pixel is on the

                                  dark or light side of an edgedark or light side of an edge

                                  4141

                                  Review of First DerivateReview of First Derivate

                                  Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                  Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                  Sobel operators 3Sobel operators 333

                                  6 5 8 5x yG z z G z z

                                  1 2 3

                                  4 5 6

                                  7 8 9

                                  z z z

                                  z z z

                                  z z z

                                  1 2 3

                                  4 5 6

                                  7 8 9

                                  z z z

                                  z z z

                                  z z z

                                  9 5 8 6x yG z z G z z 1 0 0 1

                                  0 1 1 0

                                  1 0 0 1

                                  0 1 1 0

                                  7 8 9 1 2 3

                                  3 6 9 1 4 7

                                  2 2

                                  2 2

                                  x

                                  y

                                  G z z z z z z

                                  G z z z z z z

                                  1 2 1 1 0 1

                                  0 0 0 2 0 2

                                  1 2 1 1 0 1

                                  1 2 1 1 0 1

                                  0 0 0 2 0 2

                                  1 2 1 1 0 1

                                  x yf G G

                                  4242

                                  Edge direction and strengthEdge direction and strength

                                  Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                  α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                  The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                  The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                  2 2x yf G G

                                  4343

                                  Gradient MasksGradient Masks

                                  1 0 0 1

                                  0 1 1 0

                                  Roberts

                                  1 0 0 1

                                  0 1 1 0

                                  Roberts

                                  1 2 1 1 0 1

                                  0 0 0 2 0 2

                                  1 2 1 1 0 1

                                  Sobel

                                  1 2 1 1 0 1

                                  0 0 0 2 0 2

                                  1 2 1 1 0 1

                                  Sobel

                                  1 1 1 1 0 1

                                  0 0 0 1 0 1

                                  1 1 1 1 0 1

                                  Prewitt

                                  1 1 1 1 0 1

                                  0 0 0 1 0 1

                                  1 1 1 1 0 1

                                  Prewitt

                                  4444

                                  Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                  0 1 1 1 1 0

                                  1 0 1 1 0 1

                                  1 1 0 0 1 1

                                  Prewitt

                                  0 1 1 1 1 0

                                  1 0 1 1 0 1

                                  1 1 0 0 1 1

                                  Prewitt

                                  4545

                                  Review of Second DerivateReview of Second Derivate

                                  Laplacian OperatorLaplacian Operator

                                  21 1

                                  1 1 4

                                  f x y f x yf

                                  f x y f x y f x y

                                  0 1 0

                                  1 4 1

                                  0 1 0

                                  0 1 0

                                  1 4 1

                                  0 1 0

                                  LaplacianLaplacian

                                  MaskMask

                                  1 1 1

                                  1 8 1

                                  1 1 1

                                  1 1 1

                                  1 8 1

                                  1 1 1

                                  4646

                                  Example of edge detectionExample of edge detection

                                  See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                  Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                  derivative)derivative)

                                  produces double edgesproduces double edges

                                  unable to detect edge directionunable to detect edge direction

                                  4747

                                  Canny edge detectorCanny edge detector

                                  The most powerful edge-detection The most powerful edge-detection

                                  method method

                                  It differs from the other edge-It differs from the other edge-

                                  detection methods in that detection methods in that

                                  it uses two different thresholds (to detect it uses two different thresholds (to detect

                                  strong and weak edges) strong and weak edges)

                                  and includes the weak edges in the and includes the weak edges in the

                                  output only if they are connected to output only if they are connected to

                                  strong edges strong edges

                                  This method is therefore less likely This method is therefore less likely

                                  than the others to be fooled by than the others to be fooled by

                                  noise and more likely to detect true noise and more likely to detect true

                                  weak edgesweak edges

                                  4848

                                  Laplacian of GaussianLaplacian of Gaussian

                                  Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                  2 2 22

                                  4 2

                                  2 2 2

                                  2exp

                                  r rh

                                  r x y

                                  determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                  4949

                                  Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                  0 0 1 0 0

                                  0 1 2 1 0

                                  1 2 16 2 1

                                  0 1 2 1 0

                                  0 0 1 0 0

                                  0 0 1 0 0

                                  0 1 2 1 0

                                  1 2 16 2 1

                                  0 1 2 1 0

                                  0 0 1 0 0

                                  The coefficient must sum to The coefficient must sum to

                                  zerozero

                                  5050

                                  Edge Detection and Edge Detection and SegmentationSegmentation

                                  Image resulting from edge detection cannot be used as a segmentation result

                                  Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                  5151

                                  75 Region-based 75 Region-based SegmentationSegmentation

                                  GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                  Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                  PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                  For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                  PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                  For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                  5252

                                  Two methods of Region Two methods of Region SegmentationSegmentation

                                  Region GrowingRegion Growing

                                  Region SplittingRegion Splitting

                                  Region growing is the opposite of the Region growing is the opposite of the

                                  split and merge approachsplit and merge approach

                                  5353

                                  Region GrowingRegion Growing

                                  The objective of segmentation is to The objective of segmentation is to

                                  partition an image into regionspartition an image into regions

                                  A region is a connected component with A region is a connected component with

                                  some uniformity (say gray-levels or some uniformity (say gray-levels or

                                  texture)texture)

                                  In region growing we start with a set In region growing we start with a set

                                  of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                  appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                  pixels if they have pixels if they have similar propertiessimilar properties

                                  such as specific ranges of gray level such as specific ranges of gray level

                                  and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                  Need initialization Need initialization similarity similarity

                                  criterioncriterion

                                  5454

                                  Steps of Region GrowingSteps of Region Growing

                                  Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                  pixel andpixel and compare it with neighbor compare it with neighbor

                                  ppixelsixels

                                  When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                  is grown from the seed pixel by is grown from the seed pixel by

                                  addingadding neighboneighborr pixel pixelss

                                  When the growth of one region stopsWhen the growth of one region stops

                                  choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                  againagain

                                  5555

                                  Region Region growing growing

                                  An initial set of small An initial set of small

                                  areas are iterativelyareas are iteratively

                                  merged according to merged according to

                                  similarity constraintssimilarity constraints

                                  5656

                                  Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                  X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                  We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                  On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                  Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                  SeedSeed pointspoints

                                  5757

                                  CriterionCriterion

                                  There is a valley at around 190 in the There is a valley at around 190 in the

                                  histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                  to be considered as a part of region to the to be considered as a part of region to the

                                  seed pointseed point

                                  The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                  to at least one pixel in that regionto at least one pixel in that region

                                  Result of region growing and boundaries of Result of region growing and boundaries of

                                  defectsdefects

                                  5858

                                  Region SplittingRegion Splitting

                                  The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                  ption that the entire image is homogeneousption that the entire image is homogeneous

                                  If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                  This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                  Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                  5959

                                  Region SplittingRegion Splitting

                                  DisadvantageDisadvantage

                                  they create regions that may be adjacent they create regions that may be adjacent

                                  and homogeneous but not mergedand homogeneous but not merged

                                  6060

                                  Region Splitting and MergingRegion Splitting and Merging

                                  ProcedureProcedure

                                  11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                  22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                  33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                  6161

                                  Region Splitting and Merging

                                  Quadtree

                                  (四叉树 )

                                  6262

                                  PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                  where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                  mmii is the mean gray level of that region is the mean gray level of that region

                                  σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                  ExampleExample

                                  Original Original

                                  imageimageThresholded imageThresholded image Result of Result of

                                  Splitting and Splitting and

                                  MergingMerging

                                  • Slide 1
                                  • Slide 2
                                  • Slide 3
                                  • Slide 4
                                  • Slide 5
                                  • Slide 6
                                  • Slide 7
                                  • Slide 8
                                  • Slide 9
                                  • Slide 10
                                  • Slide 11
                                  • Slide 12
                                  • Slide 13
                                  • Slide 14
                                  • Slide 15
                                  • Slide 16
                                  • Slide 17
                                  • Slide 18
                                  • Slide 19
                                  • Slide 20
                                  • Slide 21
                                  • Slide 22
                                  • Slide 23
                                  • Slide 24
                                  • Slide 25
                                  • Slide 26
                                  • Slide 27
                                  • Slide 28
                                  • Slide 29
                                  • Slide 30
                                  • Slide 31
                                  • Slide 32
                                  • Slide 33
                                  • Slide 34
                                  • Slide 35
                                  • Slide 36
                                  • Slide 37
                                  • Slide 38
                                  • Slide 39
                                  • Slide 40
                                  • Slide 41
                                  • Slide 42
                                  • Slide 43
                                  • Slide 44
                                  • Slide 45
                                  • Slide 46
                                  • Slide 47
                                  • Slide 48
                                  • Slide 49
                                  • Slide 50
                                  • Slide 51
                                  • Slide 52
                                  • Slide 53
                                  • Slide 54
                                  • Slide 55
                                  • Slide 56
                                  • Slide 57
                                  • Slide 58
                                  • Slide 59
                                  • Slide 60
                                  • Slide 61
                                  • Slide 62

                                    1818

                                    Optimal ThresholdingOptimal Thresholding

                                    If the form of the If the form of the

                                    densities is densities is

                                    known or known or

                                    assumed in assumed in

                                    terms of terms of

                                    minimum error minimum error

                                    determining an determining an

                                    optimal optimal

                                    threshold for threshold for

                                    segmenting the segmenting the

                                    image is image is

                                    possiblepossible

                                    1 1 2 2

                                    1 2 1

                                    p z P p z P p z

                                    P P

                                    1919

                                    Optimal ThresholdingOptimal Thresholding

                                    Probability of erroneouslyProbability of erroneously

                                    2020

                                    Optimal ThresholdingOptimal Thresholding

                                    Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                                    ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                                    find T which makesfind T which makes

                                    2121

                                    Optimal ThresholdingOptimal Thresholding

                                    Minimum errorMinimum error

                                    Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                                    threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                                    pp22(z) intersect(z) intersect

                                    2222

                                    Optimal ThresholdingOptimal Thresholding

                                    For exampleFor example

                                    Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                                    pp22(z)(z)

                                    where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                                    variance of the Gaussian density of one variance of the Gaussian density of one

                                    objectobject

                                    μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                                    the Gaussian density of the other objectthe Gaussian density of the other object

                                    2323

                                    Optimal ThresholdingOptimal Thresholding

                                    Quadratic equation (Quadratic equation (二次方程二次方程 ))

                                    2424

                                    Problems of ThresholdingProblems of Thresholding

                                    Original imageOriginal image Thresholded imageThresholded image

                                    2525

                                    Problems of ThresholdingProblems of Thresholding

                                    (a)(a) Exact threshold Exact threshold

                                    segmentationsegmentation

                                    (b)(b) Threshold too lowThreshold too low

                                    (c)(c) Threshold too Threshold too

                                    highhigh

                                    2626

                                    72 Point Detection72 Point Detection

                                    a point has been detected at the a point has been detected at the

                                    location on which the mark is location on which the mark is

                                    centered ifcentered if

                                    |R|geT|R|geT

                                    where T is a nonnegative thresholdwhere T is a nonnegative threshold

                                    R is the sum of products of the R is the sum of products of the

                                    coefficients with the gray levels contained coefficients with the gray levels contained

                                    in the region encompassed by the markin the region encompassed by the mark

                                    1 1 1

                                    1 8 1

                                    1 1 1

                                    1 1 1

                                    1 8 1

                                    1 1 1

                                    2727

                                    72 Point Detection72 Point Detection

                                    Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                                    The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                                    1 1 1

                                    1 8 1

                                    1 1 1

                                    1 1 1

                                    1 8 1

                                    1 1 1

                                    0 1 0

                                    1 4 1

                                    0 1 0

                                    0 1 0

                                    1 4 1

                                    0 1 0

                                    2828

                                    ExampleExample

                                    2929

                                    73 Line Detection73 Line Detection

                                    Horizontal mask will result with max Horizontal mask will result with max

                                    response when a line passed through the response when a line passed through the

                                    middle row of the mask with a constant middle row of the mask with a constant

                                    backgroundbackground

                                    the similar idea is used with other masksthe similar idea is used with other masks

                                    Note the preferred direction of each mask Note the preferred direction of each mask

                                    is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                    than other possible directionsthan other possible directions

                                    1 1 1 1 1 2 1 2 1 2 1 1

                                    2 2 2 1 2 1 1 2 1 1 2 1

                                    1 1 1 2 1 1 1 2 1 1 1 2

                                    45 45Horizontal Vertical

                                    1 1 1 1 1 2 1 2 1 2 1 1

                                    2 2 2 1 2 1 1 2 1 1 2 1

                                    1 1 1 2 1 1 1 2 1 1 1 2

                                    45 45Horizontal Vertical

                                    3030

                                    Idea 1 of Line DetectionIdea 1 of Line Detection

                                    Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                    izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                    if at a certain point in the imageif at a certain point in the image

                                    |Ri||Ri|gtgt|Rj||Rj|

                                    for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                    3131

                                    Idea 2 of Line DetectionIdea 2 of Line Detection

                                    Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                    After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                    3232

                                    ExampleExample

                                    3333

                                    74 Edge-based 74 Edge-based SegmentationSegmentation

                                    Edge-based segmentations rely on edges found in an image by edge detecting operators

                                    these edges mark image locations of discontinuities in gray level

                                    Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                    There are a large group of methods based on information about edges in the image

                                    3434

                                    What is edgeWhat is edge

                                    Edge is where change occurs Change is measured by derivative in 1D

                                    ―Biggest change derivative has maximum magnitude

                                    Or 2nd derivative is zero we discuss approaches for implementing

                                    ―first-order derivative (Gradient operator)

                                    ―second-order derivative (Laplacian operator)

                                    ―we have introduced both derivatives in chapter 3

                                    ―Here we will talk only about their properties for edge detection

                                    3535

                                    What is edgeWhat is edge

                                    In other wordsIn other words an edge is a set of an edge is a set of

                                    connected pixelsconnected pixels

                                    that lie on the boundary between two that lie on the boundary between two

                                    regions with relatively distinct gray-level regions with relatively distinct gray-level

                                    propertiesproperties

                                    Note edge vs boundaryNote edge vs boundary

                                    ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                    ―whereas a region boundary owing to whereas a region boundary owing to

                                    the way it is defined is a more global the way it is defined is a more global

                                    ideaidea

                                    3636

                                    Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                    because of because of

                                    optics optics

                                    sampling sampling

                                    image image

                                    acquisition acquisition

                                    imperfectionimperfection

                                    3737

                                    Thick and Thin EdgeThick and Thin Edge

                                    The slope of the ramp is inversely The slope of the ramp is inversely

                                    proportional to the degree of blurring in the proportional to the degree of blurring in the

                                    edgeedge

                                    Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                    pathpath

                                    Instead an edge point now is any point Instead an edge point now is any point

                                    contained in the ramp and contained in the ramp and an edge would an edge would

                                    then be a set of such points that are then be a set of such points that are

                                    connectedconnected

                                    The thickness is determined by the length of the The thickness is determined by the length of the

                                    rampramp

                                    The length is determined by the slope which is in The length is determined by the slope which is in

                                    turn determined by the degree of blurringturn determined by the degree of blurring

                                    Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                    edges tend to be thinedges tend to be thin

                                    3838

                                    First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                    the signs of the the signs of the

                                    derivatives would be derivatives would be

                                    reversed for an edge reversed for an edge

                                    that transitions from that transitions from

                                    light to darklight to dark

                                    First First derivatderivatee

                                    SeconSecond d derivatderivatee

                                    Gray-Gray-level level profileprofile

                                    3939

                                    Second derivativesSecond derivatives

                                    an undesirable featurean undesirable feature

                                    produces 2 values for every edge in an produces 2 values for every edge in an

                                    imageimage

                                    zero-crossing propertyzero-crossing property

                                    an imaginary straight line joining the an imaginary straight line joining the

                                    extreme positive and negative values of extreme positive and negative values of

                                    the second derivative would cross zero the second derivative would cross zero

                                    near the midpoint of the edgenear the midpoint of the edge

                                    quite useful for locating the centers of quite useful for locating the centers of

                                    thick edgesthick edges

                                    4040

                                    Basic idea of edge detectionBasic idea of edge detection

                                    A profile is defined perpendicularly to A profile is defined perpendicularly to

                                    the edge direction and the results are the edge direction and the results are

                                    interpretedinterpreted

                                    The magnitude of the first derivative is The magnitude of the first derivative is

                                    used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                    ramp)ramp)

                                    The sign of the second derivative can The sign of the second derivative can

                                    determine whether an edge pixel is on the determine whether an edge pixel is on the

                                    dark or light side of an edgedark or light side of an edge

                                    4141

                                    Review of First DerivateReview of First Derivate

                                    Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                    Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                    Sobel operators 3Sobel operators 333

                                    6 5 8 5x yG z z G z z

                                    1 2 3

                                    4 5 6

                                    7 8 9

                                    z z z

                                    z z z

                                    z z z

                                    1 2 3

                                    4 5 6

                                    7 8 9

                                    z z z

                                    z z z

                                    z z z

                                    9 5 8 6x yG z z G z z 1 0 0 1

                                    0 1 1 0

                                    1 0 0 1

                                    0 1 1 0

                                    7 8 9 1 2 3

                                    3 6 9 1 4 7

                                    2 2

                                    2 2

                                    x

                                    y

                                    G z z z z z z

                                    G z z z z z z

                                    1 2 1 1 0 1

                                    0 0 0 2 0 2

                                    1 2 1 1 0 1

                                    1 2 1 1 0 1

                                    0 0 0 2 0 2

                                    1 2 1 1 0 1

                                    x yf G G

                                    4242

                                    Edge direction and strengthEdge direction and strength

                                    Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                    α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                    The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                    The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                    2 2x yf G G

                                    4343

                                    Gradient MasksGradient Masks

                                    1 0 0 1

                                    0 1 1 0

                                    Roberts

                                    1 0 0 1

                                    0 1 1 0

                                    Roberts

                                    1 2 1 1 0 1

                                    0 0 0 2 0 2

                                    1 2 1 1 0 1

                                    Sobel

                                    1 2 1 1 0 1

                                    0 0 0 2 0 2

                                    1 2 1 1 0 1

                                    Sobel

                                    1 1 1 1 0 1

                                    0 0 0 1 0 1

                                    1 1 1 1 0 1

                                    Prewitt

                                    1 1 1 1 0 1

                                    0 0 0 1 0 1

                                    1 1 1 1 0 1

                                    Prewitt

                                    4444

                                    Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                    0 1 1 1 1 0

                                    1 0 1 1 0 1

                                    1 1 0 0 1 1

                                    Prewitt

                                    0 1 1 1 1 0

                                    1 0 1 1 0 1

                                    1 1 0 0 1 1

                                    Prewitt

                                    4545

                                    Review of Second DerivateReview of Second Derivate

                                    Laplacian OperatorLaplacian Operator

                                    21 1

                                    1 1 4

                                    f x y f x yf

                                    f x y f x y f x y

                                    0 1 0

                                    1 4 1

                                    0 1 0

                                    0 1 0

                                    1 4 1

                                    0 1 0

                                    LaplacianLaplacian

                                    MaskMask

                                    1 1 1

                                    1 8 1

                                    1 1 1

                                    1 1 1

                                    1 8 1

                                    1 1 1

                                    4646

                                    Example of edge detectionExample of edge detection

                                    See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                    Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                    derivative)derivative)

                                    produces double edgesproduces double edges

                                    unable to detect edge directionunable to detect edge direction

                                    4747

                                    Canny edge detectorCanny edge detector

                                    The most powerful edge-detection The most powerful edge-detection

                                    method method

                                    It differs from the other edge-It differs from the other edge-

                                    detection methods in that detection methods in that

                                    it uses two different thresholds (to detect it uses two different thresholds (to detect

                                    strong and weak edges) strong and weak edges)

                                    and includes the weak edges in the and includes the weak edges in the

                                    output only if they are connected to output only if they are connected to

                                    strong edges strong edges

                                    This method is therefore less likely This method is therefore less likely

                                    than the others to be fooled by than the others to be fooled by

                                    noise and more likely to detect true noise and more likely to detect true

                                    weak edgesweak edges

                                    4848

                                    Laplacian of GaussianLaplacian of Gaussian

                                    Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                    2 2 22

                                    4 2

                                    2 2 2

                                    2exp

                                    r rh

                                    r x y

                                    determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                    4949

                                    Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                    0 0 1 0 0

                                    0 1 2 1 0

                                    1 2 16 2 1

                                    0 1 2 1 0

                                    0 0 1 0 0

                                    0 0 1 0 0

                                    0 1 2 1 0

                                    1 2 16 2 1

                                    0 1 2 1 0

                                    0 0 1 0 0

                                    The coefficient must sum to The coefficient must sum to

                                    zerozero

                                    5050

                                    Edge Detection and Edge Detection and SegmentationSegmentation

                                    Image resulting from edge detection cannot be used as a segmentation result

                                    Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                    5151

                                    75 Region-based 75 Region-based SegmentationSegmentation

                                    GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                    Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                    5252

                                    Two methods of Region Two methods of Region SegmentationSegmentation

                                    Region GrowingRegion Growing

                                    Region SplittingRegion Splitting

                                    Region growing is the opposite of the Region growing is the opposite of the

                                    split and merge approachsplit and merge approach

                                    5353

                                    Region GrowingRegion Growing

                                    The objective of segmentation is to The objective of segmentation is to

                                    partition an image into regionspartition an image into regions

                                    A region is a connected component with A region is a connected component with

                                    some uniformity (say gray-levels or some uniformity (say gray-levels or

                                    texture)texture)

                                    In region growing we start with a set In region growing we start with a set

                                    of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                    appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                    pixels if they have pixels if they have similar propertiessimilar properties

                                    such as specific ranges of gray level such as specific ranges of gray level

                                    and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                    Need initialization Need initialization similarity similarity

                                    criterioncriterion

                                    5454

                                    Steps of Region GrowingSteps of Region Growing

                                    Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                    pixel andpixel and compare it with neighbor compare it with neighbor

                                    ppixelsixels

                                    When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                    is grown from the seed pixel by is grown from the seed pixel by

                                    addingadding neighboneighborr pixel pixelss

                                    When the growth of one region stopsWhen the growth of one region stops

                                    choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                    againagain

                                    5555

                                    Region Region growing growing

                                    An initial set of small An initial set of small

                                    areas are iterativelyareas are iteratively

                                    merged according to merged according to

                                    similarity constraintssimilarity constraints

                                    5656

                                    Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                    X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                    We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                    On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                    Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                    SeedSeed pointspoints

                                    5757

                                    CriterionCriterion

                                    There is a valley at around 190 in the There is a valley at around 190 in the

                                    histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                    to be considered as a part of region to the to be considered as a part of region to the

                                    seed pointseed point

                                    The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                    to at least one pixel in that regionto at least one pixel in that region

                                    Result of region growing and boundaries of Result of region growing and boundaries of

                                    defectsdefects

                                    5858

                                    Region SplittingRegion Splitting

                                    The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                    ption that the entire image is homogeneousption that the entire image is homogeneous

                                    If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                    This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                    Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                    5959

                                    Region SplittingRegion Splitting

                                    DisadvantageDisadvantage

                                    they create regions that may be adjacent they create regions that may be adjacent

                                    and homogeneous but not mergedand homogeneous but not merged

                                    6060

                                    Region Splitting and MergingRegion Splitting and Merging

                                    ProcedureProcedure

                                    11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                    22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                    33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                    6161

                                    Region Splitting and Merging

                                    Quadtree

                                    (四叉树 )

                                    6262

                                    PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                    where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                    mmii is the mean gray level of that region is the mean gray level of that region

                                    σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                    ExampleExample

                                    Original Original

                                    imageimageThresholded imageThresholded image Result of Result of

                                    Splitting and Splitting and

                                    MergingMerging

                                    • Slide 1
                                    • Slide 2
                                    • Slide 3
                                    • Slide 4
                                    • Slide 5
                                    • Slide 6
                                    • Slide 7
                                    • Slide 8
                                    • Slide 9
                                    • Slide 10
                                    • Slide 11
                                    • Slide 12
                                    • Slide 13
                                    • Slide 14
                                    • Slide 15
                                    • Slide 16
                                    • Slide 17
                                    • Slide 18
                                    • Slide 19
                                    • Slide 20
                                    • Slide 21
                                    • Slide 22
                                    • Slide 23
                                    • Slide 24
                                    • Slide 25
                                    • Slide 26
                                    • Slide 27
                                    • Slide 28
                                    • Slide 29
                                    • Slide 30
                                    • Slide 31
                                    • Slide 32
                                    • Slide 33
                                    • Slide 34
                                    • Slide 35
                                    • Slide 36
                                    • Slide 37
                                    • Slide 38
                                    • Slide 39
                                    • Slide 40
                                    • Slide 41
                                    • Slide 42
                                    • Slide 43
                                    • Slide 44
                                    • Slide 45
                                    • Slide 46
                                    • Slide 47
                                    • Slide 48
                                    • Slide 49
                                    • Slide 50
                                    • Slide 51
                                    • Slide 52
                                    • Slide 53
                                    • Slide 54
                                    • Slide 55
                                    • Slide 56
                                    • Slide 57
                                    • Slide 58
                                    • Slide 59
                                    • Slide 60
                                    • Slide 61
                                    • Slide 62

                                      1919

                                      Optimal ThresholdingOptimal Thresholding

                                      Probability of erroneouslyProbability of erroneously

                                      2020

                                      Optimal ThresholdingOptimal Thresholding

                                      Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                                      ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                                      find T which makesfind T which makes

                                      2121

                                      Optimal ThresholdingOptimal Thresholding

                                      Minimum errorMinimum error

                                      Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                                      threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                                      pp22(z) intersect(z) intersect

                                      2222

                                      Optimal ThresholdingOptimal Thresholding

                                      For exampleFor example

                                      Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                                      pp22(z)(z)

                                      where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                                      variance of the Gaussian density of one variance of the Gaussian density of one

                                      objectobject

                                      μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                                      the Gaussian density of the other objectthe Gaussian density of the other object

                                      2323

                                      Optimal ThresholdingOptimal Thresholding

                                      Quadratic equation (Quadratic equation (二次方程二次方程 ))

                                      2424

                                      Problems of ThresholdingProblems of Thresholding

                                      Original imageOriginal image Thresholded imageThresholded image

                                      2525

                                      Problems of ThresholdingProblems of Thresholding

                                      (a)(a) Exact threshold Exact threshold

                                      segmentationsegmentation

                                      (b)(b) Threshold too lowThreshold too low

                                      (c)(c) Threshold too Threshold too

                                      highhigh

                                      2626

                                      72 Point Detection72 Point Detection

                                      a point has been detected at the a point has been detected at the

                                      location on which the mark is location on which the mark is

                                      centered ifcentered if

                                      |R|geT|R|geT

                                      where T is a nonnegative thresholdwhere T is a nonnegative threshold

                                      R is the sum of products of the R is the sum of products of the

                                      coefficients with the gray levels contained coefficients with the gray levels contained

                                      in the region encompassed by the markin the region encompassed by the mark

                                      1 1 1

                                      1 8 1

                                      1 1 1

                                      1 1 1

                                      1 8 1

                                      1 1 1

                                      2727

                                      72 Point Detection72 Point Detection

                                      Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                                      The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                                      1 1 1

                                      1 8 1

                                      1 1 1

                                      1 1 1

                                      1 8 1

                                      1 1 1

                                      0 1 0

                                      1 4 1

                                      0 1 0

                                      0 1 0

                                      1 4 1

                                      0 1 0

                                      2828

                                      ExampleExample

                                      2929

                                      73 Line Detection73 Line Detection

                                      Horizontal mask will result with max Horizontal mask will result with max

                                      response when a line passed through the response when a line passed through the

                                      middle row of the mask with a constant middle row of the mask with a constant

                                      backgroundbackground

                                      the similar idea is used with other masksthe similar idea is used with other masks

                                      Note the preferred direction of each mask Note the preferred direction of each mask

                                      is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                      than other possible directionsthan other possible directions

                                      1 1 1 1 1 2 1 2 1 2 1 1

                                      2 2 2 1 2 1 1 2 1 1 2 1

                                      1 1 1 2 1 1 1 2 1 1 1 2

                                      45 45Horizontal Vertical

                                      1 1 1 1 1 2 1 2 1 2 1 1

                                      2 2 2 1 2 1 1 2 1 1 2 1

                                      1 1 1 2 1 1 1 2 1 1 1 2

                                      45 45Horizontal Vertical

                                      3030

                                      Idea 1 of Line DetectionIdea 1 of Line Detection

                                      Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                      izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                      if at a certain point in the imageif at a certain point in the image

                                      |Ri||Ri|gtgt|Rj||Rj|

                                      for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                      3131

                                      Idea 2 of Line DetectionIdea 2 of Line Detection

                                      Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                      After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                      3232

                                      ExampleExample

                                      3333

                                      74 Edge-based 74 Edge-based SegmentationSegmentation

                                      Edge-based segmentations rely on edges found in an image by edge detecting operators

                                      these edges mark image locations of discontinuities in gray level

                                      Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                      There are a large group of methods based on information about edges in the image

                                      3434

                                      What is edgeWhat is edge

                                      Edge is where change occurs Change is measured by derivative in 1D

                                      ―Biggest change derivative has maximum magnitude

                                      Or 2nd derivative is zero we discuss approaches for implementing

                                      ―first-order derivative (Gradient operator)

                                      ―second-order derivative (Laplacian operator)

                                      ―we have introduced both derivatives in chapter 3

                                      ―Here we will talk only about their properties for edge detection

                                      3535

                                      What is edgeWhat is edge

                                      In other wordsIn other words an edge is a set of an edge is a set of

                                      connected pixelsconnected pixels

                                      that lie on the boundary between two that lie on the boundary between two

                                      regions with relatively distinct gray-level regions with relatively distinct gray-level

                                      propertiesproperties

                                      Note edge vs boundaryNote edge vs boundary

                                      ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                      ―whereas a region boundary owing to whereas a region boundary owing to

                                      the way it is defined is a more global the way it is defined is a more global

                                      ideaidea

                                      3636

                                      Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                      because of because of

                                      optics optics

                                      sampling sampling

                                      image image

                                      acquisition acquisition

                                      imperfectionimperfection

                                      3737

                                      Thick and Thin EdgeThick and Thin Edge

                                      The slope of the ramp is inversely The slope of the ramp is inversely

                                      proportional to the degree of blurring in the proportional to the degree of blurring in the

                                      edgeedge

                                      Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                      pathpath

                                      Instead an edge point now is any point Instead an edge point now is any point

                                      contained in the ramp and contained in the ramp and an edge would an edge would

                                      then be a set of such points that are then be a set of such points that are

                                      connectedconnected

                                      The thickness is determined by the length of the The thickness is determined by the length of the

                                      rampramp

                                      The length is determined by the slope which is in The length is determined by the slope which is in

                                      turn determined by the degree of blurringturn determined by the degree of blurring

                                      Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                      edges tend to be thinedges tend to be thin

                                      3838

                                      First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                      the signs of the the signs of the

                                      derivatives would be derivatives would be

                                      reversed for an edge reversed for an edge

                                      that transitions from that transitions from

                                      light to darklight to dark

                                      First First derivatderivatee

                                      SeconSecond d derivatderivatee

                                      Gray-Gray-level level profileprofile

                                      3939

                                      Second derivativesSecond derivatives

                                      an undesirable featurean undesirable feature

                                      produces 2 values for every edge in an produces 2 values for every edge in an

                                      imageimage

                                      zero-crossing propertyzero-crossing property

                                      an imaginary straight line joining the an imaginary straight line joining the

                                      extreme positive and negative values of extreme positive and negative values of

                                      the second derivative would cross zero the second derivative would cross zero

                                      near the midpoint of the edgenear the midpoint of the edge

                                      quite useful for locating the centers of quite useful for locating the centers of

                                      thick edgesthick edges

                                      4040

                                      Basic idea of edge detectionBasic idea of edge detection

                                      A profile is defined perpendicularly to A profile is defined perpendicularly to

                                      the edge direction and the results are the edge direction and the results are

                                      interpretedinterpreted

                                      The magnitude of the first derivative is The magnitude of the first derivative is

                                      used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                      ramp)ramp)

                                      The sign of the second derivative can The sign of the second derivative can

                                      determine whether an edge pixel is on the determine whether an edge pixel is on the

                                      dark or light side of an edgedark or light side of an edge

                                      4141

                                      Review of First DerivateReview of First Derivate

                                      Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                      Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                      Sobel operators 3Sobel operators 333

                                      6 5 8 5x yG z z G z z

                                      1 2 3

                                      4 5 6

                                      7 8 9

                                      z z z

                                      z z z

                                      z z z

                                      1 2 3

                                      4 5 6

                                      7 8 9

                                      z z z

                                      z z z

                                      z z z

                                      9 5 8 6x yG z z G z z 1 0 0 1

                                      0 1 1 0

                                      1 0 0 1

                                      0 1 1 0

                                      7 8 9 1 2 3

                                      3 6 9 1 4 7

                                      2 2

                                      2 2

                                      x

                                      y

                                      G z z z z z z

                                      G z z z z z z

                                      1 2 1 1 0 1

                                      0 0 0 2 0 2

                                      1 2 1 1 0 1

                                      1 2 1 1 0 1

                                      0 0 0 2 0 2

                                      1 2 1 1 0 1

                                      x yf G G

                                      4242

                                      Edge direction and strengthEdge direction and strength

                                      Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                      α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                      The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                      The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                      2 2x yf G G

                                      4343

                                      Gradient MasksGradient Masks

                                      1 0 0 1

                                      0 1 1 0

                                      Roberts

                                      1 0 0 1

                                      0 1 1 0

                                      Roberts

                                      1 2 1 1 0 1

                                      0 0 0 2 0 2

                                      1 2 1 1 0 1

                                      Sobel

                                      1 2 1 1 0 1

                                      0 0 0 2 0 2

                                      1 2 1 1 0 1

                                      Sobel

                                      1 1 1 1 0 1

                                      0 0 0 1 0 1

                                      1 1 1 1 0 1

                                      Prewitt

                                      1 1 1 1 0 1

                                      0 0 0 1 0 1

                                      1 1 1 1 0 1

                                      Prewitt

                                      4444

                                      Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                      0 1 1 1 1 0

                                      1 0 1 1 0 1

                                      1 1 0 0 1 1

                                      Prewitt

                                      0 1 1 1 1 0

                                      1 0 1 1 0 1

                                      1 1 0 0 1 1

                                      Prewitt

                                      4545

                                      Review of Second DerivateReview of Second Derivate

                                      Laplacian OperatorLaplacian Operator

                                      21 1

                                      1 1 4

                                      f x y f x yf

                                      f x y f x y f x y

                                      0 1 0

                                      1 4 1

                                      0 1 0

                                      0 1 0

                                      1 4 1

                                      0 1 0

                                      LaplacianLaplacian

                                      MaskMask

                                      1 1 1

                                      1 8 1

                                      1 1 1

                                      1 1 1

                                      1 8 1

                                      1 1 1

                                      4646

                                      Example of edge detectionExample of edge detection

                                      See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                      Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                      derivative)derivative)

                                      produces double edgesproduces double edges

                                      unable to detect edge directionunable to detect edge direction

                                      4747

                                      Canny edge detectorCanny edge detector

                                      The most powerful edge-detection The most powerful edge-detection

                                      method method

                                      It differs from the other edge-It differs from the other edge-

                                      detection methods in that detection methods in that

                                      it uses two different thresholds (to detect it uses two different thresholds (to detect

                                      strong and weak edges) strong and weak edges)

                                      and includes the weak edges in the and includes the weak edges in the

                                      output only if they are connected to output only if they are connected to

                                      strong edges strong edges

                                      This method is therefore less likely This method is therefore less likely

                                      than the others to be fooled by than the others to be fooled by

                                      noise and more likely to detect true noise and more likely to detect true

                                      weak edgesweak edges

                                      4848

                                      Laplacian of GaussianLaplacian of Gaussian

                                      Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                      2 2 22

                                      4 2

                                      2 2 2

                                      2exp

                                      r rh

                                      r x y

                                      determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                      4949

                                      Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                      0 0 1 0 0

                                      0 1 2 1 0

                                      1 2 16 2 1

                                      0 1 2 1 0

                                      0 0 1 0 0

                                      0 0 1 0 0

                                      0 1 2 1 0

                                      1 2 16 2 1

                                      0 1 2 1 0

                                      0 0 1 0 0

                                      The coefficient must sum to The coefficient must sum to

                                      zerozero

                                      5050

                                      Edge Detection and Edge Detection and SegmentationSegmentation

                                      Image resulting from edge detection cannot be used as a segmentation result

                                      Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                      5151

                                      75 Region-based 75 Region-based SegmentationSegmentation

                                      GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                      Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                      5252

                                      Two methods of Region Two methods of Region SegmentationSegmentation

                                      Region GrowingRegion Growing

                                      Region SplittingRegion Splitting

                                      Region growing is the opposite of the Region growing is the opposite of the

                                      split and merge approachsplit and merge approach

                                      5353

                                      Region GrowingRegion Growing

                                      The objective of segmentation is to The objective of segmentation is to

                                      partition an image into regionspartition an image into regions

                                      A region is a connected component with A region is a connected component with

                                      some uniformity (say gray-levels or some uniformity (say gray-levels or

                                      texture)texture)

                                      In region growing we start with a set In region growing we start with a set

                                      of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                      appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                      pixels if they have pixels if they have similar propertiessimilar properties

                                      such as specific ranges of gray level such as specific ranges of gray level

                                      and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                      Need initialization Need initialization similarity similarity

                                      criterioncriterion

                                      5454

                                      Steps of Region GrowingSteps of Region Growing

                                      Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                      pixel andpixel and compare it with neighbor compare it with neighbor

                                      ppixelsixels

                                      When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                      is grown from the seed pixel by is grown from the seed pixel by

                                      addingadding neighboneighborr pixel pixelss

                                      When the growth of one region stopsWhen the growth of one region stops

                                      choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                      againagain

                                      5555

                                      Region Region growing growing

                                      An initial set of small An initial set of small

                                      areas are iterativelyareas are iteratively

                                      merged according to merged according to

                                      similarity constraintssimilarity constraints

                                      5656

                                      Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                      X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                      We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                      On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                      Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                      SeedSeed pointspoints

                                      5757

                                      CriterionCriterion

                                      There is a valley at around 190 in the There is a valley at around 190 in the

                                      histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                      to be considered as a part of region to the to be considered as a part of region to the

                                      seed pointseed point

                                      The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                      to at least one pixel in that regionto at least one pixel in that region

                                      Result of region growing and boundaries of Result of region growing and boundaries of

                                      defectsdefects

                                      5858

                                      Region SplittingRegion Splitting

                                      The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                      ption that the entire image is homogeneousption that the entire image is homogeneous

                                      If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                      This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                      Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                      5959

                                      Region SplittingRegion Splitting

                                      DisadvantageDisadvantage

                                      they create regions that may be adjacent they create regions that may be adjacent

                                      and homogeneous but not mergedand homogeneous but not merged

                                      6060

                                      Region Splitting and MergingRegion Splitting and Merging

                                      ProcedureProcedure

                                      11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                      22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                      33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                      6161

                                      Region Splitting and Merging

                                      Quadtree

                                      (四叉树 )

                                      6262

                                      PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                      where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                      mmii is the mean gray level of that region is the mean gray level of that region

                                      σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                      ExampleExample

                                      Original Original

                                      imageimageThresholded imageThresholded image Result of Result of

                                      Splitting and Splitting and

                                      MergingMerging

                                      • Slide 1
                                      • Slide 2
                                      • Slide 3
                                      • Slide 4
                                      • Slide 5
                                      • Slide 6
                                      • Slide 7
                                      • Slide 8
                                      • Slide 9
                                      • Slide 10
                                      • Slide 11
                                      • Slide 12
                                      • Slide 13
                                      • Slide 14
                                      • Slide 15
                                      • Slide 16
                                      • Slide 17
                                      • Slide 18
                                      • Slide 19
                                      • Slide 20
                                      • Slide 21
                                      • Slide 22
                                      • Slide 23
                                      • Slide 24
                                      • Slide 25
                                      • Slide 26
                                      • Slide 27
                                      • Slide 28
                                      • Slide 29
                                      • Slide 30
                                      • Slide 31
                                      • Slide 32
                                      • Slide 33
                                      • Slide 34
                                      • Slide 35
                                      • Slide 36
                                      • Slide 37
                                      • Slide 38
                                      • Slide 39
                                      • Slide 40
                                      • Slide 41
                                      • Slide 42
                                      • Slide 43
                                      • Slide 44
                                      • Slide 45
                                      • Slide 46
                                      • Slide 47
                                      • Slide 48
                                      • Slide 49
                                      • Slide 50
                                      • Slide 51
                                      • Slide 52
                                      • Slide 53
                                      • Slide 54
                                      • Slide 55
                                      • Slide 56
                                      • Slide 57
                                      • Slide 58
                                      • Slide 59
                                      • Slide 60
                                      • Slide 61
                                      • Slide 62

                                        2020

                                        Optimal ThresholdingOptimal Thresholding

                                        Minimum errorMinimum error Differentiating ( Differentiating ( 微分微分 ) E(T) with respect to T (usi) E(T) with respect to T (usi

                                        ng Leibnizrsquos rule) and equating the result to 0ng Leibnizrsquos rule) and equating the result to 0

                                        find T which makesfind T which makes

                                        2121

                                        Optimal ThresholdingOptimal Thresholding

                                        Minimum errorMinimum error

                                        Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                                        threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                                        pp22(z) intersect(z) intersect

                                        2222

                                        Optimal ThresholdingOptimal Thresholding

                                        For exampleFor example

                                        Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                                        pp22(z)(z)

                                        where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                                        variance of the Gaussian density of one variance of the Gaussian density of one

                                        objectobject

                                        μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                                        the Gaussian density of the other objectthe Gaussian density of the other object

                                        2323

                                        Optimal ThresholdingOptimal Thresholding

                                        Quadratic equation (Quadratic equation (二次方程二次方程 ))

                                        2424

                                        Problems of ThresholdingProblems of Thresholding

                                        Original imageOriginal image Thresholded imageThresholded image

                                        2525

                                        Problems of ThresholdingProblems of Thresholding

                                        (a)(a) Exact threshold Exact threshold

                                        segmentationsegmentation

                                        (b)(b) Threshold too lowThreshold too low

                                        (c)(c) Threshold too Threshold too

                                        highhigh

                                        2626

                                        72 Point Detection72 Point Detection

                                        a point has been detected at the a point has been detected at the

                                        location on which the mark is location on which the mark is

                                        centered ifcentered if

                                        |R|geT|R|geT

                                        where T is a nonnegative thresholdwhere T is a nonnegative threshold

                                        R is the sum of products of the R is the sum of products of the

                                        coefficients with the gray levels contained coefficients with the gray levels contained

                                        in the region encompassed by the markin the region encompassed by the mark

                                        1 1 1

                                        1 8 1

                                        1 1 1

                                        1 1 1

                                        1 8 1

                                        1 1 1

                                        2727

                                        72 Point Detection72 Point Detection

                                        Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                                        The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                                        1 1 1

                                        1 8 1

                                        1 1 1

                                        1 1 1

                                        1 8 1

                                        1 1 1

                                        0 1 0

                                        1 4 1

                                        0 1 0

                                        0 1 0

                                        1 4 1

                                        0 1 0

                                        2828

                                        ExampleExample

                                        2929

                                        73 Line Detection73 Line Detection

                                        Horizontal mask will result with max Horizontal mask will result with max

                                        response when a line passed through the response when a line passed through the

                                        middle row of the mask with a constant middle row of the mask with a constant

                                        backgroundbackground

                                        the similar idea is used with other masksthe similar idea is used with other masks

                                        Note the preferred direction of each mask Note the preferred direction of each mask

                                        is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                        than other possible directionsthan other possible directions

                                        1 1 1 1 1 2 1 2 1 2 1 1

                                        2 2 2 1 2 1 1 2 1 1 2 1

                                        1 1 1 2 1 1 1 2 1 1 1 2

                                        45 45Horizontal Vertical

                                        1 1 1 1 1 2 1 2 1 2 1 1

                                        2 2 2 1 2 1 1 2 1 1 2 1

                                        1 1 1 2 1 1 1 2 1 1 1 2

                                        45 45Horizontal Vertical

                                        3030

                                        Idea 1 of Line DetectionIdea 1 of Line Detection

                                        Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                        izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                        if at a certain point in the imageif at a certain point in the image

                                        |Ri||Ri|gtgt|Rj||Rj|

                                        for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                        3131

                                        Idea 2 of Line DetectionIdea 2 of Line Detection

                                        Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                        After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                        3232

                                        ExampleExample

                                        3333

                                        74 Edge-based 74 Edge-based SegmentationSegmentation

                                        Edge-based segmentations rely on edges found in an image by edge detecting operators

                                        these edges mark image locations of discontinuities in gray level

                                        Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                        There are a large group of methods based on information about edges in the image

                                        3434

                                        What is edgeWhat is edge

                                        Edge is where change occurs Change is measured by derivative in 1D

                                        ―Biggest change derivative has maximum magnitude

                                        Or 2nd derivative is zero we discuss approaches for implementing

                                        ―first-order derivative (Gradient operator)

                                        ―second-order derivative (Laplacian operator)

                                        ―we have introduced both derivatives in chapter 3

                                        ―Here we will talk only about their properties for edge detection

                                        3535

                                        What is edgeWhat is edge

                                        In other wordsIn other words an edge is a set of an edge is a set of

                                        connected pixelsconnected pixels

                                        that lie on the boundary between two that lie on the boundary between two

                                        regions with relatively distinct gray-level regions with relatively distinct gray-level

                                        propertiesproperties

                                        Note edge vs boundaryNote edge vs boundary

                                        ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                        ―whereas a region boundary owing to whereas a region boundary owing to

                                        the way it is defined is a more global the way it is defined is a more global

                                        ideaidea

                                        3636

                                        Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                        because of because of

                                        optics optics

                                        sampling sampling

                                        image image

                                        acquisition acquisition

                                        imperfectionimperfection

                                        3737

                                        Thick and Thin EdgeThick and Thin Edge

                                        The slope of the ramp is inversely The slope of the ramp is inversely

                                        proportional to the degree of blurring in the proportional to the degree of blurring in the

                                        edgeedge

                                        Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                        pathpath

                                        Instead an edge point now is any point Instead an edge point now is any point

                                        contained in the ramp and contained in the ramp and an edge would an edge would

                                        then be a set of such points that are then be a set of such points that are

                                        connectedconnected

                                        The thickness is determined by the length of the The thickness is determined by the length of the

                                        rampramp

                                        The length is determined by the slope which is in The length is determined by the slope which is in

                                        turn determined by the degree of blurringturn determined by the degree of blurring

                                        Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                        edges tend to be thinedges tend to be thin

                                        3838

                                        First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                        the signs of the the signs of the

                                        derivatives would be derivatives would be

                                        reversed for an edge reversed for an edge

                                        that transitions from that transitions from

                                        light to darklight to dark

                                        First First derivatderivatee

                                        SeconSecond d derivatderivatee

                                        Gray-Gray-level level profileprofile

                                        3939

                                        Second derivativesSecond derivatives

                                        an undesirable featurean undesirable feature

                                        produces 2 values for every edge in an produces 2 values for every edge in an

                                        imageimage

                                        zero-crossing propertyzero-crossing property

                                        an imaginary straight line joining the an imaginary straight line joining the

                                        extreme positive and negative values of extreme positive and negative values of

                                        the second derivative would cross zero the second derivative would cross zero

                                        near the midpoint of the edgenear the midpoint of the edge

                                        quite useful for locating the centers of quite useful for locating the centers of

                                        thick edgesthick edges

                                        4040

                                        Basic idea of edge detectionBasic idea of edge detection

                                        A profile is defined perpendicularly to A profile is defined perpendicularly to

                                        the edge direction and the results are the edge direction and the results are

                                        interpretedinterpreted

                                        The magnitude of the first derivative is The magnitude of the first derivative is

                                        used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                        ramp)ramp)

                                        The sign of the second derivative can The sign of the second derivative can

                                        determine whether an edge pixel is on the determine whether an edge pixel is on the

                                        dark or light side of an edgedark or light side of an edge

                                        4141

                                        Review of First DerivateReview of First Derivate

                                        Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                        Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                        Sobel operators 3Sobel operators 333

                                        6 5 8 5x yG z z G z z

                                        1 2 3

                                        4 5 6

                                        7 8 9

                                        z z z

                                        z z z

                                        z z z

                                        1 2 3

                                        4 5 6

                                        7 8 9

                                        z z z

                                        z z z

                                        z z z

                                        9 5 8 6x yG z z G z z 1 0 0 1

                                        0 1 1 0

                                        1 0 0 1

                                        0 1 1 0

                                        7 8 9 1 2 3

                                        3 6 9 1 4 7

                                        2 2

                                        2 2

                                        x

                                        y

                                        G z z z z z z

                                        G z z z z z z

                                        1 2 1 1 0 1

                                        0 0 0 2 0 2

                                        1 2 1 1 0 1

                                        1 2 1 1 0 1

                                        0 0 0 2 0 2

                                        1 2 1 1 0 1

                                        x yf G G

                                        4242

                                        Edge direction and strengthEdge direction and strength

                                        Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                        α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                        The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                        The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                        2 2x yf G G

                                        4343

                                        Gradient MasksGradient Masks

                                        1 0 0 1

                                        0 1 1 0

                                        Roberts

                                        1 0 0 1

                                        0 1 1 0

                                        Roberts

                                        1 2 1 1 0 1

                                        0 0 0 2 0 2

                                        1 2 1 1 0 1

                                        Sobel

                                        1 2 1 1 0 1

                                        0 0 0 2 0 2

                                        1 2 1 1 0 1

                                        Sobel

                                        1 1 1 1 0 1

                                        0 0 0 1 0 1

                                        1 1 1 1 0 1

                                        Prewitt

                                        1 1 1 1 0 1

                                        0 0 0 1 0 1

                                        1 1 1 1 0 1

                                        Prewitt

                                        4444

                                        Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                        0 1 1 1 1 0

                                        1 0 1 1 0 1

                                        1 1 0 0 1 1

                                        Prewitt

                                        0 1 1 1 1 0

                                        1 0 1 1 0 1

                                        1 1 0 0 1 1

                                        Prewitt

                                        4545

                                        Review of Second DerivateReview of Second Derivate

                                        Laplacian OperatorLaplacian Operator

                                        21 1

                                        1 1 4

                                        f x y f x yf

                                        f x y f x y f x y

                                        0 1 0

                                        1 4 1

                                        0 1 0

                                        0 1 0

                                        1 4 1

                                        0 1 0

                                        LaplacianLaplacian

                                        MaskMask

                                        1 1 1

                                        1 8 1

                                        1 1 1

                                        1 1 1

                                        1 8 1

                                        1 1 1

                                        4646

                                        Example of edge detectionExample of edge detection

                                        See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                        Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                        derivative)derivative)

                                        produces double edgesproduces double edges

                                        unable to detect edge directionunable to detect edge direction

                                        4747

                                        Canny edge detectorCanny edge detector

                                        The most powerful edge-detection The most powerful edge-detection

                                        method method

                                        It differs from the other edge-It differs from the other edge-

                                        detection methods in that detection methods in that

                                        it uses two different thresholds (to detect it uses two different thresholds (to detect

                                        strong and weak edges) strong and weak edges)

                                        and includes the weak edges in the and includes the weak edges in the

                                        output only if they are connected to output only if they are connected to

                                        strong edges strong edges

                                        This method is therefore less likely This method is therefore less likely

                                        than the others to be fooled by than the others to be fooled by

                                        noise and more likely to detect true noise and more likely to detect true

                                        weak edgesweak edges

                                        4848

                                        Laplacian of GaussianLaplacian of Gaussian

                                        Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                        2 2 22

                                        4 2

                                        2 2 2

                                        2exp

                                        r rh

                                        r x y

                                        determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                        4949

                                        Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                        0 0 1 0 0

                                        0 1 2 1 0

                                        1 2 16 2 1

                                        0 1 2 1 0

                                        0 0 1 0 0

                                        0 0 1 0 0

                                        0 1 2 1 0

                                        1 2 16 2 1

                                        0 1 2 1 0

                                        0 0 1 0 0

                                        The coefficient must sum to The coefficient must sum to

                                        zerozero

                                        5050

                                        Edge Detection and Edge Detection and SegmentationSegmentation

                                        Image resulting from edge detection cannot be used as a segmentation result

                                        Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                        5151

                                        75 Region-based 75 Region-based SegmentationSegmentation

                                        GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                        Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                        PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                        For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                        PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                        For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                        5252

                                        Two methods of Region Two methods of Region SegmentationSegmentation

                                        Region GrowingRegion Growing

                                        Region SplittingRegion Splitting

                                        Region growing is the opposite of the Region growing is the opposite of the

                                        split and merge approachsplit and merge approach

                                        5353

                                        Region GrowingRegion Growing

                                        The objective of segmentation is to The objective of segmentation is to

                                        partition an image into regionspartition an image into regions

                                        A region is a connected component with A region is a connected component with

                                        some uniformity (say gray-levels or some uniformity (say gray-levels or

                                        texture)texture)

                                        In region growing we start with a set In region growing we start with a set

                                        of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                        appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                        pixels if they have pixels if they have similar propertiessimilar properties

                                        such as specific ranges of gray level such as specific ranges of gray level

                                        and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                        Need initialization Need initialization similarity similarity

                                        criterioncriterion

                                        5454

                                        Steps of Region GrowingSteps of Region Growing

                                        Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                        pixel andpixel and compare it with neighbor compare it with neighbor

                                        ppixelsixels

                                        When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                        is grown from the seed pixel by is grown from the seed pixel by

                                        addingadding neighboneighborr pixel pixelss

                                        When the growth of one region stopsWhen the growth of one region stops

                                        choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                        againagain

                                        5555

                                        Region Region growing growing

                                        An initial set of small An initial set of small

                                        areas are iterativelyareas are iteratively

                                        merged according to merged according to

                                        similarity constraintssimilarity constraints

                                        5656

                                        Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                        X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                        We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                        On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                        Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                        SeedSeed pointspoints

                                        5757

                                        CriterionCriterion

                                        There is a valley at around 190 in the There is a valley at around 190 in the

                                        histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                        to be considered as a part of region to the to be considered as a part of region to the

                                        seed pointseed point

                                        The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                        to at least one pixel in that regionto at least one pixel in that region

                                        Result of region growing and boundaries of Result of region growing and boundaries of

                                        defectsdefects

                                        5858

                                        Region SplittingRegion Splitting

                                        The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                        ption that the entire image is homogeneousption that the entire image is homogeneous

                                        If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                        This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                        Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                        5959

                                        Region SplittingRegion Splitting

                                        DisadvantageDisadvantage

                                        they create regions that may be adjacent they create regions that may be adjacent

                                        and homogeneous but not mergedand homogeneous but not merged

                                        6060

                                        Region Splitting and MergingRegion Splitting and Merging

                                        ProcedureProcedure

                                        11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                        22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                        33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                        6161

                                        Region Splitting and Merging

                                        Quadtree

                                        (四叉树 )

                                        6262

                                        PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                        where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                        mmii is the mean gray level of that region is the mean gray level of that region

                                        σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                        ExampleExample

                                        Original Original

                                        imageimageThresholded imageThresholded image Result of Result of

                                        Splitting and Splitting and

                                        MergingMerging

                                        • Slide 1
                                        • Slide 2
                                        • Slide 3
                                        • Slide 4
                                        • Slide 5
                                        • Slide 6
                                        • Slide 7
                                        • Slide 8
                                        • Slide 9
                                        • Slide 10
                                        • Slide 11
                                        • Slide 12
                                        • Slide 13
                                        • Slide 14
                                        • Slide 15
                                        • Slide 16
                                        • Slide 17
                                        • Slide 18
                                        • Slide 19
                                        • Slide 20
                                        • Slide 21
                                        • Slide 22
                                        • Slide 23
                                        • Slide 24
                                        • Slide 25
                                        • Slide 26
                                        • Slide 27
                                        • Slide 28
                                        • Slide 29
                                        • Slide 30
                                        • Slide 31
                                        • Slide 32
                                        • Slide 33
                                        • Slide 34
                                        • Slide 35
                                        • Slide 36
                                        • Slide 37
                                        • Slide 38
                                        • Slide 39
                                        • Slide 40
                                        • Slide 41
                                        • Slide 42
                                        • Slide 43
                                        • Slide 44
                                        • Slide 45
                                        • Slide 46
                                        • Slide 47
                                        • Slide 48
                                        • Slide 49
                                        • Slide 50
                                        • Slide 51
                                        • Slide 52
                                        • Slide 53
                                        • Slide 54
                                        • Slide 55
                                        • Slide 56
                                        • Slide 57
                                        • Slide 58
                                        • Slide 59
                                        • Slide 60
                                        • Slide 61
                                        • Slide 62

                                          2121

                                          Optimal ThresholdingOptimal Thresholding

                                          Minimum errorMinimum error

                                          Specially if Specially if PP11 = P = P22 then the optimum then the optimum

                                          threshold is where the curve pthreshold is where the curve p11(z) and (z) and

                                          pp22(z) intersect(z) intersect

                                          2222

                                          Optimal ThresholdingOptimal Thresholding

                                          For exampleFor example

                                          Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                                          pp22(z)(z)

                                          where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                                          variance of the Gaussian density of one variance of the Gaussian density of one

                                          objectobject

                                          μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                                          the Gaussian density of the other objectthe Gaussian density of the other object

                                          2323

                                          Optimal ThresholdingOptimal Thresholding

                                          Quadratic equation (Quadratic equation (二次方程二次方程 ))

                                          2424

                                          Problems of ThresholdingProblems of Thresholding

                                          Original imageOriginal image Thresholded imageThresholded image

                                          2525

                                          Problems of ThresholdingProblems of Thresholding

                                          (a)(a) Exact threshold Exact threshold

                                          segmentationsegmentation

                                          (b)(b) Threshold too lowThreshold too low

                                          (c)(c) Threshold too Threshold too

                                          highhigh

                                          2626

                                          72 Point Detection72 Point Detection

                                          a point has been detected at the a point has been detected at the

                                          location on which the mark is location on which the mark is

                                          centered ifcentered if

                                          |R|geT|R|geT

                                          where T is a nonnegative thresholdwhere T is a nonnegative threshold

                                          R is the sum of products of the R is the sum of products of the

                                          coefficients with the gray levels contained coefficients with the gray levels contained

                                          in the region encompassed by the markin the region encompassed by the mark

                                          1 1 1

                                          1 8 1

                                          1 1 1

                                          1 1 1

                                          1 8 1

                                          1 1 1

                                          2727

                                          72 Point Detection72 Point Detection

                                          Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                                          The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                                          1 1 1

                                          1 8 1

                                          1 1 1

                                          1 1 1

                                          1 8 1

                                          1 1 1

                                          0 1 0

                                          1 4 1

                                          0 1 0

                                          0 1 0

                                          1 4 1

                                          0 1 0

                                          2828

                                          ExampleExample

                                          2929

                                          73 Line Detection73 Line Detection

                                          Horizontal mask will result with max Horizontal mask will result with max

                                          response when a line passed through the response when a line passed through the

                                          middle row of the mask with a constant middle row of the mask with a constant

                                          backgroundbackground

                                          the similar idea is used with other masksthe similar idea is used with other masks

                                          Note the preferred direction of each mask Note the preferred direction of each mask

                                          is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                          than other possible directionsthan other possible directions

                                          1 1 1 1 1 2 1 2 1 2 1 1

                                          2 2 2 1 2 1 1 2 1 1 2 1

                                          1 1 1 2 1 1 1 2 1 1 1 2

                                          45 45Horizontal Vertical

                                          1 1 1 1 1 2 1 2 1 2 1 1

                                          2 2 2 1 2 1 1 2 1 1 2 1

                                          1 1 1 2 1 1 1 2 1 1 1 2

                                          45 45Horizontal Vertical

                                          3030

                                          Idea 1 of Line DetectionIdea 1 of Line Detection

                                          Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                          izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                          if at a certain point in the imageif at a certain point in the image

                                          |Ri||Ri|gtgt|Rj||Rj|

                                          for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                          3131

                                          Idea 2 of Line DetectionIdea 2 of Line Detection

                                          Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                          After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                          3232

                                          ExampleExample

                                          3333

                                          74 Edge-based 74 Edge-based SegmentationSegmentation

                                          Edge-based segmentations rely on edges found in an image by edge detecting operators

                                          these edges mark image locations of discontinuities in gray level

                                          Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                          There are a large group of methods based on information about edges in the image

                                          3434

                                          What is edgeWhat is edge

                                          Edge is where change occurs Change is measured by derivative in 1D

                                          ―Biggest change derivative has maximum magnitude

                                          Or 2nd derivative is zero we discuss approaches for implementing

                                          ―first-order derivative (Gradient operator)

                                          ―second-order derivative (Laplacian operator)

                                          ―we have introduced both derivatives in chapter 3

                                          ―Here we will talk only about their properties for edge detection

                                          3535

                                          What is edgeWhat is edge

                                          In other wordsIn other words an edge is a set of an edge is a set of

                                          connected pixelsconnected pixels

                                          that lie on the boundary between two that lie on the boundary between two

                                          regions with relatively distinct gray-level regions with relatively distinct gray-level

                                          propertiesproperties

                                          Note edge vs boundaryNote edge vs boundary

                                          ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                          ―whereas a region boundary owing to whereas a region boundary owing to

                                          the way it is defined is a more global the way it is defined is a more global

                                          ideaidea

                                          3636

                                          Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                          because of because of

                                          optics optics

                                          sampling sampling

                                          image image

                                          acquisition acquisition

                                          imperfectionimperfection

                                          3737

                                          Thick and Thin EdgeThick and Thin Edge

                                          The slope of the ramp is inversely The slope of the ramp is inversely

                                          proportional to the degree of blurring in the proportional to the degree of blurring in the

                                          edgeedge

                                          Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                          pathpath

                                          Instead an edge point now is any point Instead an edge point now is any point

                                          contained in the ramp and contained in the ramp and an edge would an edge would

                                          then be a set of such points that are then be a set of such points that are

                                          connectedconnected

                                          The thickness is determined by the length of the The thickness is determined by the length of the

                                          rampramp

                                          The length is determined by the slope which is in The length is determined by the slope which is in

                                          turn determined by the degree of blurringturn determined by the degree of blurring

                                          Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                          edges tend to be thinedges tend to be thin

                                          3838

                                          First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                          the signs of the the signs of the

                                          derivatives would be derivatives would be

                                          reversed for an edge reversed for an edge

                                          that transitions from that transitions from

                                          light to darklight to dark

                                          First First derivatderivatee

                                          SeconSecond d derivatderivatee

                                          Gray-Gray-level level profileprofile

                                          3939

                                          Second derivativesSecond derivatives

                                          an undesirable featurean undesirable feature

                                          produces 2 values for every edge in an produces 2 values for every edge in an

                                          imageimage

                                          zero-crossing propertyzero-crossing property

                                          an imaginary straight line joining the an imaginary straight line joining the

                                          extreme positive and negative values of extreme positive and negative values of

                                          the second derivative would cross zero the second derivative would cross zero

                                          near the midpoint of the edgenear the midpoint of the edge

                                          quite useful for locating the centers of quite useful for locating the centers of

                                          thick edgesthick edges

                                          4040

                                          Basic idea of edge detectionBasic idea of edge detection

                                          A profile is defined perpendicularly to A profile is defined perpendicularly to

                                          the edge direction and the results are the edge direction and the results are

                                          interpretedinterpreted

                                          The magnitude of the first derivative is The magnitude of the first derivative is

                                          used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                          ramp)ramp)

                                          The sign of the second derivative can The sign of the second derivative can

                                          determine whether an edge pixel is on the determine whether an edge pixel is on the

                                          dark or light side of an edgedark or light side of an edge

                                          4141

                                          Review of First DerivateReview of First Derivate

                                          Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                          Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                          Sobel operators 3Sobel operators 333

                                          6 5 8 5x yG z z G z z

                                          1 2 3

                                          4 5 6

                                          7 8 9

                                          z z z

                                          z z z

                                          z z z

                                          1 2 3

                                          4 5 6

                                          7 8 9

                                          z z z

                                          z z z

                                          z z z

                                          9 5 8 6x yG z z G z z 1 0 0 1

                                          0 1 1 0

                                          1 0 0 1

                                          0 1 1 0

                                          7 8 9 1 2 3

                                          3 6 9 1 4 7

                                          2 2

                                          2 2

                                          x

                                          y

                                          G z z z z z z

                                          G z z z z z z

                                          1 2 1 1 0 1

                                          0 0 0 2 0 2

                                          1 2 1 1 0 1

                                          1 2 1 1 0 1

                                          0 0 0 2 0 2

                                          1 2 1 1 0 1

                                          x yf G G

                                          4242

                                          Edge direction and strengthEdge direction and strength

                                          Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                          α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                          The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                          The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                          2 2x yf G G

                                          4343

                                          Gradient MasksGradient Masks

                                          1 0 0 1

                                          0 1 1 0

                                          Roberts

                                          1 0 0 1

                                          0 1 1 0

                                          Roberts

                                          1 2 1 1 0 1

                                          0 0 0 2 0 2

                                          1 2 1 1 0 1

                                          Sobel

                                          1 2 1 1 0 1

                                          0 0 0 2 0 2

                                          1 2 1 1 0 1

                                          Sobel

                                          1 1 1 1 0 1

                                          0 0 0 1 0 1

                                          1 1 1 1 0 1

                                          Prewitt

                                          1 1 1 1 0 1

                                          0 0 0 1 0 1

                                          1 1 1 1 0 1

                                          Prewitt

                                          4444

                                          Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                          0 1 1 1 1 0

                                          1 0 1 1 0 1

                                          1 1 0 0 1 1

                                          Prewitt

                                          0 1 1 1 1 0

                                          1 0 1 1 0 1

                                          1 1 0 0 1 1

                                          Prewitt

                                          4545

                                          Review of Second DerivateReview of Second Derivate

                                          Laplacian OperatorLaplacian Operator

                                          21 1

                                          1 1 4

                                          f x y f x yf

                                          f x y f x y f x y

                                          0 1 0

                                          1 4 1

                                          0 1 0

                                          0 1 0

                                          1 4 1

                                          0 1 0

                                          LaplacianLaplacian

                                          MaskMask

                                          1 1 1

                                          1 8 1

                                          1 1 1

                                          1 1 1

                                          1 8 1

                                          1 1 1

                                          4646

                                          Example of edge detectionExample of edge detection

                                          See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                          Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                          derivative)derivative)

                                          produces double edgesproduces double edges

                                          unable to detect edge directionunable to detect edge direction

                                          4747

                                          Canny edge detectorCanny edge detector

                                          The most powerful edge-detection The most powerful edge-detection

                                          method method

                                          It differs from the other edge-It differs from the other edge-

                                          detection methods in that detection methods in that

                                          it uses two different thresholds (to detect it uses two different thresholds (to detect

                                          strong and weak edges) strong and weak edges)

                                          and includes the weak edges in the and includes the weak edges in the

                                          output only if they are connected to output only if they are connected to

                                          strong edges strong edges

                                          This method is therefore less likely This method is therefore less likely

                                          than the others to be fooled by than the others to be fooled by

                                          noise and more likely to detect true noise and more likely to detect true

                                          weak edgesweak edges

                                          4848

                                          Laplacian of GaussianLaplacian of Gaussian

                                          Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                          2 2 22

                                          4 2

                                          2 2 2

                                          2exp

                                          r rh

                                          r x y

                                          determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                          4949

                                          Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                          0 0 1 0 0

                                          0 1 2 1 0

                                          1 2 16 2 1

                                          0 1 2 1 0

                                          0 0 1 0 0

                                          0 0 1 0 0

                                          0 1 2 1 0

                                          1 2 16 2 1

                                          0 1 2 1 0

                                          0 0 1 0 0

                                          The coefficient must sum to The coefficient must sum to

                                          zerozero

                                          5050

                                          Edge Detection and Edge Detection and SegmentationSegmentation

                                          Image resulting from edge detection cannot be used as a segmentation result

                                          Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                          5151

                                          75 Region-based 75 Region-based SegmentationSegmentation

                                          GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                          Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                          PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                          For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                          PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                          For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                          5252

                                          Two methods of Region Two methods of Region SegmentationSegmentation

                                          Region GrowingRegion Growing

                                          Region SplittingRegion Splitting

                                          Region growing is the opposite of the Region growing is the opposite of the

                                          split and merge approachsplit and merge approach

                                          5353

                                          Region GrowingRegion Growing

                                          The objective of segmentation is to The objective of segmentation is to

                                          partition an image into regionspartition an image into regions

                                          A region is a connected component with A region is a connected component with

                                          some uniformity (say gray-levels or some uniformity (say gray-levels or

                                          texture)texture)

                                          In region growing we start with a set In region growing we start with a set

                                          of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                          appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                          pixels if they have pixels if they have similar propertiessimilar properties

                                          such as specific ranges of gray level such as specific ranges of gray level

                                          and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                          Need initialization Need initialization similarity similarity

                                          criterioncriterion

                                          5454

                                          Steps of Region GrowingSteps of Region Growing

                                          Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                          pixel andpixel and compare it with neighbor compare it with neighbor

                                          ppixelsixels

                                          When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                          is grown from the seed pixel by is grown from the seed pixel by

                                          addingadding neighboneighborr pixel pixelss

                                          When the growth of one region stopsWhen the growth of one region stops

                                          choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                          againagain

                                          5555

                                          Region Region growing growing

                                          An initial set of small An initial set of small

                                          areas are iterativelyareas are iteratively

                                          merged according to merged according to

                                          similarity constraintssimilarity constraints

                                          5656

                                          Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                          X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                          We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                          On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                          Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                          SeedSeed pointspoints

                                          5757

                                          CriterionCriterion

                                          There is a valley at around 190 in the There is a valley at around 190 in the

                                          histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                          to be considered as a part of region to the to be considered as a part of region to the

                                          seed pointseed point

                                          The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                          to at least one pixel in that regionto at least one pixel in that region

                                          Result of region growing and boundaries of Result of region growing and boundaries of

                                          defectsdefects

                                          5858

                                          Region SplittingRegion Splitting

                                          The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                          ption that the entire image is homogeneousption that the entire image is homogeneous

                                          If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                          This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                          Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                          5959

                                          Region SplittingRegion Splitting

                                          DisadvantageDisadvantage

                                          they create regions that may be adjacent they create regions that may be adjacent

                                          and homogeneous but not mergedand homogeneous but not merged

                                          6060

                                          Region Splitting and MergingRegion Splitting and Merging

                                          ProcedureProcedure

                                          11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                          22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                          33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                          6161

                                          Region Splitting and Merging

                                          Quadtree

                                          (四叉树 )

                                          6262

                                          PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                          where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                          mmii is the mean gray level of that region is the mean gray level of that region

                                          σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                          ExampleExample

                                          Original Original

                                          imageimageThresholded imageThresholded image Result of Result of

                                          Splitting and Splitting and

                                          MergingMerging

                                          • Slide 1
                                          • Slide 2
                                          • Slide 3
                                          • Slide 4
                                          • Slide 5
                                          • Slide 6
                                          • Slide 7
                                          • Slide 8
                                          • Slide 9
                                          • Slide 10
                                          • Slide 11
                                          • Slide 12
                                          • Slide 13
                                          • Slide 14
                                          • Slide 15
                                          • Slide 16
                                          • Slide 17
                                          • Slide 18
                                          • Slide 19
                                          • Slide 20
                                          • Slide 21
                                          • Slide 22
                                          • Slide 23
                                          • Slide 24
                                          • Slide 25
                                          • Slide 26
                                          • Slide 27
                                          • Slide 28
                                          • Slide 29
                                          • Slide 30
                                          • Slide 31
                                          • Slide 32
                                          • Slide 33
                                          • Slide 34
                                          • Slide 35
                                          • Slide 36
                                          • Slide 37
                                          • Slide 38
                                          • Slide 39
                                          • Slide 40
                                          • Slide 41
                                          • Slide 42
                                          • Slide 43
                                          • Slide 44
                                          • Slide 45
                                          • Slide 46
                                          • Slide 47
                                          • Slide 48
                                          • Slide 49
                                          • Slide 50
                                          • Slide 51
                                          • Slide 52
                                          • Slide 53
                                          • Slide 54
                                          • Slide 55
                                          • Slide 56
                                          • Slide 57
                                          • Slide 58
                                          • Slide 59
                                          • Slide 60
                                          • Slide 61
                                          • Slide 62

                                            2222

                                            Optimal ThresholdingOptimal Thresholding

                                            For exampleFor example

                                            Let Let PDF=Gaussian densityPDF=Gaussian density p p11(z) and (z) and

                                            pp22(z)(z)

                                            where μwhere μ11 and σ and σ1122 are the mean and are the mean and

                                            variance of the Gaussian density of one variance of the Gaussian density of one

                                            objectobject

                                            μμ22 and σ and σ2222 are the mean and variance of are the mean and variance of

                                            the Gaussian density of the other objectthe Gaussian density of the other object

                                            2323

                                            Optimal ThresholdingOptimal Thresholding

                                            Quadratic equation (Quadratic equation (二次方程二次方程 ))

                                            2424

                                            Problems of ThresholdingProblems of Thresholding

                                            Original imageOriginal image Thresholded imageThresholded image

                                            2525

                                            Problems of ThresholdingProblems of Thresholding

                                            (a)(a) Exact threshold Exact threshold

                                            segmentationsegmentation

                                            (b)(b) Threshold too lowThreshold too low

                                            (c)(c) Threshold too Threshold too

                                            highhigh

                                            2626

                                            72 Point Detection72 Point Detection

                                            a point has been detected at the a point has been detected at the

                                            location on which the mark is location on which the mark is

                                            centered ifcentered if

                                            |R|geT|R|geT

                                            where T is a nonnegative thresholdwhere T is a nonnegative threshold

                                            R is the sum of products of the R is the sum of products of the

                                            coefficients with the gray levels contained coefficients with the gray levels contained

                                            in the region encompassed by the markin the region encompassed by the mark

                                            1 1 1

                                            1 8 1

                                            1 1 1

                                            1 1 1

                                            1 8 1

                                            1 1 1

                                            2727

                                            72 Point Detection72 Point Detection

                                            Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                                            The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                                            1 1 1

                                            1 8 1

                                            1 1 1

                                            1 1 1

                                            1 8 1

                                            1 1 1

                                            0 1 0

                                            1 4 1

                                            0 1 0

                                            0 1 0

                                            1 4 1

                                            0 1 0

                                            2828

                                            ExampleExample

                                            2929

                                            73 Line Detection73 Line Detection

                                            Horizontal mask will result with max Horizontal mask will result with max

                                            response when a line passed through the response when a line passed through the

                                            middle row of the mask with a constant middle row of the mask with a constant

                                            backgroundbackground

                                            the similar idea is used with other masksthe similar idea is used with other masks

                                            Note the preferred direction of each mask Note the preferred direction of each mask

                                            is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                            than other possible directionsthan other possible directions

                                            1 1 1 1 1 2 1 2 1 2 1 1

                                            2 2 2 1 2 1 1 2 1 1 2 1

                                            1 1 1 2 1 1 1 2 1 1 1 2

                                            45 45Horizontal Vertical

                                            1 1 1 1 1 2 1 2 1 2 1 1

                                            2 2 2 1 2 1 1 2 1 1 2 1

                                            1 1 1 2 1 1 1 2 1 1 1 2

                                            45 45Horizontal Vertical

                                            3030

                                            Idea 1 of Line DetectionIdea 1 of Line Detection

                                            Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                            izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                            if at a certain point in the imageif at a certain point in the image

                                            |Ri||Ri|gtgt|Rj||Rj|

                                            for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                            3131

                                            Idea 2 of Line DetectionIdea 2 of Line Detection

                                            Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                            After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                            3232

                                            ExampleExample

                                            3333

                                            74 Edge-based 74 Edge-based SegmentationSegmentation

                                            Edge-based segmentations rely on edges found in an image by edge detecting operators

                                            these edges mark image locations of discontinuities in gray level

                                            Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                            There are a large group of methods based on information about edges in the image

                                            3434

                                            What is edgeWhat is edge

                                            Edge is where change occurs Change is measured by derivative in 1D

                                            ―Biggest change derivative has maximum magnitude

                                            Or 2nd derivative is zero we discuss approaches for implementing

                                            ―first-order derivative (Gradient operator)

                                            ―second-order derivative (Laplacian operator)

                                            ―we have introduced both derivatives in chapter 3

                                            ―Here we will talk only about their properties for edge detection

                                            3535

                                            What is edgeWhat is edge

                                            In other wordsIn other words an edge is a set of an edge is a set of

                                            connected pixelsconnected pixels

                                            that lie on the boundary between two that lie on the boundary between two

                                            regions with relatively distinct gray-level regions with relatively distinct gray-level

                                            propertiesproperties

                                            Note edge vs boundaryNote edge vs boundary

                                            ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                            ―whereas a region boundary owing to whereas a region boundary owing to

                                            the way it is defined is a more global the way it is defined is a more global

                                            ideaidea

                                            3636

                                            Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                            because of because of

                                            optics optics

                                            sampling sampling

                                            image image

                                            acquisition acquisition

                                            imperfectionimperfection

                                            3737

                                            Thick and Thin EdgeThick and Thin Edge

                                            The slope of the ramp is inversely The slope of the ramp is inversely

                                            proportional to the degree of blurring in the proportional to the degree of blurring in the

                                            edgeedge

                                            Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                            pathpath

                                            Instead an edge point now is any point Instead an edge point now is any point

                                            contained in the ramp and contained in the ramp and an edge would an edge would

                                            then be a set of such points that are then be a set of such points that are

                                            connectedconnected

                                            The thickness is determined by the length of the The thickness is determined by the length of the

                                            rampramp

                                            The length is determined by the slope which is in The length is determined by the slope which is in

                                            turn determined by the degree of blurringturn determined by the degree of blurring

                                            Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                            edges tend to be thinedges tend to be thin

                                            3838

                                            First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                            the signs of the the signs of the

                                            derivatives would be derivatives would be

                                            reversed for an edge reversed for an edge

                                            that transitions from that transitions from

                                            light to darklight to dark

                                            First First derivatderivatee

                                            SeconSecond d derivatderivatee

                                            Gray-Gray-level level profileprofile

                                            3939

                                            Second derivativesSecond derivatives

                                            an undesirable featurean undesirable feature

                                            produces 2 values for every edge in an produces 2 values for every edge in an

                                            imageimage

                                            zero-crossing propertyzero-crossing property

                                            an imaginary straight line joining the an imaginary straight line joining the

                                            extreme positive and negative values of extreme positive and negative values of

                                            the second derivative would cross zero the second derivative would cross zero

                                            near the midpoint of the edgenear the midpoint of the edge

                                            quite useful for locating the centers of quite useful for locating the centers of

                                            thick edgesthick edges

                                            4040

                                            Basic idea of edge detectionBasic idea of edge detection

                                            A profile is defined perpendicularly to A profile is defined perpendicularly to

                                            the edge direction and the results are the edge direction and the results are

                                            interpretedinterpreted

                                            The magnitude of the first derivative is The magnitude of the first derivative is

                                            used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                            ramp)ramp)

                                            The sign of the second derivative can The sign of the second derivative can

                                            determine whether an edge pixel is on the determine whether an edge pixel is on the

                                            dark or light side of an edgedark or light side of an edge

                                            4141

                                            Review of First DerivateReview of First Derivate

                                            Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                            Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                            Sobel operators 3Sobel operators 333

                                            6 5 8 5x yG z z G z z

                                            1 2 3

                                            4 5 6

                                            7 8 9

                                            z z z

                                            z z z

                                            z z z

                                            1 2 3

                                            4 5 6

                                            7 8 9

                                            z z z

                                            z z z

                                            z z z

                                            9 5 8 6x yG z z G z z 1 0 0 1

                                            0 1 1 0

                                            1 0 0 1

                                            0 1 1 0

                                            7 8 9 1 2 3

                                            3 6 9 1 4 7

                                            2 2

                                            2 2

                                            x

                                            y

                                            G z z z z z z

                                            G z z z z z z

                                            1 2 1 1 0 1

                                            0 0 0 2 0 2

                                            1 2 1 1 0 1

                                            1 2 1 1 0 1

                                            0 0 0 2 0 2

                                            1 2 1 1 0 1

                                            x yf G G

                                            4242

                                            Edge direction and strengthEdge direction and strength

                                            Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                            α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                            The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                            The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                            2 2x yf G G

                                            4343

                                            Gradient MasksGradient Masks

                                            1 0 0 1

                                            0 1 1 0

                                            Roberts

                                            1 0 0 1

                                            0 1 1 0

                                            Roberts

                                            1 2 1 1 0 1

                                            0 0 0 2 0 2

                                            1 2 1 1 0 1

                                            Sobel

                                            1 2 1 1 0 1

                                            0 0 0 2 0 2

                                            1 2 1 1 0 1

                                            Sobel

                                            1 1 1 1 0 1

                                            0 0 0 1 0 1

                                            1 1 1 1 0 1

                                            Prewitt

                                            1 1 1 1 0 1

                                            0 0 0 1 0 1

                                            1 1 1 1 0 1

                                            Prewitt

                                            4444

                                            Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                            0 1 1 1 1 0

                                            1 0 1 1 0 1

                                            1 1 0 0 1 1

                                            Prewitt

                                            0 1 1 1 1 0

                                            1 0 1 1 0 1

                                            1 1 0 0 1 1

                                            Prewitt

                                            4545

                                            Review of Second DerivateReview of Second Derivate

                                            Laplacian OperatorLaplacian Operator

                                            21 1

                                            1 1 4

                                            f x y f x yf

                                            f x y f x y f x y

                                            0 1 0

                                            1 4 1

                                            0 1 0

                                            0 1 0

                                            1 4 1

                                            0 1 0

                                            LaplacianLaplacian

                                            MaskMask

                                            1 1 1

                                            1 8 1

                                            1 1 1

                                            1 1 1

                                            1 8 1

                                            1 1 1

                                            4646

                                            Example of edge detectionExample of edge detection

                                            See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                            Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                            derivative)derivative)

                                            produces double edgesproduces double edges

                                            unable to detect edge directionunable to detect edge direction

                                            4747

                                            Canny edge detectorCanny edge detector

                                            The most powerful edge-detection The most powerful edge-detection

                                            method method

                                            It differs from the other edge-It differs from the other edge-

                                            detection methods in that detection methods in that

                                            it uses two different thresholds (to detect it uses two different thresholds (to detect

                                            strong and weak edges) strong and weak edges)

                                            and includes the weak edges in the and includes the weak edges in the

                                            output only if they are connected to output only if they are connected to

                                            strong edges strong edges

                                            This method is therefore less likely This method is therefore less likely

                                            than the others to be fooled by than the others to be fooled by

                                            noise and more likely to detect true noise and more likely to detect true

                                            weak edgesweak edges

                                            4848

                                            Laplacian of GaussianLaplacian of Gaussian

                                            Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                            2 2 22

                                            4 2

                                            2 2 2

                                            2exp

                                            r rh

                                            r x y

                                            determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                            4949

                                            Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                            0 0 1 0 0

                                            0 1 2 1 0

                                            1 2 16 2 1

                                            0 1 2 1 0

                                            0 0 1 0 0

                                            0 0 1 0 0

                                            0 1 2 1 0

                                            1 2 16 2 1

                                            0 1 2 1 0

                                            0 0 1 0 0

                                            The coefficient must sum to The coefficient must sum to

                                            zerozero

                                            5050

                                            Edge Detection and Edge Detection and SegmentationSegmentation

                                            Image resulting from edge detection cannot be used as a segmentation result

                                            Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                            5151

                                            75 Region-based 75 Region-based SegmentationSegmentation

                                            GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                            Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                            PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                            For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                            PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                            For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                            5252

                                            Two methods of Region Two methods of Region SegmentationSegmentation

                                            Region GrowingRegion Growing

                                            Region SplittingRegion Splitting

                                            Region growing is the opposite of the Region growing is the opposite of the

                                            split and merge approachsplit and merge approach

                                            5353

                                            Region GrowingRegion Growing

                                            The objective of segmentation is to The objective of segmentation is to

                                            partition an image into regionspartition an image into regions

                                            A region is a connected component with A region is a connected component with

                                            some uniformity (say gray-levels or some uniformity (say gray-levels or

                                            texture)texture)

                                            In region growing we start with a set In region growing we start with a set

                                            of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                            appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                            pixels if they have pixels if they have similar propertiessimilar properties

                                            such as specific ranges of gray level such as specific ranges of gray level

                                            and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                            Need initialization Need initialization similarity similarity

                                            criterioncriterion

                                            5454

                                            Steps of Region GrowingSteps of Region Growing

                                            Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                            pixel andpixel and compare it with neighbor compare it with neighbor

                                            ppixelsixels

                                            When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                            is grown from the seed pixel by is grown from the seed pixel by

                                            addingadding neighboneighborr pixel pixelss

                                            When the growth of one region stopsWhen the growth of one region stops

                                            choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                            againagain

                                            5555

                                            Region Region growing growing

                                            An initial set of small An initial set of small

                                            areas are iterativelyareas are iteratively

                                            merged according to merged according to

                                            similarity constraintssimilarity constraints

                                            5656

                                            Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                            X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                            We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                            On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                            Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                            SeedSeed pointspoints

                                            5757

                                            CriterionCriterion

                                            There is a valley at around 190 in the There is a valley at around 190 in the

                                            histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                            to be considered as a part of region to the to be considered as a part of region to the

                                            seed pointseed point

                                            The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                            to at least one pixel in that regionto at least one pixel in that region

                                            Result of region growing and boundaries of Result of region growing and boundaries of

                                            defectsdefects

                                            5858

                                            Region SplittingRegion Splitting

                                            The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                            ption that the entire image is homogeneousption that the entire image is homogeneous

                                            If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                            This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                            Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                            5959

                                            Region SplittingRegion Splitting

                                            DisadvantageDisadvantage

                                            they create regions that may be adjacent they create regions that may be adjacent

                                            and homogeneous but not mergedand homogeneous but not merged

                                            6060

                                            Region Splitting and MergingRegion Splitting and Merging

                                            ProcedureProcedure

                                            11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                            22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                            33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                            6161

                                            Region Splitting and Merging

                                            Quadtree

                                            (四叉树 )

                                            6262

                                            PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                            where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                            mmii is the mean gray level of that region is the mean gray level of that region

                                            σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                            ExampleExample

                                            Original Original

                                            imageimageThresholded imageThresholded image Result of Result of

                                            Splitting and Splitting and

                                            MergingMerging

                                            • Slide 1
                                            • Slide 2
                                            • Slide 3
                                            • Slide 4
                                            • Slide 5
                                            • Slide 6
                                            • Slide 7
                                            • Slide 8
                                            • Slide 9
                                            • Slide 10
                                            • Slide 11
                                            • Slide 12
                                            • Slide 13
                                            • Slide 14
                                            • Slide 15
                                            • Slide 16
                                            • Slide 17
                                            • Slide 18
                                            • Slide 19
                                            • Slide 20
                                            • Slide 21
                                            • Slide 22
                                            • Slide 23
                                            • Slide 24
                                            • Slide 25
                                            • Slide 26
                                            • Slide 27
                                            • Slide 28
                                            • Slide 29
                                            • Slide 30
                                            • Slide 31
                                            • Slide 32
                                            • Slide 33
                                            • Slide 34
                                            • Slide 35
                                            • Slide 36
                                            • Slide 37
                                            • Slide 38
                                            • Slide 39
                                            • Slide 40
                                            • Slide 41
                                            • Slide 42
                                            • Slide 43
                                            • Slide 44
                                            • Slide 45
                                            • Slide 46
                                            • Slide 47
                                            • Slide 48
                                            • Slide 49
                                            • Slide 50
                                            • Slide 51
                                            • Slide 52
                                            • Slide 53
                                            • Slide 54
                                            • Slide 55
                                            • Slide 56
                                            • Slide 57
                                            • Slide 58
                                            • Slide 59
                                            • Slide 60
                                            • Slide 61
                                            • Slide 62

                                              2323

                                              Optimal ThresholdingOptimal Thresholding

                                              Quadratic equation (Quadratic equation (二次方程二次方程 ))

                                              2424

                                              Problems of ThresholdingProblems of Thresholding

                                              Original imageOriginal image Thresholded imageThresholded image

                                              2525

                                              Problems of ThresholdingProblems of Thresholding

                                              (a)(a) Exact threshold Exact threshold

                                              segmentationsegmentation

                                              (b)(b) Threshold too lowThreshold too low

                                              (c)(c) Threshold too Threshold too

                                              highhigh

                                              2626

                                              72 Point Detection72 Point Detection

                                              a point has been detected at the a point has been detected at the

                                              location on which the mark is location on which the mark is

                                              centered ifcentered if

                                              |R|geT|R|geT

                                              where T is a nonnegative thresholdwhere T is a nonnegative threshold

                                              R is the sum of products of the R is the sum of products of the

                                              coefficients with the gray levels contained coefficients with the gray levels contained

                                              in the region encompassed by the markin the region encompassed by the mark

                                              1 1 1

                                              1 8 1

                                              1 1 1

                                              1 1 1

                                              1 8 1

                                              1 1 1

                                              2727

                                              72 Point Detection72 Point Detection

                                              Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                                              The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                                              1 1 1

                                              1 8 1

                                              1 1 1

                                              1 1 1

                                              1 8 1

                                              1 1 1

                                              0 1 0

                                              1 4 1

                                              0 1 0

                                              0 1 0

                                              1 4 1

                                              0 1 0

                                              2828

                                              ExampleExample

                                              2929

                                              73 Line Detection73 Line Detection

                                              Horizontal mask will result with max Horizontal mask will result with max

                                              response when a line passed through the response when a line passed through the

                                              middle row of the mask with a constant middle row of the mask with a constant

                                              backgroundbackground

                                              the similar idea is used with other masksthe similar idea is used with other masks

                                              Note the preferred direction of each mask Note the preferred direction of each mask

                                              is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                              than other possible directionsthan other possible directions

                                              1 1 1 1 1 2 1 2 1 2 1 1

                                              2 2 2 1 2 1 1 2 1 1 2 1

                                              1 1 1 2 1 1 1 2 1 1 1 2

                                              45 45Horizontal Vertical

                                              1 1 1 1 1 2 1 2 1 2 1 1

                                              2 2 2 1 2 1 1 2 1 1 2 1

                                              1 1 1 2 1 1 1 2 1 1 1 2

                                              45 45Horizontal Vertical

                                              3030

                                              Idea 1 of Line DetectionIdea 1 of Line Detection

                                              Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                              izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                              if at a certain point in the imageif at a certain point in the image

                                              |Ri||Ri|gtgt|Rj||Rj|

                                              for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                              3131

                                              Idea 2 of Line DetectionIdea 2 of Line Detection

                                              Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                              After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                              3232

                                              ExampleExample

                                              3333

                                              74 Edge-based 74 Edge-based SegmentationSegmentation

                                              Edge-based segmentations rely on edges found in an image by edge detecting operators

                                              these edges mark image locations of discontinuities in gray level

                                              Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                              There are a large group of methods based on information about edges in the image

                                              3434

                                              What is edgeWhat is edge

                                              Edge is where change occurs Change is measured by derivative in 1D

                                              ―Biggest change derivative has maximum magnitude

                                              Or 2nd derivative is zero we discuss approaches for implementing

                                              ―first-order derivative (Gradient operator)

                                              ―second-order derivative (Laplacian operator)

                                              ―we have introduced both derivatives in chapter 3

                                              ―Here we will talk only about their properties for edge detection

                                              3535

                                              What is edgeWhat is edge

                                              In other wordsIn other words an edge is a set of an edge is a set of

                                              connected pixelsconnected pixels

                                              that lie on the boundary between two that lie on the boundary between two

                                              regions with relatively distinct gray-level regions with relatively distinct gray-level

                                              propertiesproperties

                                              Note edge vs boundaryNote edge vs boundary

                                              ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                              ―whereas a region boundary owing to whereas a region boundary owing to

                                              the way it is defined is a more global the way it is defined is a more global

                                              ideaidea

                                              3636

                                              Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                              because of because of

                                              optics optics

                                              sampling sampling

                                              image image

                                              acquisition acquisition

                                              imperfectionimperfection

                                              3737

                                              Thick and Thin EdgeThick and Thin Edge

                                              The slope of the ramp is inversely The slope of the ramp is inversely

                                              proportional to the degree of blurring in the proportional to the degree of blurring in the

                                              edgeedge

                                              Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                              pathpath

                                              Instead an edge point now is any point Instead an edge point now is any point

                                              contained in the ramp and contained in the ramp and an edge would an edge would

                                              then be a set of such points that are then be a set of such points that are

                                              connectedconnected

                                              The thickness is determined by the length of the The thickness is determined by the length of the

                                              rampramp

                                              The length is determined by the slope which is in The length is determined by the slope which is in

                                              turn determined by the degree of blurringturn determined by the degree of blurring

                                              Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                              edges tend to be thinedges tend to be thin

                                              3838

                                              First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                              the signs of the the signs of the

                                              derivatives would be derivatives would be

                                              reversed for an edge reversed for an edge

                                              that transitions from that transitions from

                                              light to darklight to dark

                                              First First derivatderivatee

                                              SeconSecond d derivatderivatee

                                              Gray-Gray-level level profileprofile

                                              3939

                                              Second derivativesSecond derivatives

                                              an undesirable featurean undesirable feature

                                              produces 2 values for every edge in an produces 2 values for every edge in an

                                              imageimage

                                              zero-crossing propertyzero-crossing property

                                              an imaginary straight line joining the an imaginary straight line joining the

                                              extreme positive and negative values of extreme positive and negative values of

                                              the second derivative would cross zero the second derivative would cross zero

                                              near the midpoint of the edgenear the midpoint of the edge

                                              quite useful for locating the centers of quite useful for locating the centers of

                                              thick edgesthick edges

                                              4040

                                              Basic idea of edge detectionBasic idea of edge detection

                                              A profile is defined perpendicularly to A profile is defined perpendicularly to

                                              the edge direction and the results are the edge direction and the results are

                                              interpretedinterpreted

                                              The magnitude of the first derivative is The magnitude of the first derivative is

                                              used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                              ramp)ramp)

                                              The sign of the second derivative can The sign of the second derivative can

                                              determine whether an edge pixel is on the determine whether an edge pixel is on the

                                              dark or light side of an edgedark or light side of an edge

                                              4141

                                              Review of First DerivateReview of First Derivate

                                              Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                              Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                              Sobel operators 3Sobel operators 333

                                              6 5 8 5x yG z z G z z

                                              1 2 3

                                              4 5 6

                                              7 8 9

                                              z z z

                                              z z z

                                              z z z

                                              1 2 3

                                              4 5 6

                                              7 8 9

                                              z z z

                                              z z z

                                              z z z

                                              9 5 8 6x yG z z G z z 1 0 0 1

                                              0 1 1 0

                                              1 0 0 1

                                              0 1 1 0

                                              7 8 9 1 2 3

                                              3 6 9 1 4 7

                                              2 2

                                              2 2

                                              x

                                              y

                                              G z z z z z z

                                              G z z z z z z

                                              1 2 1 1 0 1

                                              0 0 0 2 0 2

                                              1 2 1 1 0 1

                                              1 2 1 1 0 1

                                              0 0 0 2 0 2

                                              1 2 1 1 0 1

                                              x yf G G

                                              4242

                                              Edge direction and strengthEdge direction and strength

                                              Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                              α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                              The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                              The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                              2 2x yf G G

                                              4343

                                              Gradient MasksGradient Masks

                                              1 0 0 1

                                              0 1 1 0

                                              Roberts

                                              1 0 0 1

                                              0 1 1 0

                                              Roberts

                                              1 2 1 1 0 1

                                              0 0 0 2 0 2

                                              1 2 1 1 0 1

                                              Sobel

                                              1 2 1 1 0 1

                                              0 0 0 2 0 2

                                              1 2 1 1 0 1

                                              Sobel

                                              1 1 1 1 0 1

                                              0 0 0 1 0 1

                                              1 1 1 1 0 1

                                              Prewitt

                                              1 1 1 1 0 1

                                              0 0 0 1 0 1

                                              1 1 1 1 0 1

                                              Prewitt

                                              4444

                                              Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                              0 1 1 1 1 0

                                              1 0 1 1 0 1

                                              1 1 0 0 1 1

                                              Prewitt

                                              0 1 1 1 1 0

                                              1 0 1 1 0 1

                                              1 1 0 0 1 1

                                              Prewitt

                                              4545

                                              Review of Second DerivateReview of Second Derivate

                                              Laplacian OperatorLaplacian Operator

                                              21 1

                                              1 1 4

                                              f x y f x yf

                                              f x y f x y f x y

                                              0 1 0

                                              1 4 1

                                              0 1 0

                                              0 1 0

                                              1 4 1

                                              0 1 0

                                              LaplacianLaplacian

                                              MaskMask

                                              1 1 1

                                              1 8 1

                                              1 1 1

                                              1 1 1

                                              1 8 1

                                              1 1 1

                                              4646

                                              Example of edge detectionExample of edge detection

                                              See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                              Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                              derivative)derivative)

                                              produces double edgesproduces double edges

                                              unable to detect edge directionunable to detect edge direction

                                              4747

                                              Canny edge detectorCanny edge detector

                                              The most powerful edge-detection The most powerful edge-detection

                                              method method

                                              It differs from the other edge-It differs from the other edge-

                                              detection methods in that detection methods in that

                                              it uses two different thresholds (to detect it uses two different thresholds (to detect

                                              strong and weak edges) strong and weak edges)

                                              and includes the weak edges in the and includes the weak edges in the

                                              output only if they are connected to output only if they are connected to

                                              strong edges strong edges

                                              This method is therefore less likely This method is therefore less likely

                                              than the others to be fooled by than the others to be fooled by

                                              noise and more likely to detect true noise and more likely to detect true

                                              weak edgesweak edges

                                              4848

                                              Laplacian of GaussianLaplacian of Gaussian

                                              Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                              2 2 22

                                              4 2

                                              2 2 2

                                              2exp

                                              r rh

                                              r x y

                                              determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                              4949

                                              Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                              0 0 1 0 0

                                              0 1 2 1 0

                                              1 2 16 2 1

                                              0 1 2 1 0

                                              0 0 1 0 0

                                              0 0 1 0 0

                                              0 1 2 1 0

                                              1 2 16 2 1

                                              0 1 2 1 0

                                              0 0 1 0 0

                                              The coefficient must sum to The coefficient must sum to

                                              zerozero

                                              5050

                                              Edge Detection and Edge Detection and SegmentationSegmentation

                                              Image resulting from edge detection cannot be used as a segmentation result

                                              Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                              5151

                                              75 Region-based 75 Region-based SegmentationSegmentation

                                              GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                              Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                              PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                              For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                              PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                              For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                              5252

                                              Two methods of Region Two methods of Region SegmentationSegmentation

                                              Region GrowingRegion Growing

                                              Region SplittingRegion Splitting

                                              Region growing is the opposite of the Region growing is the opposite of the

                                              split and merge approachsplit and merge approach

                                              5353

                                              Region GrowingRegion Growing

                                              The objective of segmentation is to The objective of segmentation is to

                                              partition an image into regionspartition an image into regions

                                              A region is a connected component with A region is a connected component with

                                              some uniformity (say gray-levels or some uniformity (say gray-levels or

                                              texture)texture)

                                              In region growing we start with a set In region growing we start with a set

                                              of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                              appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                              pixels if they have pixels if they have similar propertiessimilar properties

                                              such as specific ranges of gray level such as specific ranges of gray level

                                              and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                              Need initialization Need initialization similarity similarity

                                              criterioncriterion

                                              5454

                                              Steps of Region GrowingSteps of Region Growing

                                              Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                              pixel andpixel and compare it with neighbor compare it with neighbor

                                              ppixelsixels

                                              When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                              is grown from the seed pixel by is grown from the seed pixel by

                                              addingadding neighboneighborr pixel pixelss

                                              When the growth of one region stopsWhen the growth of one region stops

                                              choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                              againagain

                                              5555

                                              Region Region growing growing

                                              An initial set of small An initial set of small

                                              areas are iterativelyareas are iteratively

                                              merged according to merged according to

                                              similarity constraintssimilarity constraints

                                              5656

                                              Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                              X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                              We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                              On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                              Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                              SeedSeed pointspoints

                                              5757

                                              CriterionCriterion

                                              There is a valley at around 190 in the There is a valley at around 190 in the

                                              histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                              to be considered as a part of region to the to be considered as a part of region to the

                                              seed pointseed point

                                              The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                              to at least one pixel in that regionto at least one pixel in that region

                                              Result of region growing and boundaries of Result of region growing and boundaries of

                                              defectsdefects

                                              5858

                                              Region SplittingRegion Splitting

                                              The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                              ption that the entire image is homogeneousption that the entire image is homogeneous

                                              If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                              This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                              Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                              5959

                                              Region SplittingRegion Splitting

                                              DisadvantageDisadvantage

                                              they create regions that may be adjacent they create regions that may be adjacent

                                              and homogeneous but not mergedand homogeneous but not merged

                                              6060

                                              Region Splitting and MergingRegion Splitting and Merging

                                              ProcedureProcedure

                                              11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                              22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                              33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                              6161

                                              Region Splitting and Merging

                                              Quadtree

                                              (四叉树 )

                                              6262

                                              PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                              where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                              mmii is the mean gray level of that region is the mean gray level of that region

                                              σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                              ExampleExample

                                              Original Original

                                              imageimageThresholded imageThresholded image Result of Result of

                                              Splitting and Splitting and

                                              MergingMerging

                                              • Slide 1
                                              • Slide 2
                                              • Slide 3
                                              • Slide 4
                                              • Slide 5
                                              • Slide 6
                                              • Slide 7
                                              • Slide 8
                                              • Slide 9
                                              • Slide 10
                                              • Slide 11
                                              • Slide 12
                                              • Slide 13
                                              • Slide 14
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                                              • Slide 28
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                                              • Slide 34
                                              • Slide 35
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                                              • Slide 43
                                              • Slide 44
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                                              • Slide 54
                                              • Slide 55
                                              • Slide 56
                                              • Slide 57
                                              • Slide 58
                                              • Slide 59
                                              • Slide 60
                                              • Slide 61
                                              • Slide 62

                                                2424

                                                Problems of ThresholdingProblems of Thresholding

                                                Original imageOriginal image Thresholded imageThresholded image

                                                2525

                                                Problems of ThresholdingProblems of Thresholding

                                                (a)(a) Exact threshold Exact threshold

                                                segmentationsegmentation

                                                (b)(b) Threshold too lowThreshold too low

                                                (c)(c) Threshold too Threshold too

                                                highhigh

                                                2626

                                                72 Point Detection72 Point Detection

                                                a point has been detected at the a point has been detected at the

                                                location on which the mark is location on which the mark is

                                                centered ifcentered if

                                                |R|geT|R|geT

                                                where T is a nonnegative thresholdwhere T is a nonnegative threshold

                                                R is the sum of products of the R is the sum of products of the

                                                coefficients with the gray levels contained coefficients with the gray levels contained

                                                in the region encompassed by the markin the region encompassed by the mark

                                                1 1 1

                                                1 8 1

                                                1 1 1

                                                1 1 1

                                                1 8 1

                                                1 1 1

                                                2727

                                                72 Point Detection72 Point Detection

                                                Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                                                The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                                                1 1 1

                                                1 8 1

                                                1 1 1

                                                1 1 1

                                                1 8 1

                                                1 1 1

                                                0 1 0

                                                1 4 1

                                                0 1 0

                                                0 1 0

                                                1 4 1

                                                0 1 0

                                                2828

                                                ExampleExample

                                                2929

                                                73 Line Detection73 Line Detection

                                                Horizontal mask will result with max Horizontal mask will result with max

                                                response when a line passed through the response when a line passed through the

                                                middle row of the mask with a constant middle row of the mask with a constant

                                                backgroundbackground

                                                the similar idea is used with other masksthe similar idea is used with other masks

                                                Note the preferred direction of each mask Note the preferred direction of each mask

                                                is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                                than other possible directionsthan other possible directions

                                                1 1 1 1 1 2 1 2 1 2 1 1

                                                2 2 2 1 2 1 1 2 1 1 2 1

                                                1 1 1 2 1 1 1 2 1 1 1 2

                                                45 45Horizontal Vertical

                                                1 1 1 1 1 2 1 2 1 2 1 1

                                                2 2 2 1 2 1 1 2 1 1 2 1

                                                1 1 1 2 1 1 1 2 1 1 1 2

                                                45 45Horizontal Vertical

                                                3030

                                                Idea 1 of Line DetectionIdea 1 of Line Detection

                                                Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                                izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                                if at a certain point in the imageif at a certain point in the image

                                                |Ri||Ri|gtgt|Rj||Rj|

                                                for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                                3131

                                                Idea 2 of Line DetectionIdea 2 of Line Detection

                                                Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                                After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                                3232

                                                ExampleExample

                                                3333

                                                74 Edge-based 74 Edge-based SegmentationSegmentation

                                                Edge-based segmentations rely on edges found in an image by edge detecting operators

                                                these edges mark image locations of discontinuities in gray level

                                                Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                                There are a large group of methods based on information about edges in the image

                                                3434

                                                What is edgeWhat is edge

                                                Edge is where change occurs Change is measured by derivative in 1D

                                                ―Biggest change derivative has maximum magnitude

                                                Or 2nd derivative is zero we discuss approaches for implementing

                                                ―first-order derivative (Gradient operator)

                                                ―second-order derivative (Laplacian operator)

                                                ―we have introduced both derivatives in chapter 3

                                                ―Here we will talk only about their properties for edge detection

                                                3535

                                                What is edgeWhat is edge

                                                In other wordsIn other words an edge is a set of an edge is a set of

                                                connected pixelsconnected pixels

                                                that lie on the boundary between two that lie on the boundary between two

                                                regions with relatively distinct gray-level regions with relatively distinct gray-level

                                                propertiesproperties

                                                Note edge vs boundaryNote edge vs boundary

                                                ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                                ―whereas a region boundary owing to whereas a region boundary owing to

                                                the way it is defined is a more global the way it is defined is a more global

                                                ideaidea

                                                3636

                                                Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                because of because of

                                                optics optics

                                                sampling sampling

                                                image image

                                                acquisition acquisition

                                                imperfectionimperfection

                                                3737

                                                Thick and Thin EdgeThick and Thin Edge

                                                The slope of the ramp is inversely The slope of the ramp is inversely

                                                proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                edgeedge

                                                Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                pathpath

                                                Instead an edge point now is any point Instead an edge point now is any point

                                                contained in the ramp and contained in the ramp and an edge would an edge would

                                                then be a set of such points that are then be a set of such points that are

                                                connectedconnected

                                                The thickness is determined by the length of the The thickness is determined by the length of the

                                                rampramp

                                                The length is determined by the slope which is in The length is determined by the slope which is in

                                                turn determined by the degree of blurringturn determined by the degree of blurring

                                                Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                edges tend to be thinedges tend to be thin

                                                3838

                                                First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                the signs of the the signs of the

                                                derivatives would be derivatives would be

                                                reversed for an edge reversed for an edge

                                                that transitions from that transitions from

                                                light to darklight to dark

                                                First First derivatderivatee

                                                SeconSecond d derivatderivatee

                                                Gray-Gray-level level profileprofile

                                                3939

                                                Second derivativesSecond derivatives

                                                an undesirable featurean undesirable feature

                                                produces 2 values for every edge in an produces 2 values for every edge in an

                                                imageimage

                                                zero-crossing propertyzero-crossing property

                                                an imaginary straight line joining the an imaginary straight line joining the

                                                extreme positive and negative values of extreme positive and negative values of

                                                the second derivative would cross zero the second derivative would cross zero

                                                near the midpoint of the edgenear the midpoint of the edge

                                                quite useful for locating the centers of quite useful for locating the centers of

                                                thick edgesthick edges

                                                4040

                                                Basic idea of edge detectionBasic idea of edge detection

                                                A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                the edge direction and the results are the edge direction and the results are

                                                interpretedinterpreted

                                                The magnitude of the first derivative is The magnitude of the first derivative is

                                                used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                ramp)ramp)

                                                The sign of the second derivative can The sign of the second derivative can

                                                determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                dark or light side of an edgedark or light side of an edge

                                                4141

                                                Review of First DerivateReview of First Derivate

                                                Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                Sobel operators 3Sobel operators 333

                                                6 5 8 5x yG z z G z z

                                                1 2 3

                                                4 5 6

                                                7 8 9

                                                z z z

                                                z z z

                                                z z z

                                                1 2 3

                                                4 5 6

                                                7 8 9

                                                z z z

                                                z z z

                                                z z z

                                                9 5 8 6x yG z z G z z 1 0 0 1

                                                0 1 1 0

                                                1 0 0 1

                                                0 1 1 0

                                                7 8 9 1 2 3

                                                3 6 9 1 4 7

                                                2 2

                                                2 2

                                                x

                                                y

                                                G z z z z z z

                                                G z z z z z z

                                                1 2 1 1 0 1

                                                0 0 0 2 0 2

                                                1 2 1 1 0 1

                                                1 2 1 1 0 1

                                                0 0 0 2 0 2

                                                1 2 1 1 0 1

                                                x yf G G

                                                4242

                                                Edge direction and strengthEdge direction and strength

                                                Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                2 2x yf G G

                                                4343

                                                Gradient MasksGradient Masks

                                                1 0 0 1

                                                0 1 1 0

                                                Roberts

                                                1 0 0 1

                                                0 1 1 0

                                                Roberts

                                                1 2 1 1 0 1

                                                0 0 0 2 0 2

                                                1 2 1 1 0 1

                                                Sobel

                                                1 2 1 1 0 1

                                                0 0 0 2 0 2

                                                1 2 1 1 0 1

                                                Sobel

                                                1 1 1 1 0 1

                                                0 0 0 1 0 1

                                                1 1 1 1 0 1

                                                Prewitt

                                                1 1 1 1 0 1

                                                0 0 0 1 0 1

                                                1 1 1 1 0 1

                                                Prewitt

                                                4444

                                                Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                0 1 1 1 1 0

                                                1 0 1 1 0 1

                                                1 1 0 0 1 1

                                                Prewitt

                                                0 1 1 1 1 0

                                                1 0 1 1 0 1

                                                1 1 0 0 1 1

                                                Prewitt

                                                4545

                                                Review of Second DerivateReview of Second Derivate

                                                Laplacian OperatorLaplacian Operator

                                                21 1

                                                1 1 4

                                                f x y f x yf

                                                f x y f x y f x y

                                                0 1 0

                                                1 4 1

                                                0 1 0

                                                0 1 0

                                                1 4 1

                                                0 1 0

                                                LaplacianLaplacian

                                                MaskMask

                                                1 1 1

                                                1 8 1

                                                1 1 1

                                                1 1 1

                                                1 8 1

                                                1 1 1

                                                4646

                                                Example of edge detectionExample of edge detection

                                                See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                derivative)derivative)

                                                produces double edgesproduces double edges

                                                unable to detect edge directionunable to detect edge direction

                                                4747

                                                Canny edge detectorCanny edge detector

                                                The most powerful edge-detection The most powerful edge-detection

                                                method method

                                                It differs from the other edge-It differs from the other edge-

                                                detection methods in that detection methods in that

                                                it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                strong and weak edges) strong and weak edges)

                                                and includes the weak edges in the and includes the weak edges in the

                                                output only if they are connected to output only if they are connected to

                                                strong edges strong edges

                                                This method is therefore less likely This method is therefore less likely

                                                than the others to be fooled by than the others to be fooled by

                                                noise and more likely to detect true noise and more likely to detect true

                                                weak edgesweak edges

                                                4848

                                                Laplacian of GaussianLaplacian of Gaussian

                                                Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                2 2 22

                                                4 2

                                                2 2 2

                                                2exp

                                                r rh

                                                r x y

                                                determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                4949

                                                Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                0 0 1 0 0

                                                0 1 2 1 0

                                                1 2 16 2 1

                                                0 1 2 1 0

                                                0 0 1 0 0

                                                0 0 1 0 0

                                                0 1 2 1 0

                                                1 2 16 2 1

                                                0 1 2 1 0

                                                0 0 1 0 0

                                                The coefficient must sum to The coefficient must sum to

                                                zerozero

                                                5050

                                                Edge Detection and Edge Detection and SegmentationSegmentation

                                                Image resulting from edge detection cannot be used as a segmentation result

                                                Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                5151

                                                75 Region-based 75 Region-based SegmentationSegmentation

                                                GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                5252

                                                Two methods of Region Two methods of Region SegmentationSegmentation

                                                Region GrowingRegion Growing

                                                Region SplittingRegion Splitting

                                                Region growing is the opposite of the Region growing is the opposite of the

                                                split and merge approachsplit and merge approach

                                                5353

                                                Region GrowingRegion Growing

                                                The objective of segmentation is to The objective of segmentation is to

                                                partition an image into regionspartition an image into regions

                                                A region is a connected component with A region is a connected component with

                                                some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                texture)texture)

                                                In region growing we start with a set In region growing we start with a set

                                                of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                pixels if they have pixels if they have similar propertiessimilar properties

                                                such as specific ranges of gray level such as specific ranges of gray level

                                                and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                Need initialization Need initialization similarity similarity

                                                criterioncriterion

                                                5454

                                                Steps of Region GrowingSteps of Region Growing

                                                Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                pixel andpixel and compare it with neighbor compare it with neighbor

                                                ppixelsixels

                                                When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                is grown from the seed pixel by is grown from the seed pixel by

                                                addingadding neighboneighborr pixel pixelss

                                                When the growth of one region stopsWhen the growth of one region stops

                                                choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                againagain

                                                5555

                                                Region Region growing growing

                                                An initial set of small An initial set of small

                                                areas are iterativelyareas are iteratively

                                                merged according to merged according to

                                                similarity constraintssimilarity constraints

                                                5656

                                                Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                SeedSeed pointspoints

                                                5757

                                                CriterionCriterion

                                                There is a valley at around 190 in the There is a valley at around 190 in the

                                                histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                to be considered as a part of region to the to be considered as a part of region to the

                                                seed pointseed point

                                                The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                to at least one pixel in that regionto at least one pixel in that region

                                                Result of region growing and boundaries of Result of region growing and boundaries of

                                                defectsdefects

                                                5858

                                                Region SplittingRegion Splitting

                                                The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                ption that the entire image is homogeneousption that the entire image is homogeneous

                                                If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                5959

                                                Region SplittingRegion Splitting

                                                DisadvantageDisadvantage

                                                they create regions that may be adjacent they create regions that may be adjacent

                                                and homogeneous but not mergedand homogeneous but not merged

                                                6060

                                                Region Splitting and MergingRegion Splitting and Merging

                                                ProcedureProcedure

                                                11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                6161

                                                Region Splitting and Merging

                                                Quadtree

                                                (四叉树 )

                                                6262

                                                PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                mmii is the mean gray level of that region is the mean gray level of that region

                                                σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                ExampleExample

                                                Original Original

                                                imageimageThresholded imageThresholded image Result of Result of

                                                Splitting and Splitting and

                                                MergingMerging

                                                • Slide 1
                                                • Slide 2
                                                • Slide 3
                                                • Slide 4
                                                • Slide 5
                                                • Slide 6
                                                • Slide 7
                                                • Slide 8
                                                • Slide 9
                                                • Slide 10
                                                • Slide 11
                                                • Slide 12
                                                • Slide 13
                                                • Slide 14
                                                • Slide 15
                                                • Slide 16
                                                • Slide 17
                                                • Slide 18
                                                • Slide 19
                                                • Slide 20
                                                • Slide 21
                                                • Slide 22
                                                • Slide 23
                                                • Slide 24
                                                • Slide 25
                                                • Slide 26
                                                • Slide 27
                                                • Slide 28
                                                • Slide 29
                                                • Slide 30
                                                • Slide 31
                                                • Slide 32
                                                • Slide 33
                                                • Slide 34
                                                • Slide 35
                                                • Slide 36
                                                • Slide 37
                                                • Slide 38
                                                • Slide 39
                                                • Slide 40
                                                • Slide 41
                                                • Slide 42
                                                • Slide 43
                                                • Slide 44
                                                • Slide 45
                                                • Slide 46
                                                • Slide 47
                                                • Slide 48
                                                • Slide 49
                                                • Slide 50
                                                • Slide 51
                                                • Slide 52
                                                • Slide 53
                                                • Slide 54
                                                • Slide 55
                                                • Slide 56
                                                • Slide 57
                                                • Slide 58
                                                • Slide 59
                                                • Slide 60
                                                • Slide 61
                                                • Slide 62

                                                  2525

                                                  Problems of ThresholdingProblems of Thresholding

                                                  (a)(a) Exact threshold Exact threshold

                                                  segmentationsegmentation

                                                  (b)(b) Threshold too lowThreshold too low

                                                  (c)(c) Threshold too Threshold too

                                                  highhigh

                                                  2626

                                                  72 Point Detection72 Point Detection

                                                  a point has been detected at the a point has been detected at the

                                                  location on which the mark is location on which the mark is

                                                  centered ifcentered if

                                                  |R|geT|R|geT

                                                  where T is a nonnegative thresholdwhere T is a nonnegative threshold

                                                  R is the sum of products of the R is the sum of products of the

                                                  coefficients with the gray levels contained coefficients with the gray levels contained

                                                  in the region encompassed by the markin the region encompassed by the mark

                                                  1 1 1

                                                  1 8 1

                                                  1 1 1

                                                  1 1 1

                                                  1 8 1

                                                  1 1 1

                                                  2727

                                                  72 Point Detection72 Point Detection

                                                  Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                                                  The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                                                  1 1 1

                                                  1 8 1

                                                  1 1 1

                                                  1 1 1

                                                  1 8 1

                                                  1 1 1

                                                  0 1 0

                                                  1 4 1

                                                  0 1 0

                                                  0 1 0

                                                  1 4 1

                                                  0 1 0

                                                  2828

                                                  ExampleExample

                                                  2929

                                                  73 Line Detection73 Line Detection

                                                  Horizontal mask will result with max Horizontal mask will result with max

                                                  response when a line passed through the response when a line passed through the

                                                  middle row of the mask with a constant middle row of the mask with a constant

                                                  backgroundbackground

                                                  the similar idea is used with other masksthe similar idea is used with other masks

                                                  Note the preferred direction of each mask Note the preferred direction of each mask

                                                  is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                                  than other possible directionsthan other possible directions

                                                  1 1 1 1 1 2 1 2 1 2 1 1

                                                  2 2 2 1 2 1 1 2 1 1 2 1

                                                  1 1 1 2 1 1 1 2 1 1 1 2

                                                  45 45Horizontal Vertical

                                                  1 1 1 1 1 2 1 2 1 2 1 1

                                                  2 2 2 1 2 1 1 2 1 1 2 1

                                                  1 1 1 2 1 1 1 2 1 1 1 2

                                                  45 45Horizontal Vertical

                                                  3030

                                                  Idea 1 of Line DetectionIdea 1 of Line Detection

                                                  Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                                  izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                                  if at a certain point in the imageif at a certain point in the image

                                                  |Ri||Ri|gtgt|Rj||Rj|

                                                  for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                                  3131

                                                  Idea 2 of Line DetectionIdea 2 of Line Detection

                                                  Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                                  After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                                  3232

                                                  ExampleExample

                                                  3333

                                                  74 Edge-based 74 Edge-based SegmentationSegmentation

                                                  Edge-based segmentations rely on edges found in an image by edge detecting operators

                                                  these edges mark image locations of discontinuities in gray level

                                                  Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                                  There are a large group of methods based on information about edges in the image

                                                  3434

                                                  What is edgeWhat is edge

                                                  Edge is where change occurs Change is measured by derivative in 1D

                                                  ―Biggest change derivative has maximum magnitude

                                                  Or 2nd derivative is zero we discuss approaches for implementing

                                                  ―first-order derivative (Gradient operator)

                                                  ―second-order derivative (Laplacian operator)

                                                  ―we have introduced both derivatives in chapter 3

                                                  ―Here we will talk only about their properties for edge detection

                                                  3535

                                                  What is edgeWhat is edge

                                                  In other wordsIn other words an edge is a set of an edge is a set of

                                                  connected pixelsconnected pixels

                                                  that lie on the boundary between two that lie on the boundary between two

                                                  regions with relatively distinct gray-level regions with relatively distinct gray-level

                                                  propertiesproperties

                                                  Note edge vs boundaryNote edge vs boundary

                                                  ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                                  ―whereas a region boundary owing to whereas a region boundary owing to

                                                  the way it is defined is a more global the way it is defined is a more global

                                                  ideaidea

                                                  3636

                                                  Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                  because of because of

                                                  optics optics

                                                  sampling sampling

                                                  image image

                                                  acquisition acquisition

                                                  imperfectionimperfection

                                                  3737

                                                  Thick and Thin EdgeThick and Thin Edge

                                                  The slope of the ramp is inversely The slope of the ramp is inversely

                                                  proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                  edgeedge

                                                  Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                  pathpath

                                                  Instead an edge point now is any point Instead an edge point now is any point

                                                  contained in the ramp and contained in the ramp and an edge would an edge would

                                                  then be a set of such points that are then be a set of such points that are

                                                  connectedconnected

                                                  The thickness is determined by the length of the The thickness is determined by the length of the

                                                  rampramp

                                                  The length is determined by the slope which is in The length is determined by the slope which is in

                                                  turn determined by the degree of blurringturn determined by the degree of blurring

                                                  Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                  edges tend to be thinedges tend to be thin

                                                  3838

                                                  First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                  the signs of the the signs of the

                                                  derivatives would be derivatives would be

                                                  reversed for an edge reversed for an edge

                                                  that transitions from that transitions from

                                                  light to darklight to dark

                                                  First First derivatderivatee

                                                  SeconSecond d derivatderivatee

                                                  Gray-Gray-level level profileprofile

                                                  3939

                                                  Second derivativesSecond derivatives

                                                  an undesirable featurean undesirable feature

                                                  produces 2 values for every edge in an produces 2 values for every edge in an

                                                  imageimage

                                                  zero-crossing propertyzero-crossing property

                                                  an imaginary straight line joining the an imaginary straight line joining the

                                                  extreme positive and negative values of extreme positive and negative values of

                                                  the second derivative would cross zero the second derivative would cross zero

                                                  near the midpoint of the edgenear the midpoint of the edge

                                                  quite useful for locating the centers of quite useful for locating the centers of

                                                  thick edgesthick edges

                                                  4040

                                                  Basic idea of edge detectionBasic idea of edge detection

                                                  A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                  the edge direction and the results are the edge direction and the results are

                                                  interpretedinterpreted

                                                  The magnitude of the first derivative is The magnitude of the first derivative is

                                                  used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                  ramp)ramp)

                                                  The sign of the second derivative can The sign of the second derivative can

                                                  determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                  dark or light side of an edgedark or light side of an edge

                                                  4141

                                                  Review of First DerivateReview of First Derivate

                                                  Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                  Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                  Sobel operators 3Sobel operators 333

                                                  6 5 8 5x yG z z G z z

                                                  1 2 3

                                                  4 5 6

                                                  7 8 9

                                                  z z z

                                                  z z z

                                                  z z z

                                                  1 2 3

                                                  4 5 6

                                                  7 8 9

                                                  z z z

                                                  z z z

                                                  z z z

                                                  9 5 8 6x yG z z G z z 1 0 0 1

                                                  0 1 1 0

                                                  1 0 0 1

                                                  0 1 1 0

                                                  7 8 9 1 2 3

                                                  3 6 9 1 4 7

                                                  2 2

                                                  2 2

                                                  x

                                                  y

                                                  G z z z z z z

                                                  G z z z z z z

                                                  1 2 1 1 0 1

                                                  0 0 0 2 0 2

                                                  1 2 1 1 0 1

                                                  1 2 1 1 0 1

                                                  0 0 0 2 0 2

                                                  1 2 1 1 0 1

                                                  x yf G G

                                                  4242

                                                  Edge direction and strengthEdge direction and strength

                                                  Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                  α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                  The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                  The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                  2 2x yf G G

                                                  4343

                                                  Gradient MasksGradient Masks

                                                  1 0 0 1

                                                  0 1 1 0

                                                  Roberts

                                                  1 0 0 1

                                                  0 1 1 0

                                                  Roberts

                                                  1 2 1 1 0 1

                                                  0 0 0 2 0 2

                                                  1 2 1 1 0 1

                                                  Sobel

                                                  1 2 1 1 0 1

                                                  0 0 0 2 0 2

                                                  1 2 1 1 0 1

                                                  Sobel

                                                  1 1 1 1 0 1

                                                  0 0 0 1 0 1

                                                  1 1 1 1 0 1

                                                  Prewitt

                                                  1 1 1 1 0 1

                                                  0 0 0 1 0 1

                                                  1 1 1 1 0 1

                                                  Prewitt

                                                  4444

                                                  Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                  0 1 1 1 1 0

                                                  1 0 1 1 0 1

                                                  1 1 0 0 1 1

                                                  Prewitt

                                                  0 1 1 1 1 0

                                                  1 0 1 1 0 1

                                                  1 1 0 0 1 1

                                                  Prewitt

                                                  4545

                                                  Review of Second DerivateReview of Second Derivate

                                                  Laplacian OperatorLaplacian Operator

                                                  21 1

                                                  1 1 4

                                                  f x y f x yf

                                                  f x y f x y f x y

                                                  0 1 0

                                                  1 4 1

                                                  0 1 0

                                                  0 1 0

                                                  1 4 1

                                                  0 1 0

                                                  LaplacianLaplacian

                                                  MaskMask

                                                  1 1 1

                                                  1 8 1

                                                  1 1 1

                                                  1 1 1

                                                  1 8 1

                                                  1 1 1

                                                  4646

                                                  Example of edge detectionExample of edge detection

                                                  See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                  Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                  derivative)derivative)

                                                  produces double edgesproduces double edges

                                                  unable to detect edge directionunable to detect edge direction

                                                  4747

                                                  Canny edge detectorCanny edge detector

                                                  The most powerful edge-detection The most powerful edge-detection

                                                  method method

                                                  It differs from the other edge-It differs from the other edge-

                                                  detection methods in that detection methods in that

                                                  it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                  strong and weak edges) strong and weak edges)

                                                  and includes the weak edges in the and includes the weak edges in the

                                                  output only if they are connected to output only if they are connected to

                                                  strong edges strong edges

                                                  This method is therefore less likely This method is therefore less likely

                                                  than the others to be fooled by than the others to be fooled by

                                                  noise and more likely to detect true noise and more likely to detect true

                                                  weak edgesweak edges

                                                  4848

                                                  Laplacian of GaussianLaplacian of Gaussian

                                                  Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                  2 2 22

                                                  4 2

                                                  2 2 2

                                                  2exp

                                                  r rh

                                                  r x y

                                                  determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                  4949

                                                  Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                  0 0 1 0 0

                                                  0 1 2 1 0

                                                  1 2 16 2 1

                                                  0 1 2 1 0

                                                  0 0 1 0 0

                                                  0 0 1 0 0

                                                  0 1 2 1 0

                                                  1 2 16 2 1

                                                  0 1 2 1 0

                                                  0 0 1 0 0

                                                  The coefficient must sum to The coefficient must sum to

                                                  zerozero

                                                  5050

                                                  Edge Detection and Edge Detection and SegmentationSegmentation

                                                  Image resulting from edge detection cannot be used as a segmentation result

                                                  Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                  5151

                                                  75 Region-based 75 Region-based SegmentationSegmentation

                                                  GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                  Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                  PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                  For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                  PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                  For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                  5252

                                                  Two methods of Region Two methods of Region SegmentationSegmentation

                                                  Region GrowingRegion Growing

                                                  Region SplittingRegion Splitting

                                                  Region growing is the opposite of the Region growing is the opposite of the

                                                  split and merge approachsplit and merge approach

                                                  5353

                                                  Region GrowingRegion Growing

                                                  The objective of segmentation is to The objective of segmentation is to

                                                  partition an image into regionspartition an image into regions

                                                  A region is a connected component with A region is a connected component with

                                                  some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                  texture)texture)

                                                  In region growing we start with a set In region growing we start with a set

                                                  of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                  appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                  pixels if they have pixels if they have similar propertiessimilar properties

                                                  such as specific ranges of gray level such as specific ranges of gray level

                                                  and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                  Need initialization Need initialization similarity similarity

                                                  criterioncriterion

                                                  5454

                                                  Steps of Region GrowingSteps of Region Growing

                                                  Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                  pixel andpixel and compare it with neighbor compare it with neighbor

                                                  ppixelsixels

                                                  When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                  is grown from the seed pixel by is grown from the seed pixel by

                                                  addingadding neighboneighborr pixel pixelss

                                                  When the growth of one region stopsWhen the growth of one region stops

                                                  choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                  againagain

                                                  5555

                                                  Region Region growing growing

                                                  An initial set of small An initial set of small

                                                  areas are iterativelyareas are iteratively

                                                  merged according to merged according to

                                                  similarity constraintssimilarity constraints

                                                  5656

                                                  Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                  X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                  We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                  On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                  Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                  SeedSeed pointspoints

                                                  5757

                                                  CriterionCriterion

                                                  There is a valley at around 190 in the There is a valley at around 190 in the

                                                  histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                  to be considered as a part of region to the to be considered as a part of region to the

                                                  seed pointseed point

                                                  The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                  to at least one pixel in that regionto at least one pixel in that region

                                                  Result of region growing and boundaries of Result of region growing and boundaries of

                                                  defectsdefects

                                                  5858

                                                  Region SplittingRegion Splitting

                                                  The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                  ption that the entire image is homogeneousption that the entire image is homogeneous

                                                  If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                  This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                  Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                  5959

                                                  Region SplittingRegion Splitting

                                                  DisadvantageDisadvantage

                                                  they create regions that may be adjacent they create regions that may be adjacent

                                                  and homogeneous but not mergedand homogeneous but not merged

                                                  6060

                                                  Region Splitting and MergingRegion Splitting and Merging

                                                  ProcedureProcedure

                                                  11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                  22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                  33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                  6161

                                                  Region Splitting and Merging

                                                  Quadtree

                                                  (四叉树 )

                                                  6262

                                                  PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                  where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                  mmii is the mean gray level of that region is the mean gray level of that region

                                                  σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                  ExampleExample

                                                  Original Original

                                                  imageimageThresholded imageThresholded image Result of Result of

                                                  Splitting and Splitting and

                                                  MergingMerging

                                                  • Slide 1
                                                  • Slide 2
                                                  • Slide 3
                                                  • Slide 4
                                                  • Slide 5
                                                  • Slide 6
                                                  • Slide 7
                                                  • Slide 8
                                                  • Slide 9
                                                  • Slide 10
                                                  • Slide 11
                                                  • Slide 12
                                                  • Slide 13
                                                  • Slide 14
                                                  • Slide 15
                                                  • Slide 16
                                                  • Slide 17
                                                  • Slide 18
                                                  • Slide 19
                                                  • Slide 20
                                                  • Slide 21
                                                  • Slide 22
                                                  • Slide 23
                                                  • Slide 24
                                                  • Slide 25
                                                  • Slide 26
                                                  • Slide 27
                                                  • Slide 28
                                                  • Slide 29
                                                  • Slide 30
                                                  • Slide 31
                                                  • Slide 32
                                                  • Slide 33
                                                  • Slide 34
                                                  • Slide 35
                                                  • Slide 36
                                                  • Slide 37
                                                  • Slide 38
                                                  • Slide 39
                                                  • Slide 40
                                                  • Slide 41
                                                  • Slide 42
                                                  • Slide 43
                                                  • Slide 44
                                                  • Slide 45
                                                  • Slide 46
                                                  • Slide 47
                                                  • Slide 48
                                                  • Slide 49
                                                  • Slide 50
                                                  • Slide 51
                                                  • Slide 52
                                                  • Slide 53
                                                  • Slide 54
                                                  • Slide 55
                                                  • Slide 56
                                                  • Slide 57
                                                  • Slide 58
                                                  • Slide 59
                                                  • Slide 60
                                                  • Slide 61
                                                  • Slide 62

                                                    2626

                                                    72 Point Detection72 Point Detection

                                                    a point has been detected at the a point has been detected at the

                                                    location on which the mark is location on which the mark is

                                                    centered ifcentered if

                                                    |R|geT|R|geT

                                                    where T is a nonnegative thresholdwhere T is a nonnegative threshold

                                                    R is the sum of products of the R is the sum of products of the

                                                    coefficients with the gray levels contained coefficients with the gray levels contained

                                                    in the region encompassed by the markin the region encompassed by the mark

                                                    1 1 1

                                                    1 8 1

                                                    1 1 1

                                                    1 1 1

                                                    1 8 1

                                                    1 1 1

                                                    2727

                                                    72 Point Detection72 Point Detection

                                                    Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                                                    The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                                                    1 1 1

                                                    1 8 1

                                                    1 1 1

                                                    1 1 1

                                                    1 8 1

                                                    1 1 1

                                                    0 1 0

                                                    1 4 1

                                                    0 1 0

                                                    0 1 0

                                                    1 4 1

                                                    0 1 0

                                                    2828

                                                    ExampleExample

                                                    2929

                                                    73 Line Detection73 Line Detection

                                                    Horizontal mask will result with max Horizontal mask will result with max

                                                    response when a line passed through the response when a line passed through the

                                                    middle row of the mask with a constant middle row of the mask with a constant

                                                    backgroundbackground

                                                    the similar idea is used with other masksthe similar idea is used with other masks

                                                    Note the preferred direction of each mask Note the preferred direction of each mask

                                                    is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                                    than other possible directionsthan other possible directions

                                                    1 1 1 1 1 2 1 2 1 2 1 1

                                                    2 2 2 1 2 1 1 2 1 1 2 1

                                                    1 1 1 2 1 1 1 2 1 1 1 2

                                                    45 45Horizontal Vertical

                                                    1 1 1 1 1 2 1 2 1 2 1 1

                                                    2 2 2 1 2 1 1 2 1 1 2 1

                                                    1 1 1 2 1 1 1 2 1 1 1 2

                                                    45 45Horizontal Vertical

                                                    3030

                                                    Idea 1 of Line DetectionIdea 1 of Line Detection

                                                    Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                                    izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                                    if at a certain point in the imageif at a certain point in the image

                                                    |Ri||Ri|gtgt|Rj||Rj|

                                                    for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                                    3131

                                                    Idea 2 of Line DetectionIdea 2 of Line Detection

                                                    Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                                    After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                                    3232

                                                    ExampleExample

                                                    3333

                                                    74 Edge-based 74 Edge-based SegmentationSegmentation

                                                    Edge-based segmentations rely on edges found in an image by edge detecting operators

                                                    these edges mark image locations of discontinuities in gray level

                                                    Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                                    There are a large group of methods based on information about edges in the image

                                                    3434

                                                    What is edgeWhat is edge

                                                    Edge is where change occurs Change is measured by derivative in 1D

                                                    ―Biggest change derivative has maximum magnitude

                                                    Or 2nd derivative is zero we discuss approaches for implementing

                                                    ―first-order derivative (Gradient operator)

                                                    ―second-order derivative (Laplacian operator)

                                                    ―we have introduced both derivatives in chapter 3

                                                    ―Here we will talk only about their properties for edge detection

                                                    3535

                                                    What is edgeWhat is edge

                                                    In other wordsIn other words an edge is a set of an edge is a set of

                                                    connected pixelsconnected pixels

                                                    that lie on the boundary between two that lie on the boundary between two

                                                    regions with relatively distinct gray-level regions with relatively distinct gray-level

                                                    propertiesproperties

                                                    Note edge vs boundaryNote edge vs boundary

                                                    ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                                    ―whereas a region boundary owing to whereas a region boundary owing to

                                                    the way it is defined is a more global the way it is defined is a more global

                                                    ideaidea

                                                    3636

                                                    Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                    because of because of

                                                    optics optics

                                                    sampling sampling

                                                    image image

                                                    acquisition acquisition

                                                    imperfectionimperfection

                                                    3737

                                                    Thick and Thin EdgeThick and Thin Edge

                                                    The slope of the ramp is inversely The slope of the ramp is inversely

                                                    proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                    edgeedge

                                                    Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                    pathpath

                                                    Instead an edge point now is any point Instead an edge point now is any point

                                                    contained in the ramp and contained in the ramp and an edge would an edge would

                                                    then be a set of such points that are then be a set of such points that are

                                                    connectedconnected

                                                    The thickness is determined by the length of the The thickness is determined by the length of the

                                                    rampramp

                                                    The length is determined by the slope which is in The length is determined by the slope which is in

                                                    turn determined by the degree of blurringturn determined by the degree of blurring

                                                    Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                    edges tend to be thinedges tend to be thin

                                                    3838

                                                    First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                    the signs of the the signs of the

                                                    derivatives would be derivatives would be

                                                    reversed for an edge reversed for an edge

                                                    that transitions from that transitions from

                                                    light to darklight to dark

                                                    First First derivatderivatee

                                                    SeconSecond d derivatderivatee

                                                    Gray-Gray-level level profileprofile

                                                    3939

                                                    Second derivativesSecond derivatives

                                                    an undesirable featurean undesirable feature

                                                    produces 2 values for every edge in an produces 2 values for every edge in an

                                                    imageimage

                                                    zero-crossing propertyzero-crossing property

                                                    an imaginary straight line joining the an imaginary straight line joining the

                                                    extreme positive and negative values of extreme positive and negative values of

                                                    the second derivative would cross zero the second derivative would cross zero

                                                    near the midpoint of the edgenear the midpoint of the edge

                                                    quite useful for locating the centers of quite useful for locating the centers of

                                                    thick edgesthick edges

                                                    4040

                                                    Basic idea of edge detectionBasic idea of edge detection

                                                    A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                    the edge direction and the results are the edge direction and the results are

                                                    interpretedinterpreted

                                                    The magnitude of the first derivative is The magnitude of the first derivative is

                                                    used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                    ramp)ramp)

                                                    The sign of the second derivative can The sign of the second derivative can

                                                    determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                    dark or light side of an edgedark or light side of an edge

                                                    4141

                                                    Review of First DerivateReview of First Derivate

                                                    Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                    Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                    Sobel operators 3Sobel operators 333

                                                    6 5 8 5x yG z z G z z

                                                    1 2 3

                                                    4 5 6

                                                    7 8 9

                                                    z z z

                                                    z z z

                                                    z z z

                                                    1 2 3

                                                    4 5 6

                                                    7 8 9

                                                    z z z

                                                    z z z

                                                    z z z

                                                    9 5 8 6x yG z z G z z 1 0 0 1

                                                    0 1 1 0

                                                    1 0 0 1

                                                    0 1 1 0

                                                    7 8 9 1 2 3

                                                    3 6 9 1 4 7

                                                    2 2

                                                    2 2

                                                    x

                                                    y

                                                    G z z z z z z

                                                    G z z z z z z

                                                    1 2 1 1 0 1

                                                    0 0 0 2 0 2

                                                    1 2 1 1 0 1

                                                    1 2 1 1 0 1

                                                    0 0 0 2 0 2

                                                    1 2 1 1 0 1

                                                    x yf G G

                                                    4242

                                                    Edge direction and strengthEdge direction and strength

                                                    Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                    α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                    The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                    The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                    2 2x yf G G

                                                    4343

                                                    Gradient MasksGradient Masks

                                                    1 0 0 1

                                                    0 1 1 0

                                                    Roberts

                                                    1 0 0 1

                                                    0 1 1 0

                                                    Roberts

                                                    1 2 1 1 0 1

                                                    0 0 0 2 0 2

                                                    1 2 1 1 0 1

                                                    Sobel

                                                    1 2 1 1 0 1

                                                    0 0 0 2 0 2

                                                    1 2 1 1 0 1

                                                    Sobel

                                                    1 1 1 1 0 1

                                                    0 0 0 1 0 1

                                                    1 1 1 1 0 1

                                                    Prewitt

                                                    1 1 1 1 0 1

                                                    0 0 0 1 0 1

                                                    1 1 1 1 0 1

                                                    Prewitt

                                                    4444

                                                    Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                    0 1 1 1 1 0

                                                    1 0 1 1 0 1

                                                    1 1 0 0 1 1

                                                    Prewitt

                                                    0 1 1 1 1 0

                                                    1 0 1 1 0 1

                                                    1 1 0 0 1 1

                                                    Prewitt

                                                    4545

                                                    Review of Second DerivateReview of Second Derivate

                                                    Laplacian OperatorLaplacian Operator

                                                    21 1

                                                    1 1 4

                                                    f x y f x yf

                                                    f x y f x y f x y

                                                    0 1 0

                                                    1 4 1

                                                    0 1 0

                                                    0 1 0

                                                    1 4 1

                                                    0 1 0

                                                    LaplacianLaplacian

                                                    MaskMask

                                                    1 1 1

                                                    1 8 1

                                                    1 1 1

                                                    1 1 1

                                                    1 8 1

                                                    1 1 1

                                                    4646

                                                    Example of edge detectionExample of edge detection

                                                    See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                    Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                    derivative)derivative)

                                                    produces double edgesproduces double edges

                                                    unable to detect edge directionunable to detect edge direction

                                                    4747

                                                    Canny edge detectorCanny edge detector

                                                    The most powerful edge-detection The most powerful edge-detection

                                                    method method

                                                    It differs from the other edge-It differs from the other edge-

                                                    detection methods in that detection methods in that

                                                    it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                    strong and weak edges) strong and weak edges)

                                                    and includes the weak edges in the and includes the weak edges in the

                                                    output only if they are connected to output only if they are connected to

                                                    strong edges strong edges

                                                    This method is therefore less likely This method is therefore less likely

                                                    than the others to be fooled by than the others to be fooled by

                                                    noise and more likely to detect true noise and more likely to detect true

                                                    weak edgesweak edges

                                                    4848

                                                    Laplacian of GaussianLaplacian of Gaussian

                                                    Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                    2 2 22

                                                    4 2

                                                    2 2 2

                                                    2exp

                                                    r rh

                                                    r x y

                                                    determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                    4949

                                                    Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                    0 0 1 0 0

                                                    0 1 2 1 0

                                                    1 2 16 2 1

                                                    0 1 2 1 0

                                                    0 0 1 0 0

                                                    0 0 1 0 0

                                                    0 1 2 1 0

                                                    1 2 16 2 1

                                                    0 1 2 1 0

                                                    0 0 1 0 0

                                                    The coefficient must sum to The coefficient must sum to

                                                    zerozero

                                                    5050

                                                    Edge Detection and Edge Detection and SegmentationSegmentation

                                                    Image resulting from edge detection cannot be used as a segmentation result

                                                    Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                    5151

                                                    75 Region-based 75 Region-based SegmentationSegmentation

                                                    GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                    Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                    5252

                                                    Two methods of Region Two methods of Region SegmentationSegmentation

                                                    Region GrowingRegion Growing

                                                    Region SplittingRegion Splitting

                                                    Region growing is the opposite of the Region growing is the opposite of the

                                                    split and merge approachsplit and merge approach

                                                    5353

                                                    Region GrowingRegion Growing

                                                    The objective of segmentation is to The objective of segmentation is to

                                                    partition an image into regionspartition an image into regions

                                                    A region is a connected component with A region is a connected component with

                                                    some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                    texture)texture)

                                                    In region growing we start with a set In region growing we start with a set

                                                    of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                    appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                    pixels if they have pixels if they have similar propertiessimilar properties

                                                    such as specific ranges of gray level such as specific ranges of gray level

                                                    and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                    Need initialization Need initialization similarity similarity

                                                    criterioncriterion

                                                    5454

                                                    Steps of Region GrowingSteps of Region Growing

                                                    Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                    pixel andpixel and compare it with neighbor compare it with neighbor

                                                    ppixelsixels

                                                    When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                    is grown from the seed pixel by is grown from the seed pixel by

                                                    addingadding neighboneighborr pixel pixelss

                                                    When the growth of one region stopsWhen the growth of one region stops

                                                    choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                    againagain

                                                    5555

                                                    Region Region growing growing

                                                    An initial set of small An initial set of small

                                                    areas are iterativelyareas are iteratively

                                                    merged according to merged according to

                                                    similarity constraintssimilarity constraints

                                                    5656

                                                    Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                    X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                    We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                    On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                    Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                    SeedSeed pointspoints

                                                    5757

                                                    CriterionCriterion

                                                    There is a valley at around 190 in the There is a valley at around 190 in the

                                                    histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                    to be considered as a part of region to the to be considered as a part of region to the

                                                    seed pointseed point

                                                    The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                    to at least one pixel in that regionto at least one pixel in that region

                                                    Result of region growing and boundaries of Result of region growing and boundaries of

                                                    defectsdefects

                                                    5858

                                                    Region SplittingRegion Splitting

                                                    The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                    ption that the entire image is homogeneousption that the entire image is homogeneous

                                                    If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                    This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                    Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                    5959

                                                    Region SplittingRegion Splitting

                                                    DisadvantageDisadvantage

                                                    they create regions that may be adjacent they create regions that may be adjacent

                                                    and homogeneous but not mergedand homogeneous but not merged

                                                    6060

                                                    Region Splitting and MergingRegion Splitting and Merging

                                                    ProcedureProcedure

                                                    11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                    22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                    33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                    6161

                                                    Region Splitting and Merging

                                                    Quadtree

                                                    (四叉树 )

                                                    6262

                                                    PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                    where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                    mmii is the mean gray level of that region is the mean gray level of that region

                                                    σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                    ExampleExample

                                                    Original Original

                                                    imageimageThresholded imageThresholded image Result of Result of

                                                    Splitting and Splitting and

                                                    MergingMerging

                                                    • Slide 1
                                                    • Slide 2
                                                    • Slide 3
                                                    • Slide 4
                                                    • Slide 5
                                                    • Slide 6
                                                    • Slide 7
                                                    • Slide 8
                                                    • Slide 9
                                                    • Slide 10
                                                    • Slide 11
                                                    • Slide 12
                                                    • Slide 13
                                                    • Slide 14
                                                    • Slide 15
                                                    • Slide 16
                                                    • Slide 17
                                                    • Slide 18
                                                    • Slide 19
                                                    • Slide 20
                                                    • Slide 21
                                                    • Slide 22
                                                    • Slide 23
                                                    • Slide 24
                                                    • Slide 25
                                                    • Slide 26
                                                    • Slide 27
                                                    • Slide 28
                                                    • Slide 29
                                                    • Slide 30
                                                    • Slide 31
                                                    • Slide 32
                                                    • Slide 33
                                                    • Slide 34
                                                    • Slide 35
                                                    • Slide 36
                                                    • Slide 37
                                                    • Slide 38
                                                    • Slide 39
                                                    • Slide 40
                                                    • Slide 41
                                                    • Slide 42
                                                    • Slide 43
                                                    • Slide 44
                                                    • Slide 45
                                                    • Slide 46
                                                    • Slide 47
                                                    • Slide 48
                                                    • Slide 49
                                                    • Slide 50
                                                    • Slide 51
                                                    • Slide 52
                                                    • Slide 53
                                                    • Slide 54
                                                    • Slide 55
                                                    • Slide 56
                                                    • Slide 57
                                                    • Slide 58
                                                    • Slide 59
                                                    • Slide 60
                                                    • Slide 61
                                                    • Slide 62

                                                      2727

                                                      72 Point Detection72 Point Detection

                                                      Note that the mark is the same as the mask Note that the mark is the same as the mask of Laplacian Operation (in chapter 3)of Laplacian Operation (in chapter 3)

                                                      The only differences that are considered intThe only differences that are considered interest are those large enough (as determined erest are those large enough (as determined by T) to be considered isolated pointsby T) to be considered isolated points

                                                      1 1 1

                                                      1 8 1

                                                      1 1 1

                                                      1 1 1

                                                      1 8 1

                                                      1 1 1

                                                      0 1 0

                                                      1 4 1

                                                      0 1 0

                                                      0 1 0

                                                      1 4 1

                                                      0 1 0

                                                      2828

                                                      ExampleExample

                                                      2929

                                                      73 Line Detection73 Line Detection

                                                      Horizontal mask will result with max Horizontal mask will result with max

                                                      response when a line passed through the response when a line passed through the

                                                      middle row of the mask with a constant middle row of the mask with a constant

                                                      backgroundbackground

                                                      the similar idea is used with other masksthe similar idea is used with other masks

                                                      Note the preferred direction of each mask Note the preferred direction of each mask

                                                      is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                                      than other possible directionsthan other possible directions

                                                      1 1 1 1 1 2 1 2 1 2 1 1

                                                      2 2 2 1 2 1 1 2 1 1 2 1

                                                      1 1 1 2 1 1 1 2 1 1 1 2

                                                      45 45Horizontal Vertical

                                                      1 1 1 1 1 2 1 2 1 2 1 1

                                                      2 2 2 1 2 1 1 2 1 1 2 1

                                                      1 1 1 2 1 1 1 2 1 1 1 2

                                                      45 45Horizontal Vertical

                                                      3030

                                                      Idea 1 of Line DetectionIdea 1 of Line Detection

                                                      Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                                      izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                                      if at a certain point in the imageif at a certain point in the image

                                                      |Ri||Ri|gtgt|Rj||Rj|

                                                      for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                                      3131

                                                      Idea 2 of Line DetectionIdea 2 of Line Detection

                                                      Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                                      After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                                      3232

                                                      ExampleExample

                                                      3333

                                                      74 Edge-based 74 Edge-based SegmentationSegmentation

                                                      Edge-based segmentations rely on edges found in an image by edge detecting operators

                                                      these edges mark image locations of discontinuities in gray level

                                                      Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                                      There are a large group of methods based on information about edges in the image

                                                      3434

                                                      What is edgeWhat is edge

                                                      Edge is where change occurs Change is measured by derivative in 1D

                                                      ―Biggest change derivative has maximum magnitude

                                                      Or 2nd derivative is zero we discuss approaches for implementing

                                                      ―first-order derivative (Gradient operator)

                                                      ―second-order derivative (Laplacian operator)

                                                      ―we have introduced both derivatives in chapter 3

                                                      ―Here we will talk only about their properties for edge detection

                                                      3535

                                                      What is edgeWhat is edge

                                                      In other wordsIn other words an edge is a set of an edge is a set of

                                                      connected pixelsconnected pixels

                                                      that lie on the boundary between two that lie on the boundary between two

                                                      regions with relatively distinct gray-level regions with relatively distinct gray-level

                                                      propertiesproperties

                                                      Note edge vs boundaryNote edge vs boundary

                                                      ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                                      ―whereas a region boundary owing to whereas a region boundary owing to

                                                      the way it is defined is a more global the way it is defined is a more global

                                                      ideaidea

                                                      3636

                                                      Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                      because of because of

                                                      optics optics

                                                      sampling sampling

                                                      image image

                                                      acquisition acquisition

                                                      imperfectionimperfection

                                                      3737

                                                      Thick and Thin EdgeThick and Thin Edge

                                                      The slope of the ramp is inversely The slope of the ramp is inversely

                                                      proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                      edgeedge

                                                      Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                      pathpath

                                                      Instead an edge point now is any point Instead an edge point now is any point

                                                      contained in the ramp and contained in the ramp and an edge would an edge would

                                                      then be a set of such points that are then be a set of such points that are

                                                      connectedconnected

                                                      The thickness is determined by the length of the The thickness is determined by the length of the

                                                      rampramp

                                                      The length is determined by the slope which is in The length is determined by the slope which is in

                                                      turn determined by the degree of blurringturn determined by the degree of blurring

                                                      Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                      edges tend to be thinedges tend to be thin

                                                      3838

                                                      First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                      the signs of the the signs of the

                                                      derivatives would be derivatives would be

                                                      reversed for an edge reversed for an edge

                                                      that transitions from that transitions from

                                                      light to darklight to dark

                                                      First First derivatderivatee

                                                      SeconSecond d derivatderivatee

                                                      Gray-Gray-level level profileprofile

                                                      3939

                                                      Second derivativesSecond derivatives

                                                      an undesirable featurean undesirable feature

                                                      produces 2 values for every edge in an produces 2 values for every edge in an

                                                      imageimage

                                                      zero-crossing propertyzero-crossing property

                                                      an imaginary straight line joining the an imaginary straight line joining the

                                                      extreme positive and negative values of extreme positive and negative values of

                                                      the second derivative would cross zero the second derivative would cross zero

                                                      near the midpoint of the edgenear the midpoint of the edge

                                                      quite useful for locating the centers of quite useful for locating the centers of

                                                      thick edgesthick edges

                                                      4040

                                                      Basic idea of edge detectionBasic idea of edge detection

                                                      A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                      the edge direction and the results are the edge direction and the results are

                                                      interpretedinterpreted

                                                      The magnitude of the first derivative is The magnitude of the first derivative is

                                                      used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                      ramp)ramp)

                                                      The sign of the second derivative can The sign of the second derivative can

                                                      determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                      dark or light side of an edgedark or light side of an edge

                                                      4141

                                                      Review of First DerivateReview of First Derivate

                                                      Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                      Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                      Sobel operators 3Sobel operators 333

                                                      6 5 8 5x yG z z G z z

                                                      1 2 3

                                                      4 5 6

                                                      7 8 9

                                                      z z z

                                                      z z z

                                                      z z z

                                                      1 2 3

                                                      4 5 6

                                                      7 8 9

                                                      z z z

                                                      z z z

                                                      z z z

                                                      9 5 8 6x yG z z G z z 1 0 0 1

                                                      0 1 1 0

                                                      1 0 0 1

                                                      0 1 1 0

                                                      7 8 9 1 2 3

                                                      3 6 9 1 4 7

                                                      2 2

                                                      2 2

                                                      x

                                                      y

                                                      G z z z z z z

                                                      G z z z z z z

                                                      1 2 1 1 0 1

                                                      0 0 0 2 0 2

                                                      1 2 1 1 0 1

                                                      1 2 1 1 0 1

                                                      0 0 0 2 0 2

                                                      1 2 1 1 0 1

                                                      x yf G G

                                                      4242

                                                      Edge direction and strengthEdge direction and strength

                                                      Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                      α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                      The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                      The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                      2 2x yf G G

                                                      4343

                                                      Gradient MasksGradient Masks

                                                      1 0 0 1

                                                      0 1 1 0

                                                      Roberts

                                                      1 0 0 1

                                                      0 1 1 0

                                                      Roberts

                                                      1 2 1 1 0 1

                                                      0 0 0 2 0 2

                                                      1 2 1 1 0 1

                                                      Sobel

                                                      1 2 1 1 0 1

                                                      0 0 0 2 0 2

                                                      1 2 1 1 0 1

                                                      Sobel

                                                      1 1 1 1 0 1

                                                      0 0 0 1 0 1

                                                      1 1 1 1 0 1

                                                      Prewitt

                                                      1 1 1 1 0 1

                                                      0 0 0 1 0 1

                                                      1 1 1 1 0 1

                                                      Prewitt

                                                      4444

                                                      Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                      0 1 1 1 1 0

                                                      1 0 1 1 0 1

                                                      1 1 0 0 1 1

                                                      Prewitt

                                                      0 1 1 1 1 0

                                                      1 0 1 1 0 1

                                                      1 1 0 0 1 1

                                                      Prewitt

                                                      4545

                                                      Review of Second DerivateReview of Second Derivate

                                                      Laplacian OperatorLaplacian Operator

                                                      21 1

                                                      1 1 4

                                                      f x y f x yf

                                                      f x y f x y f x y

                                                      0 1 0

                                                      1 4 1

                                                      0 1 0

                                                      0 1 0

                                                      1 4 1

                                                      0 1 0

                                                      LaplacianLaplacian

                                                      MaskMask

                                                      1 1 1

                                                      1 8 1

                                                      1 1 1

                                                      1 1 1

                                                      1 8 1

                                                      1 1 1

                                                      4646

                                                      Example of edge detectionExample of edge detection

                                                      See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                      Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                      derivative)derivative)

                                                      produces double edgesproduces double edges

                                                      unable to detect edge directionunable to detect edge direction

                                                      4747

                                                      Canny edge detectorCanny edge detector

                                                      The most powerful edge-detection The most powerful edge-detection

                                                      method method

                                                      It differs from the other edge-It differs from the other edge-

                                                      detection methods in that detection methods in that

                                                      it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                      strong and weak edges) strong and weak edges)

                                                      and includes the weak edges in the and includes the weak edges in the

                                                      output only if they are connected to output only if they are connected to

                                                      strong edges strong edges

                                                      This method is therefore less likely This method is therefore less likely

                                                      than the others to be fooled by than the others to be fooled by

                                                      noise and more likely to detect true noise and more likely to detect true

                                                      weak edgesweak edges

                                                      4848

                                                      Laplacian of GaussianLaplacian of Gaussian

                                                      Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                      2 2 22

                                                      4 2

                                                      2 2 2

                                                      2exp

                                                      r rh

                                                      r x y

                                                      determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                      4949

                                                      Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                      0 0 1 0 0

                                                      0 1 2 1 0

                                                      1 2 16 2 1

                                                      0 1 2 1 0

                                                      0 0 1 0 0

                                                      0 0 1 0 0

                                                      0 1 2 1 0

                                                      1 2 16 2 1

                                                      0 1 2 1 0

                                                      0 0 1 0 0

                                                      The coefficient must sum to The coefficient must sum to

                                                      zerozero

                                                      5050

                                                      Edge Detection and Edge Detection and SegmentationSegmentation

                                                      Image resulting from edge detection cannot be used as a segmentation result

                                                      Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                      5151

                                                      75 Region-based 75 Region-based SegmentationSegmentation

                                                      GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                      Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                      5252

                                                      Two methods of Region Two methods of Region SegmentationSegmentation

                                                      Region GrowingRegion Growing

                                                      Region SplittingRegion Splitting

                                                      Region growing is the opposite of the Region growing is the opposite of the

                                                      split and merge approachsplit and merge approach

                                                      5353

                                                      Region GrowingRegion Growing

                                                      The objective of segmentation is to The objective of segmentation is to

                                                      partition an image into regionspartition an image into regions

                                                      A region is a connected component with A region is a connected component with

                                                      some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                      texture)texture)

                                                      In region growing we start with a set In region growing we start with a set

                                                      of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                      appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                      pixels if they have pixels if they have similar propertiessimilar properties

                                                      such as specific ranges of gray level such as specific ranges of gray level

                                                      and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                      Need initialization Need initialization similarity similarity

                                                      criterioncriterion

                                                      5454

                                                      Steps of Region GrowingSteps of Region Growing

                                                      Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                      pixel andpixel and compare it with neighbor compare it with neighbor

                                                      ppixelsixels

                                                      When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                      is grown from the seed pixel by is grown from the seed pixel by

                                                      addingadding neighboneighborr pixel pixelss

                                                      When the growth of one region stopsWhen the growth of one region stops

                                                      choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                      againagain

                                                      5555

                                                      Region Region growing growing

                                                      An initial set of small An initial set of small

                                                      areas are iterativelyareas are iteratively

                                                      merged according to merged according to

                                                      similarity constraintssimilarity constraints

                                                      5656

                                                      Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                      X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                      We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                      On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                      Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                      SeedSeed pointspoints

                                                      5757

                                                      CriterionCriterion

                                                      There is a valley at around 190 in the There is a valley at around 190 in the

                                                      histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                      to be considered as a part of region to the to be considered as a part of region to the

                                                      seed pointseed point

                                                      The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                      to at least one pixel in that regionto at least one pixel in that region

                                                      Result of region growing and boundaries of Result of region growing and boundaries of

                                                      defectsdefects

                                                      5858

                                                      Region SplittingRegion Splitting

                                                      The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                      ption that the entire image is homogeneousption that the entire image is homogeneous

                                                      If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                      This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                      Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                      5959

                                                      Region SplittingRegion Splitting

                                                      DisadvantageDisadvantage

                                                      they create regions that may be adjacent they create regions that may be adjacent

                                                      and homogeneous but not mergedand homogeneous but not merged

                                                      6060

                                                      Region Splitting and MergingRegion Splitting and Merging

                                                      ProcedureProcedure

                                                      11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                      22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                      33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                      6161

                                                      Region Splitting and Merging

                                                      Quadtree

                                                      (四叉树 )

                                                      6262

                                                      PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                      where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                      mmii is the mean gray level of that region is the mean gray level of that region

                                                      σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                      ExampleExample

                                                      Original Original

                                                      imageimageThresholded imageThresholded image Result of Result of

                                                      Splitting and Splitting and

                                                      MergingMerging

                                                      • Slide 1
                                                      • Slide 2
                                                      • Slide 3
                                                      • Slide 4
                                                      • Slide 5
                                                      • Slide 6
                                                      • Slide 7
                                                      • Slide 8
                                                      • Slide 9
                                                      • Slide 10
                                                      • Slide 11
                                                      • Slide 12
                                                      • Slide 13
                                                      • Slide 14
                                                      • Slide 15
                                                      • Slide 16
                                                      • Slide 17
                                                      • Slide 18
                                                      • Slide 19
                                                      • Slide 20
                                                      • Slide 21
                                                      • Slide 22
                                                      • Slide 23
                                                      • Slide 24
                                                      • Slide 25
                                                      • Slide 26
                                                      • Slide 27
                                                      • Slide 28
                                                      • Slide 29
                                                      • Slide 30
                                                      • Slide 31
                                                      • Slide 32
                                                      • Slide 33
                                                      • Slide 34
                                                      • Slide 35
                                                      • Slide 36
                                                      • Slide 37
                                                      • Slide 38
                                                      • Slide 39
                                                      • Slide 40
                                                      • Slide 41
                                                      • Slide 42
                                                      • Slide 43
                                                      • Slide 44
                                                      • Slide 45
                                                      • Slide 46
                                                      • Slide 47
                                                      • Slide 48
                                                      • Slide 49
                                                      • Slide 50
                                                      • Slide 51
                                                      • Slide 52
                                                      • Slide 53
                                                      • Slide 54
                                                      • Slide 55
                                                      • Slide 56
                                                      • Slide 57
                                                      • Slide 58
                                                      • Slide 59
                                                      • Slide 60
                                                      • Slide 61
                                                      • Slide 62

                                                        2828

                                                        ExampleExample

                                                        2929

                                                        73 Line Detection73 Line Detection

                                                        Horizontal mask will result with max Horizontal mask will result with max

                                                        response when a line passed through the response when a line passed through the

                                                        middle row of the mask with a constant middle row of the mask with a constant

                                                        backgroundbackground

                                                        the similar idea is used with other masksthe similar idea is used with other masks

                                                        Note the preferred direction of each mask Note the preferred direction of each mask

                                                        is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                                        than other possible directionsthan other possible directions

                                                        1 1 1 1 1 2 1 2 1 2 1 1

                                                        2 2 2 1 2 1 1 2 1 1 2 1

                                                        1 1 1 2 1 1 1 2 1 1 1 2

                                                        45 45Horizontal Vertical

                                                        1 1 1 1 1 2 1 2 1 2 1 1

                                                        2 2 2 1 2 1 1 2 1 1 2 1

                                                        1 1 1 2 1 1 1 2 1 1 1 2

                                                        45 45Horizontal Vertical

                                                        3030

                                                        Idea 1 of Line DetectionIdea 1 of Line Detection

                                                        Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                                        izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                                        if at a certain point in the imageif at a certain point in the image

                                                        |Ri||Ri|gtgt|Rj||Rj|

                                                        for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                                        3131

                                                        Idea 2 of Line DetectionIdea 2 of Line Detection

                                                        Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                                        After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                                        3232

                                                        ExampleExample

                                                        3333

                                                        74 Edge-based 74 Edge-based SegmentationSegmentation

                                                        Edge-based segmentations rely on edges found in an image by edge detecting operators

                                                        these edges mark image locations of discontinuities in gray level

                                                        Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                                        There are a large group of methods based on information about edges in the image

                                                        3434

                                                        What is edgeWhat is edge

                                                        Edge is where change occurs Change is measured by derivative in 1D

                                                        ―Biggest change derivative has maximum magnitude

                                                        Or 2nd derivative is zero we discuss approaches for implementing

                                                        ―first-order derivative (Gradient operator)

                                                        ―second-order derivative (Laplacian operator)

                                                        ―we have introduced both derivatives in chapter 3

                                                        ―Here we will talk only about their properties for edge detection

                                                        3535

                                                        What is edgeWhat is edge

                                                        In other wordsIn other words an edge is a set of an edge is a set of

                                                        connected pixelsconnected pixels

                                                        that lie on the boundary between two that lie on the boundary between two

                                                        regions with relatively distinct gray-level regions with relatively distinct gray-level

                                                        propertiesproperties

                                                        Note edge vs boundaryNote edge vs boundary

                                                        ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                                        ―whereas a region boundary owing to whereas a region boundary owing to

                                                        the way it is defined is a more global the way it is defined is a more global

                                                        ideaidea

                                                        3636

                                                        Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                        because of because of

                                                        optics optics

                                                        sampling sampling

                                                        image image

                                                        acquisition acquisition

                                                        imperfectionimperfection

                                                        3737

                                                        Thick and Thin EdgeThick and Thin Edge

                                                        The slope of the ramp is inversely The slope of the ramp is inversely

                                                        proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                        edgeedge

                                                        Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                        pathpath

                                                        Instead an edge point now is any point Instead an edge point now is any point

                                                        contained in the ramp and contained in the ramp and an edge would an edge would

                                                        then be a set of such points that are then be a set of such points that are

                                                        connectedconnected

                                                        The thickness is determined by the length of the The thickness is determined by the length of the

                                                        rampramp

                                                        The length is determined by the slope which is in The length is determined by the slope which is in

                                                        turn determined by the degree of blurringturn determined by the degree of blurring

                                                        Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                        edges tend to be thinedges tend to be thin

                                                        3838

                                                        First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                        the signs of the the signs of the

                                                        derivatives would be derivatives would be

                                                        reversed for an edge reversed for an edge

                                                        that transitions from that transitions from

                                                        light to darklight to dark

                                                        First First derivatderivatee

                                                        SeconSecond d derivatderivatee

                                                        Gray-Gray-level level profileprofile

                                                        3939

                                                        Second derivativesSecond derivatives

                                                        an undesirable featurean undesirable feature

                                                        produces 2 values for every edge in an produces 2 values for every edge in an

                                                        imageimage

                                                        zero-crossing propertyzero-crossing property

                                                        an imaginary straight line joining the an imaginary straight line joining the

                                                        extreme positive and negative values of extreme positive and negative values of

                                                        the second derivative would cross zero the second derivative would cross zero

                                                        near the midpoint of the edgenear the midpoint of the edge

                                                        quite useful for locating the centers of quite useful for locating the centers of

                                                        thick edgesthick edges

                                                        4040

                                                        Basic idea of edge detectionBasic idea of edge detection

                                                        A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                        the edge direction and the results are the edge direction and the results are

                                                        interpretedinterpreted

                                                        The magnitude of the first derivative is The magnitude of the first derivative is

                                                        used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                        ramp)ramp)

                                                        The sign of the second derivative can The sign of the second derivative can

                                                        determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                        dark or light side of an edgedark or light side of an edge

                                                        4141

                                                        Review of First DerivateReview of First Derivate

                                                        Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                        Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                        Sobel operators 3Sobel operators 333

                                                        6 5 8 5x yG z z G z z

                                                        1 2 3

                                                        4 5 6

                                                        7 8 9

                                                        z z z

                                                        z z z

                                                        z z z

                                                        1 2 3

                                                        4 5 6

                                                        7 8 9

                                                        z z z

                                                        z z z

                                                        z z z

                                                        9 5 8 6x yG z z G z z 1 0 0 1

                                                        0 1 1 0

                                                        1 0 0 1

                                                        0 1 1 0

                                                        7 8 9 1 2 3

                                                        3 6 9 1 4 7

                                                        2 2

                                                        2 2

                                                        x

                                                        y

                                                        G z z z z z z

                                                        G z z z z z z

                                                        1 2 1 1 0 1

                                                        0 0 0 2 0 2

                                                        1 2 1 1 0 1

                                                        1 2 1 1 0 1

                                                        0 0 0 2 0 2

                                                        1 2 1 1 0 1

                                                        x yf G G

                                                        4242

                                                        Edge direction and strengthEdge direction and strength

                                                        Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                        α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                        The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                        The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                        2 2x yf G G

                                                        4343

                                                        Gradient MasksGradient Masks

                                                        1 0 0 1

                                                        0 1 1 0

                                                        Roberts

                                                        1 0 0 1

                                                        0 1 1 0

                                                        Roberts

                                                        1 2 1 1 0 1

                                                        0 0 0 2 0 2

                                                        1 2 1 1 0 1

                                                        Sobel

                                                        1 2 1 1 0 1

                                                        0 0 0 2 0 2

                                                        1 2 1 1 0 1

                                                        Sobel

                                                        1 1 1 1 0 1

                                                        0 0 0 1 0 1

                                                        1 1 1 1 0 1

                                                        Prewitt

                                                        1 1 1 1 0 1

                                                        0 0 0 1 0 1

                                                        1 1 1 1 0 1

                                                        Prewitt

                                                        4444

                                                        Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                        0 1 1 1 1 0

                                                        1 0 1 1 0 1

                                                        1 1 0 0 1 1

                                                        Prewitt

                                                        0 1 1 1 1 0

                                                        1 0 1 1 0 1

                                                        1 1 0 0 1 1

                                                        Prewitt

                                                        4545

                                                        Review of Second DerivateReview of Second Derivate

                                                        Laplacian OperatorLaplacian Operator

                                                        21 1

                                                        1 1 4

                                                        f x y f x yf

                                                        f x y f x y f x y

                                                        0 1 0

                                                        1 4 1

                                                        0 1 0

                                                        0 1 0

                                                        1 4 1

                                                        0 1 0

                                                        LaplacianLaplacian

                                                        MaskMask

                                                        1 1 1

                                                        1 8 1

                                                        1 1 1

                                                        1 1 1

                                                        1 8 1

                                                        1 1 1

                                                        4646

                                                        Example of edge detectionExample of edge detection

                                                        See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                        Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                        derivative)derivative)

                                                        produces double edgesproduces double edges

                                                        unable to detect edge directionunable to detect edge direction

                                                        4747

                                                        Canny edge detectorCanny edge detector

                                                        The most powerful edge-detection The most powerful edge-detection

                                                        method method

                                                        It differs from the other edge-It differs from the other edge-

                                                        detection methods in that detection methods in that

                                                        it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                        strong and weak edges) strong and weak edges)

                                                        and includes the weak edges in the and includes the weak edges in the

                                                        output only if they are connected to output only if they are connected to

                                                        strong edges strong edges

                                                        This method is therefore less likely This method is therefore less likely

                                                        than the others to be fooled by than the others to be fooled by

                                                        noise and more likely to detect true noise and more likely to detect true

                                                        weak edgesweak edges

                                                        4848

                                                        Laplacian of GaussianLaplacian of Gaussian

                                                        Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                        2 2 22

                                                        4 2

                                                        2 2 2

                                                        2exp

                                                        r rh

                                                        r x y

                                                        determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                        4949

                                                        Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                        0 0 1 0 0

                                                        0 1 2 1 0

                                                        1 2 16 2 1

                                                        0 1 2 1 0

                                                        0 0 1 0 0

                                                        0 0 1 0 0

                                                        0 1 2 1 0

                                                        1 2 16 2 1

                                                        0 1 2 1 0

                                                        0 0 1 0 0

                                                        The coefficient must sum to The coefficient must sum to

                                                        zerozero

                                                        5050

                                                        Edge Detection and Edge Detection and SegmentationSegmentation

                                                        Image resulting from edge detection cannot be used as a segmentation result

                                                        Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                        5151

                                                        75 Region-based 75 Region-based SegmentationSegmentation

                                                        GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                        Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                        PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                        For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                        PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                        For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                        5252

                                                        Two methods of Region Two methods of Region SegmentationSegmentation

                                                        Region GrowingRegion Growing

                                                        Region SplittingRegion Splitting

                                                        Region growing is the opposite of the Region growing is the opposite of the

                                                        split and merge approachsplit and merge approach

                                                        5353

                                                        Region GrowingRegion Growing

                                                        The objective of segmentation is to The objective of segmentation is to

                                                        partition an image into regionspartition an image into regions

                                                        A region is a connected component with A region is a connected component with

                                                        some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                        texture)texture)

                                                        In region growing we start with a set In region growing we start with a set

                                                        of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                        appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                        pixels if they have pixels if they have similar propertiessimilar properties

                                                        such as specific ranges of gray level such as specific ranges of gray level

                                                        and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                        Need initialization Need initialization similarity similarity

                                                        criterioncriterion

                                                        5454

                                                        Steps of Region GrowingSteps of Region Growing

                                                        Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                        pixel andpixel and compare it with neighbor compare it with neighbor

                                                        ppixelsixels

                                                        When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                        is grown from the seed pixel by is grown from the seed pixel by

                                                        addingadding neighboneighborr pixel pixelss

                                                        When the growth of one region stopsWhen the growth of one region stops

                                                        choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                        againagain

                                                        5555

                                                        Region Region growing growing

                                                        An initial set of small An initial set of small

                                                        areas are iterativelyareas are iteratively

                                                        merged according to merged according to

                                                        similarity constraintssimilarity constraints

                                                        5656

                                                        Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                        X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                        We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                        On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                        Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                        SeedSeed pointspoints

                                                        5757

                                                        CriterionCriterion

                                                        There is a valley at around 190 in the There is a valley at around 190 in the

                                                        histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                        to be considered as a part of region to the to be considered as a part of region to the

                                                        seed pointseed point

                                                        The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                        to at least one pixel in that regionto at least one pixel in that region

                                                        Result of region growing and boundaries of Result of region growing and boundaries of

                                                        defectsdefects

                                                        5858

                                                        Region SplittingRegion Splitting

                                                        The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                        ption that the entire image is homogeneousption that the entire image is homogeneous

                                                        If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                        This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                        Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                        5959

                                                        Region SplittingRegion Splitting

                                                        DisadvantageDisadvantage

                                                        they create regions that may be adjacent they create regions that may be adjacent

                                                        and homogeneous but not mergedand homogeneous but not merged

                                                        6060

                                                        Region Splitting and MergingRegion Splitting and Merging

                                                        ProcedureProcedure

                                                        11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                        22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                        33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                        6161

                                                        Region Splitting and Merging

                                                        Quadtree

                                                        (四叉树 )

                                                        6262

                                                        PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                        where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                        mmii is the mean gray level of that region is the mean gray level of that region

                                                        σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                        ExampleExample

                                                        Original Original

                                                        imageimageThresholded imageThresholded image Result of Result of

                                                        Splitting and Splitting and

                                                        MergingMerging

                                                        • Slide 1
                                                        • Slide 2
                                                        • Slide 3
                                                        • Slide 4
                                                        • Slide 5
                                                        • Slide 6
                                                        • Slide 7
                                                        • Slide 8
                                                        • Slide 9
                                                        • Slide 10
                                                        • Slide 11
                                                        • Slide 12
                                                        • Slide 13
                                                        • Slide 14
                                                        • Slide 15
                                                        • Slide 16
                                                        • Slide 17
                                                        • Slide 18
                                                        • Slide 19
                                                        • Slide 20
                                                        • Slide 21
                                                        • Slide 22
                                                        • Slide 23
                                                        • Slide 24
                                                        • Slide 25
                                                        • Slide 26
                                                        • Slide 27
                                                        • Slide 28
                                                        • Slide 29
                                                        • Slide 30
                                                        • Slide 31
                                                        • Slide 32
                                                        • Slide 33
                                                        • Slide 34
                                                        • Slide 35
                                                        • Slide 36
                                                        • Slide 37
                                                        • Slide 38
                                                        • Slide 39
                                                        • Slide 40
                                                        • Slide 41
                                                        • Slide 42
                                                        • Slide 43
                                                        • Slide 44
                                                        • Slide 45
                                                        • Slide 46
                                                        • Slide 47
                                                        • Slide 48
                                                        • Slide 49
                                                        • Slide 50
                                                        • Slide 51
                                                        • Slide 52
                                                        • Slide 53
                                                        • Slide 54
                                                        • Slide 55
                                                        • Slide 56
                                                        • Slide 57
                                                        • Slide 58
                                                        • Slide 59
                                                        • Slide 60
                                                        • Slide 61
                                                        • Slide 62

                                                          2929

                                                          73 Line Detection73 Line Detection

                                                          Horizontal mask will result with max Horizontal mask will result with max

                                                          response when a line passed through the response when a line passed through the

                                                          middle row of the mask with a constant middle row of the mask with a constant

                                                          backgroundbackground

                                                          the similar idea is used with other masksthe similar idea is used with other masks

                                                          Note the preferred direction of each mask Note the preferred direction of each mask

                                                          is weighted with a larger coefficient (ie2) is weighted with a larger coefficient (ie2)

                                                          than other possible directionsthan other possible directions

                                                          1 1 1 1 1 2 1 2 1 2 1 1

                                                          2 2 2 1 2 1 1 2 1 1 2 1

                                                          1 1 1 2 1 1 1 2 1 1 1 2

                                                          45 45Horizontal Vertical

                                                          1 1 1 1 1 2 1 2 1 2 1 1

                                                          2 2 2 1 2 1 1 2 1 1 2 1

                                                          1 1 1 2 1 1 1 2 1 1 1 2

                                                          45 45Horizontal Vertical

                                                          3030

                                                          Idea 1 of Line DetectionIdea 1 of Line Detection

                                                          Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                                          izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                                          if at a certain point in the imageif at a certain point in the image

                                                          |Ri||Ri|gtgt|Rj||Rj|

                                                          for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                                          3131

                                                          Idea 2 of Line DetectionIdea 2 of Line Detection

                                                          Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                                          After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                                          3232

                                                          ExampleExample

                                                          3333

                                                          74 Edge-based 74 Edge-based SegmentationSegmentation

                                                          Edge-based segmentations rely on edges found in an image by edge detecting operators

                                                          these edges mark image locations of discontinuities in gray level

                                                          Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                                          There are a large group of methods based on information about edges in the image

                                                          3434

                                                          What is edgeWhat is edge

                                                          Edge is where change occurs Change is measured by derivative in 1D

                                                          ―Biggest change derivative has maximum magnitude

                                                          Or 2nd derivative is zero we discuss approaches for implementing

                                                          ―first-order derivative (Gradient operator)

                                                          ―second-order derivative (Laplacian operator)

                                                          ―we have introduced both derivatives in chapter 3

                                                          ―Here we will talk only about their properties for edge detection

                                                          3535

                                                          What is edgeWhat is edge

                                                          In other wordsIn other words an edge is a set of an edge is a set of

                                                          connected pixelsconnected pixels

                                                          that lie on the boundary between two that lie on the boundary between two

                                                          regions with relatively distinct gray-level regions with relatively distinct gray-level

                                                          propertiesproperties

                                                          Note edge vs boundaryNote edge vs boundary

                                                          ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                                          ―whereas a region boundary owing to whereas a region boundary owing to

                                                          the way it is defined is a more global the way it is defined is a more global

                                                          ideaidea

                                                          3636

                                                          Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                          because of because of

                                                          optics optics

                                                          sampling sampling

                                                          image image

                                                          acquisition acquisition

                                                          imperfectionimperfection

                                                          3737

                                                          Thick and Thin EdgeThick and Thin Edge

                                                          The slope of the ramp is inversely The slope of the ramp is inversely

                                                          proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                          edgeedge

                                                          Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                          pathpath

                                                          Instead an edge point now is any point Instead an edge point now is any point

                                                          contained in the ramp and contained in the ramp and an edge would an edge would

                                                          then be a set of such points that are then be a set of such points that are

                                                          connectedconnected

                                                          The thickness is determined by the length of the The thickness is determined by the length of the

                                                          rampramp

                                                          The length is determined by the slope which is in The length is determined by the slope which is in

                                                          turn determined by the degree of blurringturn determined by the degree of blurring

                                                          Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                          edges tend to be thinedges tend to be thin

                                                          3838

                                                          First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                          the signs of the the signs of the

                                                          derivatives would be derivatives would be

                                                          reversed for an edge reversed for an edge

                                                          that transitions from that transitions from

                                                          light to darklight to dark

                                                          First First derivatderivatee

                                                          SeconSecond d derivatderivatee

                                                          Gray-Gray-level level profileprofile

                                                          3939

                                                          Second derivativesSecond derivatives

                                                          an undesirable featurean undesirable feature

                                                          produces 2 values for every edge in an produces 2 values for every edge in an

                                                          imageimage

                                                          zero-crossing propertyzero-crossing property

                                                          an imaginary straight line joining the an imaginary straight line joining the

                                                          extreme positive and negative values of extreme positive and negative values of

                                                          the second derivative would cross zero the second derivative would cross zero

                                                          near the midpoint of the edgenear the midpoint of the edge

                                                          quite useful for locating the centers of quite useful for locating the centers of

                                                          thick edgesthick edges

                                                          4040

                                                          Basic idea of edge detectionBasic idea of edge detection

                                                          A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                          the edge direction and the results are the edge direction and the results are

                                                          interpretedinterpreted

                                                          The magnitude of the first derivative is The magnitude of the first derivative is

                                                          used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                          ramp)ramp)

                                                          The sign of the second derivative can The sign of the second derivative can

                                                          determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                          dark or light side of an edgedark or light side of an edge

                                                          4141

                                                          Review of First DerivateReview of First Derivate

                                                          Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                          Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                          Sobel operators 3Sobel operators 333

                                                          6 5 8 5x yG z z G z z

                                                          1 2 3

                                                          4 5 6

                                                          7 8 9

                                                          z z z

                                                          z z z

                                                          z z z

                                                          1 2 3

                                                          4 5 6

                                                          7 8 9

                                                          z z z

                                                          z z z

                                                          z z z

                                                          9 5 8 6x yG z z G z z 1 0 0 1

                                                          0 1 1 0

                                                          1 0 0 1

                                                          0 1 1 0

                                                          7 8 9 1 2 3

                                                          3 6 9 1 4 7

                                                          2 2

                                                          2 2

                                                          x

                                                          y

                                                          G z z z z z z

                                                          G z z z z z z

                                                          1 2 1 1 0 1

                                                          0 0 0 2 0 2

                                                          1 2 1 1 0 1

                                                          1 2 1 1 0 1

                                                          0 0 0 2 0 2

                                                          1 2 1 1 0 1

                                                          x yf G G

                                                          4242

                                                          Edge direction and strengthEdge direction and strength

                                                          Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                          α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                          The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                          The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                          2 2x yf G G

                                                          4343

                                                          Gradient MasksGradient Masks

                                                          1 0 0 1

                                                          0 1 1 0

                                                          Roberts

                                                          1 0 0 1

                                                          0 1 1 0

                                                          Roberts

                                                          1 2 1 1 0 1

                                                          0 0 0 2 0 2

                                                          1 2 1 1 0 1

                                                          Sobel

                                                          1 2 1 1 0 1

                                                          0 0 0 2 0 2

                                                          1 2 1 1 0 1

                                                          Sobel

                                                          1 1 1 1 0 1

                                                          0 0 0 1 0 1

                                                          1 1 1 1 0 1

                                                          Prewitt

                                                          1 1 1 1 0 1

                                                          0 0 0 1 0 1

                                                          1 1 1 1 0 1

                                                          Prewitt

                                                          4444

                                                          Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                          0 1 1 1 1 0

                                                          1 0 1 1 0 1

                                                          1 1 0 0 1 1

                                                          Prewitt

                                                          0 1 1 1 1 0

                                                          1 0 1 1 0 1

                                                          1 1 0 0 1 1

                                                          Prewitt

                                                          4545

                                                          Review of Second DerivateReview of Second Derivate

                                                          Laplacian OperatorLaplacian Operator

                                                          21 1

                                                          1 1 4

                                                          f x y f x yf

                                                          f x y f x y f x y

                                                          0 1 0

                                                          1 4 1

                                                          0 1 0

                                                          0 1 0

                                                          1 4 1

                                                          0 1 0

                                                          LaplacianLaplacian

                                                          MaskMask

                                                          1 1 1

                                                          1 8 1

                                                          1 1 1

                                                          1 1 1

                                                          1 8 1

                                                          1 1 1

                                                          4646

                                                          Example of edge detectionExample of edge detection

                                                          See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                          Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                          derivative)derivative)

                                                          produces double edgesproduces double edges

                                                          unable to detect edge directionunable to detect edge direction

                                                          4747

                                                          Canny edge detectorCanny edge detector

                                                          The most powerful edge-detection The most powerful edge-detection

                                                          method method

                                                          It differs from the other edge-It differs from the other edge-

                                                          detection methods in that detection methods in that

                                                          it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                          strong and weak edges) strong and weak edges)

                                                          and includes the weak edges in the and includes the weak edges in the

                                                          output only if they are connected to output only if they are connected to

                                                          strong edges strong edges

                                                          This method is therefore less likely This method is therefore less likely

                                                          than the others to be fooled by than the others to be fooled by

                                                          noise and more likely to detect true noise and more likely to detect true

                                                          weak edgesweak edges

                                                          4848

                                                          Laplacian of GaussianLaplacian of Gaussian

                                                          Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                          2 2 22

                                                          4 2

                                                          2 2 2

                                                          2exp

                                                          r rh

                                                          r x y

                                                          determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                          4949

                                                          Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                          0 0 1 0 0

                                                          0 1 2 1 0

                                                          1 2 16 2 1

                                                          0 1 2 1 0

                                                          0 0 1 0 0

                                                          0 0 1 0 0

                                                          0 1 2 1 0

                                                          1 2 16 2 1

                                                          0 1 2 1 0

                                                          0 0 1 0 0

                                                          The coefficient must sum to The coefficient must sum to

                                                          zerozero

                                                          5050

                                                          Edge Detection and Edge Detection and SegmentationSegmentation

                                                          Image resulting from edge detection cannot be used as a segmentation result

                                                          Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                          5151

                                                          75 Region-based 75 Region-based SegmentationSegmentation

                                                          GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                          Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                          PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                          For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                          PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                          For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                          5252

                                                          Two methods of Region Two methods of Region SegmentationSegmentation

                                                          Region GrowingRegion Growing

                                                          Region SplittingRegion Splitting

                                                          Region growing is the opposite of the Region growing is the opposite of the

                                                          split and merge approachsplit and merge approach

                                                          5353

                                                          Region GrowingRegion Growing

                                                          The objective of segmentation is to The objective of segmentation is to

                                                          partition an image into regionspartition an image into regions

                                                          A region is a connected component with A region is a connected component with

                                                          some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                          texture)texture)

                                                          In region growing we start with a set In region growing we start with a set

                                                          of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                          appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                          pixels if they have pixels if they have similar propertiessimilar properties

                                                          such as specific ranges of gray level such as specific ranges of gray level

                                                          and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                          Need initialization Need initialization similarity similarity

                                                          criterioncriterion

                                                          5454

                                                          Steps of Region GrowingSteps of Region Growing

                                                          Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                          pixel andpixel and compare it with neighbor compare it with neighbor

                                                          ppixelsixels

                                                          When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                          is grown from the seed pixel by is grown from the seed pixel by

                                                          addingadding neighboneighborr pixel pixelss

                                                          When the growth of one region stopsWhen the growth of one region stops

                                                          choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                          againagain

                                                          5555

                                                          Region Region growing growing

                                                          An initial set of small An initial set of small

                                                          areas are iterativelyareas are iteratively

                                                          merged according to merged according to

                                                          similarity constraintssimilarity constraints

                                                          5656

                                                          Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                          X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                          We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                          On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                          Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                          SeedSeed pointspoints

                                                          5757

                                                          CriterionCriterion

                                                          There is a valley at around 190 in the There is a valley at around 190 in the

                                                          histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                          to be considered as a part of region to the to be considered as a part of region to the

                                                          seed pointseed point

                                                          The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                          to at least one pixel in that regionto at least one pixel in that region

                                                          Result of region growing and boundaries of Result of region growing and boundaries of

                                                          defectsdefects

                                                          5858

                                                          Region SplittingRegion Splitting

                                                          The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                          ption that the entire image is homogeneousption that the entire image is homogeneous

                                                          If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                          This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                          Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                          5959

                                                          Region SplittingRegion Splitting

                                                          DisadvantageDisadvantage

                                                          they create regions that may be adjacent they create regions that may be adjacent

                                                          and homogeneous but not mergedand homogeneous but not merged

                                                          6060

                                                          Region Splitting and MergingRegion Splitting and Merging

                                                          ProcedureProcedure

                                                          11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                          22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                          33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                          6161

                                                          Region Splitting and Merging

                                                          Quadtree

                                                          (四叉树 )

                                                          6262

                                                          PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                          where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                          mmii is the mean gray level of that region is the mean gray level of that region

                                                          σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                          ExampleExample

                                                          Original Original

                                                          imageimageThresholded imageThresholded image Result of Result of

                                                          Splitting and Splitting and

                                                          MergingMerging

                                                          • Slide 1
                                                          • Slide 2
                                                          • Slide 3
                                                          • Slide 4
                                                          • Slide 5
                                                          • Slide 6
                                                          • Slide 7
                                                          • Slide 8
                                                          • Slide 9
                                                          • Slide 10
                                                          • Slide 11
                                                          • Slide 12
                                                          • Slide 13
                                                          • Slide 14
                                                          • Slide 15
                                                          • Slide 16
                                                          • Slide 17
                                                          • Slide 18
                                                          • Slide 19
                                                          • Slide 20
                                                          • Slide 21
                                                          • Slide 22
                                                          • Slide 23
                                                          • Slide 24
                                                          • Slide 25
                                                          • Slide 26
                                                          • Slide 27
                                                          • Slide 28
                                                          • Slide 29
                                                          • Slide 30
                                                          • Slide 31
                                                          • Slide 32
                                                          • Slide 33
                                                          • Slide 34
                                                          • Slide 35
                                                          • Slide 36
                                                          • Slide 37
                                                          • Slide 38
                                                          • Slide 39
                                                          • Slide 40
                                                          • Slide 41
                                                          • Slide 42
                                                          • Slide 43
                                                          • Slide 44
                                                          • Slide 45
                                                          • Slide 46
                                                          • Slide 47
                                                          • Slide 48
                                                          • Slide 49
                                                          • Slide 50
                                                          • Slide 51
                                                          • Slide 52
                                                          • Slide 53
                                                          • Slide 54
                                                          • Slide 55
                                                          • Slide 56
                                                          • Slide 57
                                                          • Slide 58
                                                          • Slide 59
                                                          • Slide 60
                                                          • Slide 61
                                                          • Slide 62

                                                            3030

                                                            Idea 1 of Line DetectionIdea 1 of Line Detection

                                                            Apply every masks on the imageApply every masks on the image let R1 R2 R3 R4 denotes the response of the horlet R1 R2 R3 R4 denotes the response of the hor

                                                            izontal +45 degree vertical and -45 degree maskizontal +45 degree vertical and -45 degree masks respectivelys respectively

                                                            if at a certain point in the imageif at a certain point in the image

                                                            |Ri||Ri|gtgt|Rj||Rj|

                                                            for all jnei that point is said to be more likelfor all jnei that point is said to be more likely associated with a line in the direction of my associated with a line in the direction of mask iask i

                                                            3131

                                                            Idea 2 of Line DetectionIdea 2 of Line Detection

                                                            Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                                            After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                                            3232

                                                            ExampleExample

                                                            3333

                                                            74 Edge-based 74 Edge-based SegmentationSegmentation

                                                            Edge-based segmentations rely on edges found in an image by edge detecting operators

                                                            these edges mark image locations of discontinuities in gray level

                                                            Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                                            There are a large group of methods based on information about edges in the image

                                                            3434

                                                            What is edgeWhat is edge

                                                            Edge is where change occurs Change is measured by derivative in 1D

                                                            ―Biggest change derivative has maximum magnitude

                                                            Or 2nd derivative is zero we discuss approaches for implementing

                                                            ―first-order derivative (Gradient operator)

                                                            ―second-order derivative (Laplacian operator)

                                                            ―we have introduced both derivatives in chapter 3

                                                            ―Here we will talk only about their properties for edge detection

                                                            3535

                                                            What is edgeWhat is edge

                                                            In other wordsIn other words an edge is a set of an edge is a set of

                                                            connected pixelsconnected pixels

                                                            that lie on the boundary between two that lie on the boundary between two

                                                            regions with relatively distinct gray-level regions with relatively distinct gray-level

                                                            propertiesproperties

                                                            Note edge vs boundaryNote edge vs boundary

                                                            ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                                            ―whereas a region boundary owing to whereas a region boundary owing to

                                                            the way it is defined is a more global the way it is defined is a more global

                                                            ideaidea

                                                            3636

                                                            Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                            because of because of

                                                            optics optics

                                                            sampling sampling

                                                            image image

                                                            acquisition acquisition

                                                            imperfectionimperfection

                                                            3737

                                                            Thick and Thin EdgeThick and Thin Edge

                                                            The slope of the ramp is inversely The slope of the ramp is inversely

                                                            proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                            edgeedge

                                                            Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                            pathpath

                                                            Instead an edge point now is any point Instead an edge point now is any point

                                                            contained in the ramp and contained in the ramp and an edge would an edge would

                                                            then be a set of such points that are then be a set of such points that are

                                                            connectedconnected

                                                            The thickness is determined by the length of the The thickness is determined by the length of the

                                                            rampramp

                                                            The length is determined by the slope which is in The length is determined by the slope which is in

                                                            turn determined by the degree of blurringturn determined by the degree of blurring

                                                            Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                            edges tend to be thinedges tend to be thin

                                                            3838

                                                            First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                            the signs of the the signs of the

                                                            derivatives would be derivatives would be

                                                            reversed for an edge reversed for an edge

                                                            that transitions from that transitions from

                                                            light to darklight to dark

                                                            First First derivatderivatee

                                                            SeconSecond d derivatderivatee

                                                            Gray-Gray-level level profileprofile

                                                            3939

                                                            Second derivativesSecond derivatives

                                                            an undesirable featurean undesirable feature

                                                            produces 2 values for every edge in an produces 2 values for every edge in an

                                                            imageimage

                                                            zero-crossing propertyzero-crossing property

                                                            an imaginary straight line joining the an imaginary straight line joining the

                                                            extreme positive and negative values of extreme positive and negative values of

                                                            the second derivative would cross zero the second derivative would cross zero

                                                            near the midpoint of the edgenear the midpoint of the edge

                                                            quite useful for locating the centers of quite useful for locating the centers of

                                                            thick edgesthick edges

                                                            4040

                                                            Basic idea of edge detectionBasic idea of edge detection

                                                            A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                            the edge direction and the results are the edge direction and the results are

                                                            interpretedinterpreted

                                                            The magnitude of the first derivative is The magnitude of the first derivative is

                                                            used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                            ramp)ramp)

                                                            The sign of the second derivative can The sign of the second derivative can

                                                            determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                            dark or light side of an edgedark or light side of an edge

                                                            4141

                                                            Review of First DerivateReview of First Derivate

                                                            Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                            Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                            Sobel operators 3Sobel operators 333

                                                            6 5 8 5x yG z z G z z

                                                            1 2 3

                                                            4 5 6

                                                            7 8 9

                                                            z z z

                                                            z z z

                                                            z z z

                                                            1 2 3

                                                            4 5 6

                                                            7 8 9

                                                            z z z

                                                            z z z

                                                            z z z

                                                            9 5 8 6x yG z z G z z 1 0 0 1

                                                            0 1 1 0

                                                            1 0 0 1

                                                            0 1 1 0

                                                            7 8 9 1 2 3

                                                            3 6 9 1 4 7

                                                            2 2

                                                            2 2

                                                            x

                                                            y

                                                            G z z z z z z

                                                            G z z z z z z

                                                            1 2 1 1 0 1

                                                            0 0 0 2 0 2

                                                            1 2 1 1 0 1

                                                            1 2 1 1 0 1

                                                            0 0 0 2 0 2

                                                            1 2 1 1 0 1

                                                            x yf G G

                                                            4242

                                                            Edge direction and strengthEdge direction and strength

                                                            Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                            α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                            The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                            The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                            2 2x yf G G

                                                            4343

                                                            Gradient MasksGradient Masks

                                                            1 0 0 1

                                                            0 1 1 0

                                                            Roberts

                                                            1 0 0 1

                                                            0 1 1 0

                                                            Roberts

                                                            1 2 1 1 0 1

                                                            0 0 0 2 0 2

                                                            1 2 1 1 0 1

                                                            Sobel

                                                            1 2 1 1 0 1

                                                            0 0 0 2 0 2

                                                            1 2 1 1 0 1

                                                            Sobel

                                                            1 1 1 1 0 1

                                                            0 0 0 1 0 1

                                                            1 1 1 1 0 1

                                                            Prewitt

                                                            1 1 1 1 0 1

                                                            0 0 0 1 0 1

                                                            1 1 1 1 0 1

                                                            Prewitt

                                                            4444

                                                            Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                            0 1 1 1 1 0

                                                            1 0 1 1 0 1

                                                            1 1 0 0 1 1

                                                            Prewitt

                                                            0 1 1 1 1 0

                                                            1 0 1 1 0 1

                                                            1 1 0 0 1 1

                                                            Prewitt

                                                            4545

                                                            Review of Second DerivateReview of Second Derivate

                                                            Laplacian OperatorLaplacian Operator

                                                            21 1

                                                            1 1 4

                                                            f x y f x yf

                                                            f x y f x y f x y

                                                            0 1 0

                                                            1 4 1

                                                            0 1 0

                                                            0 1 0

                                                            1 4 1

                                                            0 1 0

                                                            LaplacianLaplacian

                                                            MaskMask

                                                            1 1 1

                                                            1 8 1

                                                            1 1 1

                                                            1 1 1

                                                            1 8 1

                                                            1 1 1

                                                            4646

                                                            Example of edge detectionExample of edge detection

                                                            See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                            Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                            derivative)derivative)

                                                            produces double edgesproduces double edges

                                                            unable to detect edge directionunable to detect edge direction

                                                            4747

                                                            Canny edge detectorCanny edge detector

                                                            The most powerful edge-detection The most powerful edge-detection

                                                            method method

                                                            It differs from the other edge-It differs from the other edge-

                                                            detection methods in that detection methods in that

                                                            it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                            strong and weak edges) strong and weak edges)

                                                            and includes the weak edges in the and includes the weak edges in the

                                                            output only if they are connected to output only if they are connected to

                                                            strong edges strong edges

                                                            This method is therefore less likely This method is therefore less likely

                                                            than the others to be fooled by than the others to be fooled by

                                                            noise and more likely to detect true noise and more likely to detect true

                                                            weak edgesweak edges

                                                            4848

                                                            Laplacian of GaussianLaplacian of Gaussian

                                                            Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                            2 2 22

                                                            4 2

                                                            2 2 2

                                                            2exp

                                                            r rh

                                                            r x y

                                                            determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                            4949

                                                            Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                            0 0 1 0 0

                                                            0 1 2 1 0

                                                            1 2 16 2 1

                                                            0 1 2 1 0

                                                            0 0 1 0 0

                                                            0 0 1 0 0

                                                            0 1 2 1 0

                                                            1 2 16 2 1

                                                            0 1 2 1 0

                                                            0 0 1 0 0

                                                            The coefficient must sum to The coefficient must sum to

                                                            zerozero

                                                            5050

                                                            Edge Detection and Edge Detection and SegmentationSegmentation

                                                            Image resulting from edge detection cannot be used as a segmentation result

                                                            Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                            5151

                                                            75 Region-based 75 Region-based SegmentationSegmentation

                                                            GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                            Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                            PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                            For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                            PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                            For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                            5252

                                                            Two methods of Region Two methods of Region SegmentationSegmentation

                                                            Region GrowingRegion Growing

                                                            Region SplittingRegion Splitting

                                                            Region growing is the opposite of the Region growing is the opposite of the

                                                            split and merge approachsplit and merge approach

                                                            5353

                                                            Region GrowingRegion Growing

                                                            The objective of segmentation is to The objective of segmentation is to

                                                            partition an image into regionspartition an image into regions

                                                            A region is a connected component with A region is a connected component with

                                                            some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                            texture)texture)

                                                            In region growing we start with a set In region growing we start with a set

                                                            of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                            appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                            pixels if they have pixels if they have similar propertiessimilar properties

                                                            such as specific ranges of gray level such as specific ranges of gray level

                                                            and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                            Need initialization Need initialization similarity similarity

                                                            criterioncriterion

                                                            5454

                                                            Steps of Region GrowingSteps of Region Growing

                                                            Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                            pixel andpixel and compare it with neighbor compare it with neighbor

                                                            ppixelsixels

                                                            When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                            is grown from the seed pixel by is grown from the seed pixel by

                                                            addingadding neighboneighborr pixel pixelss

                                                            When the growth of one region stopsWhen the growth of one region stops

                                                            choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                            againagain

                                                            5555

                                                            Region Region growing growing

                                                            An initial set of small An initial set of small

                                                            areas are iterativelyareas are iteratively

                                                            merged according to merged according to

                                                            similarity constraintssimilarity constraints

                                                            5656

                                                            Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                            X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                            We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                            On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                            Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                            SeedSeed pointspoints

                                                            5757

                                                            CriterionCriterion

                                                            There is a valley at around 190 in the There is a valley at around 190 in the

                                                            histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                            to be considered as a part of region to the to be considered as a part of region to the

                                                            seed pointseed point

                                                            The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                            to at least one pixel in that regionto at least one pixel in that region

                                                            Result of region growing and boundaries of Result of region growing and boundaries of

                                                            defectsdefects

                                                            5858

                                                            Region SplittingRegion Splitting

                                                            The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                            ption that the entire image is homogeneousption that the entire image is homogeneous

                                                            If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                            This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                            Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                            5959

                                                            Region SplittingRegion Splitting

                                                            DisadvantageDisadvantage

                                                            they create regions that may be adjacent they create regions that may be adjacent

                                                            and homogeneous but not mergedand homogeneous but not merged

                                                            6060

                                                            Region Splitting and MergingRegion Splitting and Merging

                                                            ProcedureProcedure

                                                            11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                            22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                            33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                            6161

                                                            Region Splitting and Merging

                                                            Quadtree

                                                            (四叉树 )

                                                            6262

                                                            PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                            where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                            mmii is the mean gray level of that region is the mean gray level of that region

                                                            σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                            ExampleExample

                                                            Original Original

                                                            imageimageThresholded imageThresholded image Result of Result of

                                                            Splitting and Splitting and

                                                            MergingMerging

                                                            • Slide 1
                                                            • Slide 2
                                                            • Slide 3
                                                            • Slide 4
                                                            • Slide 5
                                                            • Slide 6
                                                            • Slide 7
                                                            • Slide 8
                                                            • Slide 9
                                                            • Slide 10
                                                            • Slide 11
                                                            • Slide 12
                                                            • Slide 13
                                                            • Slide 14
                                                            • Slide 15
                                                            • Slide 16
                                                            • Slide 17
                                                            • Slide 18
                                                            • Slide 19
                                                            • Slide 20
                                                            • Slide 21
                                                            • Slide 22
                                                            • Slide 23
                                                            • Slide 24
                                                            • Slide 25
                                                            • Slide 26
                                                            • Slide 27
                                                            • Slide 28
                                                            • Slide 29
                                                            • Slide 30
                                                            • Slide 31
                                                            • Slide 32
                                                            • Slide 33
                                                            • Slide 34
                                                            • Slide 35
                                                            • Slide 36
                                                            • Slide 37
                                                            • Slide 38
                                                            • Slide 39
                                                            • Slide 40
                                                            • Slide 41
                                                            • Slide 42
                                                            • Slide 43
                                                            • Slide 44
                                                            • Slide 45
                                                            • Slide 46
                                                            • Slide 47
                                                            • Slide 48
                                                            • Slide 49
                                                            • Slide 50
                                                            • Slide 51
                                                            • Slide 52
                                                            • Slide 53
                                                            • Slide 54
                                                            • Slide 55
                                                            • Slide 56
                                                            • Slide 57
                                                            • Slide 58
                                                            • Slide 59
                                                            • Slide 60
                                                            • Slide 61
                                                            • Slide 62

                                                              3131

                                                              Idea 2 of Line DetectionIdea 2 of Line Detection

                                                              Alternatively if we are interested in detectiAlternatively if we are interested in detecting all lines in an image in the direction definng all lines in an image in the direction defined by a given mask we simply run the mask ed by a given mask we simply run the mask through the image and threshold the absoluthrough the image and threshold the absolute value of the resultte value of the result

                                                              After thresholding the points that are left aAfter thresholding the points that are left are the strongest responses which for lines re the strongest responses which for lines one pixel thick correspond closest to the dione pixel thick correspond closest to the direction defined by the maskrection defined by the mask

                                                              3232

                                                              ExampleExample

                                                              3333

                                                              74 Edge-based 74 Edge-based SegmentationSegmentation

                                                              Edge-based segmentations rely on edges found in an image by edge detecting operators

                                                              these edges mark image locations of discontinuities in gray level

                                                              Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                                              There are a large group of methods based on information about edges in the image

                                                              3434

                                                              What is edgeWhat is edge

                                                              Edge is where change occurs Change is measured by derivative in 1D

                                                              ―Biggest change derivative has maximum magnitude

                                                              Or 2nd derivative is zero we discuss approaches for implementing

                                                              ―first-order derivative (Gradient operator)

                                                              ―second-order derivative (Laplacian operator)

                                                              ―we have introduced both derivatives in chapter 3

                                                              ―Here we will talk only about their properties for edge detection

                                                              3535

                                                              What is edgeWhat is edge

                                                              In other wordsIn other words an edge is a set of an edge is a set of

                                                              connected pixelsconnected pixels

                                                              that lie on the boundary between two that lie on the boundary between two

                                                              regions with relatively distinct gray-level regions with relatively distinct gray-level

                                                              propertiesproperties

                                                              Note edge vs boundaryNote edge vs boundary

                                                              ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                                              ―whereas a region boundary owing to whereas a region boundary owing to

                                                              the way it is defined is a more global the way it is defined is a more global

                                                              ideaidea

                                                              3636

                                                              Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                              because of because of

                                                              optics optics

                                                              sampling sampling

                                                              image image

                                                              acquisition acquisition

                                                              imperfectionimperfection

                                                              3737

                                                              Thick and Thin EdgeThick and Thin Edge

                                                              The slope of the ramp is inversely The slope of the ramp is inversely

                                                              proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                              edgeedge

                                                              Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                              pathpath

                                                              Instead an edge point now is any point Instead an edge point now is any point

                                                              contained in the ramp and contained in the ramp and an edge would an edge would

                                                              then be a set of such points that are then be a set of such points that are

                                                              connectedconnected

                                                              The thickness is determined by the length of the The thickness is determined by the length of the

                                                              rampramp

                                                              The length is determined by the slope which is in The length is determined by the slope which is in

                                                              turn determined by the degree of blurringturn determined by the degree of blurring

                                                              Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                              edges tend to be thinedges tend to be thin

                                                              3838

                                                              First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                              the signs of the the signs of the

                                                              derivatives would be derivatives would be

                                                              reversed for an edge reversed for an edge

                                                              that transitions from that transitions from

                                                              light to darklight to dark

                                                              First First derivatderivatee

                                                              SeconSecond d derivatderivatee

                                                              Gray-Gray-level level profileprofile

                                                              3939

                                                              Second derivativesSecond derivatives

                                                              an undesirable featurean undesirable feature

                                                              produces 2 values for every edge in an produces 2 values for every edge in an

                                                              imageimage

                                                              zero-crossing propertyzero-crossing property

                                                              an imaginary straight line joining the an imaginary straight line joining the

                                                              extreme positive and negative values of extreme positive and negative values of

                                                              the second derivative would cross zero the second derivative would cross zero

                                                              near the midpoint of the edgenear the midpoint of the edge

                                                              quite useful for locating the centers of quite useful for locating the centers of

                                                              thick edgesthick edges

                                                              4040

                                                              Basic idea of edge detectionBasic idea of edge detection

                                                              A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                              the edge direction and the results are the edge direction and the results are

                                                              interpretedinterpreted

                                                              The magnitude of the first derivative is The magnitude of the first derivative is

                                                              used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                              ramp)ramp)

                                                              The sign of the second derivative can The sign of the second derivative can

                                                              determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                              dark or light side of an edgedark or light side of an edge

                                                              4141

                                                              Review of First DerivateReview of First Derivate

                                                              Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                              Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                              Sobel operators 3Sobel operators 333

                                                              6 5 8 5x yG z z G z z

                                                              1 2 3

                                                              4 5 6

                                                              7 8 9

                                                              z z z

                                                              z z z

                                                              z z z

                                                              1 2 3

                                                              4 5 6

                                                              7 8 9

                                                              z z z

                                                              z z z

                                                              z z z

                                                              9 5 8 6x yG z z G z z 1 0 0 1

                                                              0 1 1 0

                                                              1 0 0 1

                                                              0 1 1 0

                                                              7 8 9 1 2 3

                                                              3 6 9 1 4 7

                                                              2 2

                                                              2 2

                                                              x

                                                              y

                                                              G z z z z z z

                                                              G z z z z z z

                                                              1 2 1 1 0 1

                                                              0 0 0 2 0 2

                                                              1 2 1 1 0 1

                                                              1 2 1 1 0 1

                                                              0 0 0 2 0 2

                                                              1 2 1 1 0 1

                                                              x yf G G

                                                              4242

                                                              Edge direction and strengthEdge direction and strength

                                                              Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                              α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                              The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                              The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                              2 2x yf G G

                                                              4343

                                                              Gradient MasksGradient Masks

                                                              1 0 0 1

                                                              0 1 1 0

                                                              Roberts

                                                              1 0 0 1

                                                              0 1 1 0

                                                              Roberts

                                                              1 2 1 1 0 1

                                                              0 0 0 2 0 2

                                                              1 2 1 1 0 1

                                                              Sobel

                                                              1 2 1 1 0 1

                                                              0 0 0 2 0 2

                                                              1 2 1 1 0 1

                                                              Sobel

                                                              1 1 1 1 0 1

                                                              0 0 0 1 0 1

                                                              1 1 1 1 0 1

                                                              Prewitt

                                                              1 1 1 1 0 1

                                                              0 0 0 1 0 1

                                                              1 1 1 1 0 1

                                                              Prewitt

                                                              4444

                                                              Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                              0 1 1 1 1 0

                                                              1 0 1 1 0 1

                                                              1 1 0 0 1 1

                                                              Prewitt

                                                              0 1 1 1 1 0

                                                              1 0 1 1 0 1

                                                              1 1 0 0 1 1

                                                              Prewitt

                                                              4545

                                                              Review of Second DerivateReview of Second Derivate

                                                              Laplacian OperatorLaplacian Operator

                                                              21 1

                                                              1 1 4

                                                              f x y f x yf

                                                              f x y f x y f x y

                                                              0 1 0

                                                              1 4 1

                                                              0 1 0

                                                              0 1 0

                                                              1 4 1

                                                              0 1 0

                                                              LaplacianLaplacian

                                                              MaskMask

                                                              1 1 1

                                                              1 8 1

                                                              1 1 1

                                                              1 1 1

                                                              1 8 1

                                                              1 1 1

                                                              4646

                                                              Example of edge detectionExample of edge detection

                                                              See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                              Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                              derivative)derivative)

                                                              produces double edgesproduces double edges

                                                              unable to detect edge directionunable to detect edge direction

                                                              4747

                                                              Canny edge detectorCanny edge detector

                                                              The most powerful edge-detection The most powerful edge-detection

                                                              method method

                                                              It differs from the other edge-It differs from the other edge-

                                                              detection methods in that detection methods in that

                                                              it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                              strong and weak edges) strong and weak edges)

                                                              and includes the weak edges in the and includes the weak edges in the

                                                              output only if they are connected to output only if they are connected to

                                                              strong edges strong edges

                                                              This method is therefore less likely This method is therefore less likely

                                                              than the others to be fooled by than the others to be fooled by

                                                              noise and more likely to detect true noise and more likely to detect true

                                                              weak edgesweak edges

                                                              4848

                                                              Laplacian of GaussianLaplacian of Gaussian

                                                              Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                              2 2 22

                                                              4 2

                                                              2 2 2

                                                              2exp

                                                              r rh

                                                              r x y

                                                              determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                              4949

                                                              Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                              0 0 1 0 0

                                                              0 1 2 1 0

                                                              1 2 16 2 1

                                                              0 1 2 1 0

                                                              0 0 1 0 0

                                                              0 0 1 0 0

                                                              0 1 2 1 0

                                                              1 2 16 2 1

                                                              0 1 2 1 0

                                                              0 0 1 0 0

                                                              The coefficient must sum to The coefficient must sum to

                                                              zerozero

                                                              5050

                                                              Edge Detection and Edge Detection and SegmentationSegmentation

                                                              Image resulting from edge detection cannot be used as a segmentation result

                                                              Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                              5151

                                                              75 Region-based 75 Region-based SegmentationSegmentation

                                                              GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                              Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                              PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                              For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                              PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                              For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                              5252

                                                              Two methods of Region Two methods of Region SegmentationSegmentation

                                                              Region GrowingRegion Growing

                                                              Region SplittingRegion Splitting

                                                              Region growing is the opposite of the Region growing is the opposite of the

                                                              split and merge approachsplit and merge approach

                                                              5353

                                                              Region GrowingRegion Growing

                                                              The objective of segmentation is to The objective of segmentation is to

                                                              partition an image into regionspartition an image into regions

                                                              A region is a connected component with A region is a connected component with

                                                              some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                              texture)texture)

                                                              In region growing we start with a set In region growing we start with a set

                                                              of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                              appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                              pixels if they have pixels if they have similar propertiessimilar properties

                                                              such as specific ranges of gray level such as specific ranges of gray level

                                                              and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                              Need initialization Need initialization similarity similarity

                                                              criterioncriterion

                                                              5454

                                                              Steps of Region GrowingSteps of Region Growing

                                                              Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                              pixel andpixel and compare it with neighbor compare it with neighbor

                                                              ppixelsixels

                                                              When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                              is grown from the seed pixel by is grown from the seed pixel by

                                                              addingadding neighboneighborr pixel pixelss

                                                              When the growth of one region stopsWhen the growth of one region stops

                                                              choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                              againagain

                                                              5555

                                                              Region Region growing growing

                                                              An initial set of small An initial set of small

                                                              areas are iterativelyareas are iteratively

                                                              merged according to merged according to

                                                              similarity constraintssimilarity constraints

                                                              5656

                                                              Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                              X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                              We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                              On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                              Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                              SeedSeed pointspoints

                                                              5757

                                                              CriterionCriterion

                                                              There is a valley at around 190 in the There is a valley at around 190 in the

                                                              histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                              to be considered as a part of region to the to be considered as a part of region to the

                                                              seed pointseed point

                                                              The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                              to at least one pixel in that regionto at least one pixel in that region

                                                              Result of region growing and boundaries of Result of region growing and boundaries of

                                                              defectsdefects

                                                              5858

                                                              Region SplittingRegion Splitting

                                                              The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                              ption that the entire image is homogeneousption that the entire image is homogeneous

                                                              If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                              This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                              Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                              5959

                                                              Region SplittingRegion Splitting

                                                              DisadvantageDisadvantage

                                                              they create regions that may be adjacent they create regions that may be adjacent

                                                              and homogeneous but not mergedand homogeneous but not merged

                                                              6060

                                                              Region Splitting and MergingRegion Splitting and Merging

                                                              ProcedureProcedure

                                                              11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                              22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                              33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                              6161

                                                              Region Splitting and Merging

                                                              Quadtree

                                                              (四叉树 )

                                                              6262

                                                              PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                              where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                              mmii is the mean gray level of that region is the mean gray level of that region

                                                              σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                              ExampleExample

                                                              Original Original

                                                              imageimageThresholded imageThresholded image Result of Result of

                                                              Splitting and Splitting and

                                                              MergingMerging

                                                              • Slide 1
                                                              • Slide 2
                                                              • Slide 3
                                                              • Slide 4
                                                              • Slide 5
                                                              • Slide 6
                                                              • Slide 7
                                                              • Slide 8
                                                              • Slide 9
                                                              • Slide 10
                                                              • Slide 11
                                                              • Slide 12
                                                              • Slide 13
                                                              • Slide 14
                                                              • Slide 15
                                                              • Slide 16
                                                              • Slide 17
                                                              • Slide 18
                                                              • Slide 19
                                                              • Slide 20
                                                              • Slide 21
                                                              • Slide 22
                                                              • Slide 23
                                                              • Slide 24
                                                              • Slide 25
                                                              • Slide 26
                                                              • Slide 27
                                                              • Slide 28
                                                              • Slide 29
                                                              • Slide 30
                                                              • Slide 31
                                                              • Slide 32
                                                              • Slide 33
                                                              • Slide 34
                                                              • Slide 35
                                                              • Slide 36
                                                              • Slide 37
                                                              • Slide 38
                                                              • Slide 39
                                                              • Slide 40
                                                              • Slide 41
                                                              • Slide 42
                                                              • Slide 43
                                                              • Slide 44
                                                              • Slide 45
                                                              • Slide 46
                                                              • Slide 47
                                                              • Slide 48
                                                              • Slide 49
                                                              • Slide 50
                                                              • Slide 51
                                                              • Slide 52
                                                              • Slide 53
                                                              • Slide 54
                                                              • Slide 55
                                                              • Slide 56
                                                              • Slide 57
                                                              • Slide 58
                                                              • Slide 59
                                                              • Slide 60
                                                              • Slide 61
                                                              • Slide 62

                                                                3232

                                                                ExampleExample

                                                                3333

                                                                74 Edge-based 74 Edge-based SegmentationSegmentation

                                                                Edge-based segmentations rely on edges found in an image by edge detecting operators

                                                                these edges mark image locations of discontinuities in gray level

                                                                Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                                                There are a large group of methods based on information about edges in the image

                                                                3434

                                                                What is edgeWhat is edge

                                                                Edge is where change occurs Change is measured by derivative in 1D

                                                                ―Biggest change derivative has maximum magnitude

                                                                Or 2nd derivative is zero we discuss approaches for implementing

                                                                ―first-order derivative (Gradient operator)

                                                                ―second-order derivative (Laplacian operator)

                                                                ―we have introduced both derivatives in chapter 3

                                                                ―Here we will talk only about their properties for edge detection

                                                                3535

                                                                What is edgeWhat is edge

                                                                In other wordsIn other words an edge is a set of an edge is a set of

                                                                connected pixelsconnected pixels

                                                                that lie on the boundary between two that lie on the boundary between two

                                                                regions with relatively distinct gray-level regions with relatively distinct gray-level

                                                                propertiesproperties

                                                                Note edge vs boundaryNote edge vs boundary

                                                                ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                                                ―whereas a region boundary owing to whereas a region boundary owing to

                                                                the way it is defined is a more global the way it is defined is a more global

                                                                ideaidea

                                                                3636

                                                                Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                                because of because of

                                                                optics optics

                                                                sampling sampling

                                                                image image

                                                                acquisition acquisition

                                                                imperfectionimperfection

                                                                3737

                                                                Thick and Thin EdgeThick and Thin Edge

                                                                The slope of the ramp is inversely The slope of the ramp is inversely

                                                                proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                                edgeedge

                                                                Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                                pathpath

                                                                Instead an edge point now is any point Instead an edge point now is any point

                                                                contained in the ramp and contained in the ramp and an edge would an edge would

                                                                then be a set of such points that are then be a set of such points that are

                                                                connectedconnected

                                                                The thickness is determined by the length of the The thickness is determined by the length of the

                                                                rampramp

                                                                The length is determined by the slope which is in The length is determined by the slope which is in

                                                                turn determined by the degree of blurringturn determined by the degree of blurring

                                                                Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                                edges tend to be thinedges tend to be thin

                                                                3838

                                                                First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                                the signs of the the signs of the

                                                                derivatives would be derivatives would be

                                                                reversed for an edge reversed for an edge

                                                                that transitions from that transitions from

                                                                light to darklight to dark

                                                                First First derivatderivatee

                                                                SeconSecond d derivatderivatee

                                                                Gray-Gray-level level profileprofile

                                                                3939

                                                                Second derivativesSecond derivatives

                                                                an undesirable featurean undesirable feature

                                                                produces 2 values for every edge in an produces 2 values for every edge in an

                                                                imageimage

                                                                zero-crossing propertyzero-crossing property

                                                                an imaginary straight line joining the an imaginary straight line joining the

                                                                extreme positive and negative values of extreme positive and negative values of

                                                                the second derivative would cross zero the second derivative would cross zero

                                                                near the midpoint of the edgenear the midpoint of the edge

                                                                quite useful for locating the centers of quite useful for locating the centers of

                                                                thick edgesthick edges

                                                                4040

                                                                Basic idea of edge detectionBasic idea of edge detection

                                                                A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                                the edge direction and the results are the edge direction and the results are

                                                                interpretedinterpreted

                                                                The magnitude of the first derivative is The magnitude of the first derivative is

                                                                used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                                ramp)ramp)

                                                                The sign of the second derivative can The sign of the second derivative can

                                                                determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                                dark or light side of an edgedark or light side of an edge

                                                                4141

                                                                Review of First DerivateReview of First Derivate

                                                                Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                                Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                                Sobel operators 3Sobel operators 333

                                                                6 5 8 5x yG z z G z z

                                                                1 2 3

                                                                4 5 6

                                                                7 8 9

                                                                z z z

                                                                z z z

                                                                z z z

                                                                1 2 3

                                                                4 5 6

                                                                7 8 9

                                                                z z z

                                                                z z z

                                                                z z z

                                                                9 5 8 6x yG z z G z z 1 0 0 1

                                                                0 1 1 0

                                                                1 0 0 1

                                                                0 1 1 0

                                                                7 8 9 1 2 3

                                                                3 6 9 1 4 7

                                                                2 2

                                                                2 2

                                                                x

                                                                y

                                                                G z z z z z z

                                                                G z z z z z z

                                                                1 2 1 1 0 1

                                                                0 0 0 2 0 2

                                                                1 2 1 1 0 1

                                                                1 2 1 1 0 1

                                                                0 0 0 2 0 2

                                                                1 2 1 1 0 1

                                                                x yf G G

                                                                4242

                                                                Edge direction and strengthEdge direction and strength

                                                                Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                                α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                                The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                                The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                                2 2x yf G G

                                                                4343

                                                                Gradient MasksGradient Masks

                                                                1 0 0 1

                                                                0 1 1 0

                                                                Roberts

                                                                1 0 0 1

                                                                0 1 1 0

                                                                Roberts

                                                                1 2 1 1 0 1

                                                                0 0 0 2 0 2

                                                                1 2 1 1 0 1

                                                                Sobel

                                                                1 2 1 1 0 1

                                                                0 0 0 2 0 2

                                                                1 2 1 1 0 1

                                                                Sobel

                                                                1 1 1 1 0 1

                                                                0 0 0 1 0 1

                                                                1 1 1 1 0 1

                                                                Prewitt

                                                                1 1 1 1 0 1

                                                                0 0 0 1 0 1

                                                                1 1 1 1 0 1

                                                                Prewitt

                                                                4444

                                                                Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                0 1 1 1 1 0

                                                                1 0 1 1 0 1

                                                                1 1 0 0 1 1

                                                                Prewitt

                                                                0 1 1 1 1 0

                                                                1 0 1 1 0 1

                                                                1 1 0 0 1 1

                                                                Prewitt

                                                                4545

                                                                Review of Second DerivateReview of Second Derivate

                                                                Laplacian OperatorLaplacian Operator

                                                                21 1

                                                                1 1 4

                                                                f x y f x yf

                                                                f x y f x y f x y

                                                                0 1 0

                                                                1 4 1

                                                                0 1 0

                                                                0 1 0

                                                                1 4 1

                                                                0 1 0

                                                                LaplacianLaplacian

                                                                MaskMask

                                                                1 1 1

                                                                1 8 1

                                                                1 1 1

                                                                1 1 1

                                                                1 8 1

                                                                1 1 1

                                                                4646

                                                                Example of edge detectionExample of edge detection

                                                                See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                derivative)derivative)

                                                                produces double edgesproduces double edges

                                                                unable to detect edge directionunable to detect edge direction

                                                                4747

                                                                Canny edge detectorCanny edge detector

                                                                The most powerful edge-detection The most powerful edge-detection

                                                                method method

                                                                It differs from the other edge-It differs from the other edge-

                                                                detection methods in that detection methods in that

                                                                it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                strong and weak edges) strong and weak edges)

                                                                and includes the weak edges in the and includes the weak edges in the

                                                                output only if they are connected to output only if they are connected to

                                                                strong edges strong edges

                                                                This method is therefore less likely This method is therefore less likely

                                                                than the others to be fooled by than the others to be fooled by

                                                                noise and more likely to detect true noise and more likely to detect true

                                                                weak edgesweak edges

                                                                4848

                                                                Laplacian of GaussianLaplacian of Gaussian

                                                                Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                2 2 22

                                                                4 2

                                                                2 2 2

                                                                2exp

                                                                r rh

                                                                r x y

                                                                determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                4949

                                                                Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                0 0 1 0 0

                                                                0 1 2 1 0

                                                                1 2 16 2 1

                                                                0 1 2 1 0

                                                                0 0 1 0 0

                                                                0 0 1 0 0

                                                                0 1 2 1 0

                                                                1 2 16 2 1

                                                                0 1 2 1 0

                                                                0 0 1 0 0

                                                                The coefficient must sum to The coefficient must sum to

                                                                zerozero

                                                                5050

                                                                Edge Detection and Edge Detection and SegmentationSegmentation

                                                                Image resulting from edge detection cannot be used as a segmentation result

                                                                Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                5151

                                                                75 Region-based 75 Region-based SegmentationSegmentation

                                                                GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                5252

                                                                Two methods of Region Two methods of Region SegmentationSegmentation

                                                                Region GrowingRegion Growing

                                                                Region SplittingRegion Splitting

                                                                Region growing is the opposite of the Region growing is the opposite of the

                                                                split and merge approachsplit and merge approach

                                                                5353

                                                                Region GrowingRegion Growing

                                                                The objective of segmentation is to The objective of segmentation is to

                                                                partition an image into regionspartition an image into regions

                                                                A region is a connected component with A region is a connected component with

                                                                some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                texture)texture)

                                                                In region growing we start with a set In region growing we start with a set

                                                                of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                pixels if they have pixels if they have similar propertiessimilar properties

                                                                such as specific ranges of gray level such as specific ranges of gray level

                                                                and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                Need initialization Need initialization similarity similarity

                                                                criterioncriterion

                                                                5454

                                                                Steps of Region GrowingSteps of Region Growing

                                                                Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                pixel andpixel and compare it with neighbor compare it with neighbor

                                                                ppixelsixels

                                                                When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                is grown from the seed pixel by is grown from the seed pixel by

                                                                addingadding neighboneighborr pixel pixelss

                                                                When the growth of one region stopsWhen the growth of one region stops

                                                                choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                againagain

                                                                5555

                                                                Region Region growing growing

                                                                An initial set of small An initial set of small

                                                                areas are iterativelyareas are iteratively

                                                                merged according to merged according to

                                                                similarity constraintssimilarity constraints

                                                                5656

                                                                Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                SeedSeed pointspoints

                                                                5757

                                                                CriterionCriterion

                                                                There is a valley at around 190 in the There is a valley at around 190 in the

                                                                histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                to be considered as a part of region to the to be considered as a part of region to the

                                                                seed pointseed point

                                                                The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                to at least one pixel in that regionto at least one pixel in that region

                                                                Result of region growing and boundaries of Result of region growing and boundaries of

                                                                defectsdefects

                                                                5858

                                                                Region SplittingRegion Splitting

                                                                The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                5959

                                                                Region SplittingRegion Splitting

                                                                DisadvantageDisadvantage

                                                                they create regions that may be adjacent they create regions that may be adjacent

                                                                and homogeneous but not mergedand homogeneous but not merged

                                                                6060

                                                                Region Splitting and MergingRegion Splitting and Merging

                                                                ProcedureProcedure

                                                                11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                6161

                                                                Region Splitting and Merging

                                                                Quadtree

                                                                (四叉树 )

                                                                6262

                                                                PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                mmii is the mean gray level of that region is the mean gray level of that region

                                                                σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                ExampleExample

                                                                Original Original

                                                                imageimageThresholded imageThresholded image Result of Result of

                                                                Splitting and Splitting and

                                                                MergingMerging

                                                                • Slide 1
                                                                • Slide 2
                                                                • Slide 3
                                                                • Slide 4
                                                                • Slide 5
                                                                • Slide 6
                                                                • Slide 7
                                                                • Slide 8
                                                                • Slide 9
                                                                • Slide 10
                                                                • Slide 11
                                                                • Slide 12
                                                                • Slide 13
                                                                • Slide 14
                                                                • Slide 15
                                                                • Slide 16
                                                                • Slide 17
                                                                • Slide 18
                                                                • Slide 19
                                                                • Slide 20
                                                                • Slide 21
                                                                • Slide 22
                                                                • Slide 23
                                                                • Slide 24
                                                                • Slide 25
                                                                • Slide 26
                                                                • Slide 27
                                                                • Slide 28
                                                                • Slide 29
                                                                • Slide 30
                                                                • Slide 31
                                                                • Slide 32
                                                                • Slide 33
                                                                • Slide 34
                                                                • Slide 35
                                                                • Slide 36
                                                                • Slide 37
                                                                • Slide 38
                                                                • Slide 39
                                                                • Slide 40
                                                                • Slide 41
                                                                • Slide 42
                                                                • Slide 43
                                                                • Slide 44
                                                                • Slide 45
                                                                • Slide 46
                                                                • Slide 47
                                                                • Slide 48
                                                                • Slide 49
                                                                • Slide 50
                                                                • Slide 51
                                                                • Slide 52
                                                                • Slide 53
                                                                • Slide 54
                                                                • Slide 55
                                                                • Slide 56
                                                                • Slide 57
                                                                • Slide 58
                                                                • Slide 59
                                                                • Slide 60
                                                                • Slide 61
                                                                • Slide 62

                                                                  3333

                                                                  74 Edge-based 74 Edge-based SegmentationSegmentation

                                                                  Edge-based segmentations rely on edges found in an image by edge detecting operators

                                                                  these edges mark image locations of discontinuities in gray level

                                                                  Edge detection is the most common approach for detecting meaningful discontinuities in gray level

                                                                  There are a large group of methods based on information about edges in the image

                                                                  3434

                                                                  What is edgeWhat is edge

                                                                  Edge is where change occurs Change is measured by derivative in 1D

                                                                  ―Biggest change derivative has maximum magnitude

                                                                  Or 2nd derivative is zero we discuss approaches for implementing

                                                                  ―first-order derivative (Gradient operator)

                                                                  ―second-order derivative (Laplacian operator)

                                                                  ―we have introduced both derivatives in chapter 3

                                                                  ―Here we will talk only about their properties for edge detection

                                                                  3535

                                                                  What is edgeWhat is edge

                                                                  In other wordsIn other words an edge is a set of an edge is a set of

                                                                  connected pixelsconnected pixels

                                                                  that lie on the boundary between two that lie on the boundary between two

                                                                  regions with relatively distinct gray-level regions with relatively distinct gray-level

                                                                  propertiesproperties

                                                                  Note edge vs boundaryNote edge vs boundary

                                                                  ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                                                  ―whereas a region boundary owing to whereas a region boundary owing to

                                                                  the way it is defined is a more global the way it is defined is a more global

                                                                  ideaidea

                                                                  3636

                                                                  Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                                  because of because of

                                                                  optics optics

                                                                  sampling sampling

                                                                  image image

                                                                  acquisition acquisition

                                                                  imperfectionimperfection

                                                                  3737

                                                                  Thick and Thin EdgeThick and Thin Edge

                                                                  The slope of the ramp is inversely The slope of the ramp is inversely

                                                                  proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                                  edgeedge

                                                                  Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                                  pathpath

                                                                  Instead an edge point now is any point Instead an edge point now is any point

                                                                  contained in the ramp and contained in the ramp and an edge would an edge would

                                                                  then be a set of such points that are then be a set of such points that are

                                                                  connectedconnected

                                                                  The thickness is determined by the length of the The thickness is determined by the length of the

                                                                  rampramp

                                                                  The length is determined by the slope which is in The length is determined by the slope which is in

                                                                  turn determined by the degree of blurringturn determined by the degree of blurring

                                                                  Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                                  edges tend to be thinedges tend to be thin

                                                                  3838

                                                                  First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                                  the signs of the the signs of the

                                                                  derivatives would be derivatives would be

                                                                  reversed for an edge reversed for an edge

                                                                  that transitions from that transitions from

                                                                  light to darklight to dark

                                                                  First First derivatderivatee

                                                                  SeconSecond d derivatderivatee

                                                                  Gray-Gray-level level profileprofile

                                                                  3939

                                                                  Second derivativesSecond derivatives

                                                                  an undesirable featurean undesirable feature

                                                                  produces 2 values for every edge in an produces 2 values for every edge in an

                                                                  imageimage

                                                                  zero-crossing propertyzero-crossing property

                                                                  an imaginary straight line joining the an imaginary straight line joining the

                                                                  extreme positive and negative values of extreme positive and negative values of

                                                                  the second derivative would cross zero the second derivative would cross zero

                                                                  near the midpoint of the edgenear the midpoint of the edge

                                                                  quite useful for locating the centers of quite useful for locating the centers of

                                                                  thick edgesthick edges

                                                                  4040

                                                                  Basic idea of edge detectionBasic idea of edge detection

                                                                  A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                                  the edge direction and the results are the edge direction and the results are

                                                                  interpretedinterpreted

                                                                  The magnitude of the first derivative is The magnitude of the first derivative is

                                                                  used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                                  ramp)ramp)

                                                                  The sign of the second derivative can The sign of the second derivative can

                                                                  determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                                  dark or light side of an edgedark or light side of an edge

                                                                  4141

                                                                  Review of First DerivateReview of First Derivate

                                                                  Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                                  Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                                  Sobel operators 3Sobel operators 333

                                                                  6 5 8 5x yG z z G z z

                                                                  1 2 3

                                                                  4 5 6

                                                                  7 8 9

                                                                  z z z

                                                                  z z z

                                                                  z z z

                                                                  1 2 3

                                                                  4 5 6

                                                                  7 8 9

                                                                  z z z

                                                                  z z z

                                                                  z z z

                                                                  9 5 8 6x yG z z G z z 1 0 0 1

                                                                  0 1 1 0

                                                                  1 0 0 1

                                                                  0 1 1 0

                                                                  7 8 9 1 2 3

                                                                  3 6 9 1 4 7

                                                                  2 2

                                                                  2 2

                                                                  x

                                                                  y

                                                                  G z z z z z z

                                                                  G z z z z z z

                                                                  1 2 1 1 0 1

                                                                  0 0 0 2 0 2

                                                                  1 2 1 1 0 1

                                                                  1 2 1 1 0 1

                                                                  0 0 0 2 0 2

                                                                  1 2 1 1 0 1

                                                                  x yf G G

                                                                  4242

                                                                  Edge direction and strengthEdge direction and strength

                                                                  Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                                  α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                                  The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                                  The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                                  2 2x yf G G

                                                                  4343

                                                                  Gradient MasksGradient Masks

                                                                  1 0 0 1

                                                                  0 1 1 0

                                                                  Roberts

                                                                  1 0 0 1

                                                                  0 1 1 0

                                                                  Roberts

                                                                  1 2 1 1 0 1

                                                                  0 0 0 2 0 2

                                                                  1 2 1 1 0 1

                                                                  Sobel

                                                                  1 2 1 1 0 1

                                                                  0 0 0 2 0 2

                                                                  1 2 1 1 0 1

                                                                  Sobel

                                                                  1 1 1 1 0 1

                                                                  0 0 0 1 0 1

                                                                  1 1 1 1 0 1

                                                                  Prewitt

                                                                  1 1 1 1 0 1

                                                                  0 0 0 1 0 1

                                                                  1 1 1 1 0 1

                                                                  Prewitt

                                                                  4444

                                                                  Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                  0 1 1 1 1 0

                                                                  1 0 1 1 0 1

                                                                  1 1 0 0 1 1

                                                                  Prewitt

                                                                  0 1 1 1 1 0

                                                                  1 0 1 1 0 1

                                                                  1 1 0 0 1 1

                                                                  Prewitt

                                                                  4545

                                                                  Review of Second DerivateReview of Second Derivate

                                                                  Laplacian OperatorLaplacian Operator

                                                                  21 1

                                                                  1 1 4

                                                                  f x y f x yf

                                                                  f x y f x y f x y

                                                                  0 1 0

                                                                  1 4 1

                                                                  0 1 0

                                                                  0 1 0

                                                                  1 4 1

                                                                  0 1 0

                                                                  LaplacianLaplacian

                                                                  MaskMask

                                                                  1 1 1

                                                                  1 8 1

                                                                  1 1 1

                                                                  1 1 1

                                                                  1 8 1

                                                                  1 1 1

                                                                  4646

                                                                  Example of edge detectionExample of edge detection

                                                                  See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                  Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                  derivative)derivative)

                                                                  produces double edgesproduces double edges

                                                                  unable to detect edge directionunable to detect edge direction

                                                                  4747

                                                                  Canny edge detectorCanny edge detector

                                                                  The most powerful edge-detection The most powerful edge-detection

                                                                  method method

                                                                  It differs from the other edge-It differs from the other edge-

                                                                  detection methods in that detection methods in that

                                                                  it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                  strong and weak edges) strong and weak edges)

                                                                  and includes the weak edges in the and includes the weak edges in the

                                                                  output only if they are connected to output only if they are connected to

                                                                  strong edges strong edges

                                                                  This method is therefore less likely This method is therefore less likely

                                                                  than the others to be fooled by than the others to be fooled by

                                                                  noise and more likely to detect true noise and more likely to detect true

                                                                  weak edgesweak edges

                                                                  4848

                                                                  Laplacian of GaussianLaplacian of Gaussian

                                                                  Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                  2 2 22

                                                                  4 2

                                                                  2 2 2

                                                                  2exp

                                                                  r rh

                                                                  r x y

                                                                  determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                  4949

                                                                  Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                  0 0 1 0 0

                                                                  0 1 2 1 0

                                                                  1 2 16 2 1

                                                                  0 1 2 1 0

                                                                  0 0 1 0 0

                                                                  0 0 1 0 0

                                                                  0 1 2 1 0

                                                                  1 2 16 2 1

                                                                  0 1 2 1 0

                                                                  0 0 1 0 0

                                                                  The coefficient must sum to The coefficient must sum to

                                                                  zerozero

                                                                  5050

                                                                  Edge Detection and Edge Detection and SegmentationSegmentation

                                                                  Image resulting from edge detection cannot be used as a segmentation result

                                                                  Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                  5151

                                                                  75 Region-based 75 Region-based SegmentationSegmentation

                                                                  GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                  Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                  PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                  For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                  PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                  For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                  5252

                                                                  Two methods of Region Two methods of Region SegmentationSegmentation

                                                                  Region GrowingRegion Growing

                                                                  Region SplittingRegion Splitting

                                                                  Region growing is the opposite of the Region growing is the opposite of the

                                                                  split and merge approachsplit and merge approach

                                                                  5353

                                                                  Region GrowingRegion Growing

                                                                  The objective of segmentation is to The objective of segmentation is to

                                                                  partition an image into regionspartition an image into regions

                                                                  A region is a connected component with A region is a connected component with

                                                                  some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                  texture)texture)

                                                                  In region growing we start with a set In region growing we start with a set

                                                                  of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                  appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                  pixels if they have pixels if they have similar propertiessimilar properties

                                                                  such as specific ranges of gray level such as specific ranges of gray level

                                                                  and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                  Need initialization Need initialization similarity similarity

                                                                  criterioncriterion

                                                                  5454

                                                                  Steps of Region GrowingSteps of Region Growing

                                                                  Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                  pixel andpixel and compare it with neighbor compare it with neighbor

                                                                  ppixelsixels

                                                                  When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                  is grown from the seed pixel by is grown from the seed pixel by

                                                                  addingadding neighboneighborr pixel pixelss

                                                                  When the growth of one region stopsWhen the growth of one region stops

                                                                  choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                  againagain

                                                                  5555

                                                                  Region Region growing growing

                                                                  An initial set of small An initial set of small

                                                                  areas are iterativelyareas are iteratively

                                                                  merged according to merged according to

                                                                  similarity constraintssimilarity constraints

                                                                  5656

                                                                  Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                  X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                  We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                  On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                  Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                  SeedSeed pointspoints

                                                                  5757

                                                                  CriterionCriterion

                                                                  There is a valley at around 190 in the There is a valley at around 190 in the

                                                                  histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                  to be considered as a part of region to the to be considered as a part of region to the

                                                                  seed pointseed point

                                                                  The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                  to at least one pixel in that regionto at least one pixel in that region

                                                                  Result of region growing and boundaries of Result of region growing and boundaries of

                                                                  defectsdefects

                                                                  5858

                                                                  Region SplittingRegion Splitting

                                                                  The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                  ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                  If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                  This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                  Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                  5959

                                                                  Region SplittingRegion Splitting

                                                                  DisadvantageDisadvantage

                                                                  they create regions that may be adjacent they create regions that may be adjacent

                                                                  and homogeneous but not mergedand homogeneous but not merged

                                                                  6060

                                                                  Region Splitting and MergingRegion Splitting and Merging

                                                                  ProcedureProcedure

                                                                  11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                  22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                  33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                  6161

                                                                  Region Splitting and Merging

                                                                  Quadtree

                                                                  (四叉树 )

                                                                  6262

                                                                  PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                  where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                  mmii is the mean gray level of that region is the mean gray level of that region

                                                                  σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                  ExampleExample

                                                                  Original Original

                                                                  imageimageThresholded imageThresholded image Result of Result of

                                                                  Splitting and Splitting and

                                                                  MergingMerging

                                                                  • Slide 1
                                                                  • Slide 2
                                                                  • Slide 3
                                                                  • Slide 4
                                                                  • Slide 5
                                                                  • Slide 6
                                                                  • Slide 7
                                                                  • Slide 8
                                                                  • Slide 9
                                                                  • Slide 10
                                                                  • Slide 11
                                                                  • Slide 12
                                                                  • Slide 13
                                                                  • Slide 14
                                                                  • Slide 15
                                                                  • Slide 16
                                                                  • Slide 17
                                                                  • Slide 18
                                                                  • Slide 19
                                                                  • Slide 20
                                                                  • Slide 21
                                                                  • Slide 22
                                                                  • Slide 23
                                                                  • Slide 24
                                                                  • Slide 25
                                                                  • Slide 26
                                                                  • Slide 27
                                                                  • Slide 28
                                                                  • Slide 29
                                                                  • Slide 30
                                                                  • Slide 31
                                                                  • Slide 32
                                                                  • Slide 33
                                                                  • Slide 34
                                                                  • Slide 35
                                                                  • Slide 36
                                                                  • Slide 37
                                                                  • Slide 38
                                                                  • Slide 39
                                                                  • Slide 40
                                                                  • Slide 41
                                                                  • Slide 42
                                                                  • Slide 43
                                                                  • Slide 44
                                                                  • Slide 45
                                                                  • Slide 46
                                                                  • Slide 47
                                                                  • Slide 48
                                                                  • Slide 49
                                                                  • Slide 50
                                                                  • Slide 51
                                                                  • Slide 52
                                                                  • Slide 53
                                                                  • Slide 54
                                                                  • Slide 55
                                                                  • Slide 56
                                                                  • Slide 57
                                                                  • Slide 58
                                                                  • Slide 59
                                                                  • Slide 60
                                                                  • Slide 61
                                                                  • Slide 62

                                                                    3434

                                                                    What is edgeWhat is edge

                                                                    Edge is where change occurs Change is measured by derivative in 1D

                                                                    ―Biggest change derivative has maximum magnitude

                                                                    Or 2nd derivative is zero we discuss approaches for implementing

                                                                    ―first-order derivative (Gradient operator)

                                                                    ―second-order derivative (Laplacian operator)

                                                                    ―we have introduced both derivatives in chapter 3

                                                                    ―Here we will talk only about their properties for edge detection

                                                                    3535

                                                                    What is edgeWhat is edge

                                                                    In other wordsIn other words an edge is a set of an edge is a set of

                                                                    connected pixelsconnected pixels

                                                                    that lie on the boundary between two that lie on the boundary between two

                                                                    regions with relatively distinct gray-level regions with relatively distinct gray-level

                                                                    propertiesproperties

                                                                    Note edge vs boundaryNote edge vs boundary

                                                                    ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                                                    ―whereas a region boundary owing to whereas a region boundary owing to

                                                                    the way it is defined is a more global the way it is defined is a more global

                                                                    ideaidea

                                                                    3636

                                                                    Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                                    because of because of

                                                                    optics optics

                                                                    sampling sampling

                                                                    image image

                                                                    acquisition acquisition

                                                                    imperfectionimperfection

                                                                    3737

                                                                    Thick and Thin EdgeThick and Thin Edge

                                                                    The slope of the ramp is inversely The slope of the ramp is inversely

                                                                    proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                                    edgeedge

                                                                    Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                                    pathpath

                                                                    Instead an edge point now is any point Instead an edge point now is any point

                                                                    contained in the ramp and contained in the ramp and an edge would an edge would

                                                                    then be a set of such points that are then be a set of such points that are

                                                                    connectedconnected

                                                                    The thickness is determined by the length of the The thickness is determined by the length of the

                                                                    rampramp

                                                                    The length is determined by the slope which is in The length is determined by the slope which is in

                                                                    turn determined by the degree of blurringturn determined by the degree of blurring

                                                                    Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                                    edges tend to be thinedges tend to be thin

                                                                    3838

                                                                    First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                                    the signs of the the signs of the

                                                                    derivatives would be derivatives would be

                                                                    reversed for an edge reversed for an edge

                                                                    that transitions from that transitions from

                                                                    light to darklight to dark

                                                                    First First derivatderivatee

                                                                    SeconSecond d derivatderivatee

                                                                    Gray-Gray-level level profileprofile

                                                                    3939

                                                                    Second derivativesSecond derivatives

                                                                    an undesirable featurean undesirable feature

                                                                    produces 2 values for every edge in an produces 2 values for every edge in an

                                                                    imageimage

                                                                    zero-crossing propertyzero-crossing property

                                                                    an imaginary straight line joining the an imaginary straight line joining the

                                                                    extreme positive and negative values of extreme positive and negative values of

                                                                    the second derivative would cross zero the second derivative would cross zero

                                                                    near the midpoint of the edgenear the midpoint of the edge

                                                                    quite useful for locating the centers of quite useful for locating the centers of

                                                                    thick edgesthick edges

                                                                    4040

                                                                    Basic idea of edge detectionBasic idea of edge detection

                                                                    A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                                    the edge direction and the results are the edge direction and the results are

                                                                    interpretedinterpreted

                                                                    The magnitude of the first derivative is The magnitude of the first derivative is

                                                                    used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                                    ramp)ramp)

                                                                    The sign of the second derivative can The sign of the second derivative can

                                                                    determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                                    dark or light side of an edgedark or light side of an edge

                                                                    4141

                                                                    Review of First DerivateReview of First Derivate

                                                                    Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                                    Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                                    Sobel operators 3Sobel operators 333

                                                                    6 5 8 5x yG z z G z z

                                                                    1 2 3

                                                                    4 5 6

                                                                    7 8 9

                                                                    z z z

                                                                    z z z

                                                                    z z z

                                                                    1 2 3

                                                                    4 5 6

                                                                    7 8 9

                                                                    z z z

                                                                    z z z

                                                                    z z z

                                                                    9 5 8 6x yG z z G z z 1 0 0 1

                                                                    0 1 1 0

                                                                    1 0 0 1

                                                                    0 1 1 0

                                                                    7 8 9 1 2 3

                                                                    3 6 9 1 4 7

                                                                    2 2

                                                                    2 2

                                                                    x

                                                                    y

                                                                    G z z z z z z

                                                                    G z z z z z z

                                                                    1 2 1 1 0 1

                                                                    0 0 0 2 0 2

                                                                    1 2 1 1 0 1

                                                                    1 2 1 1 0 1

                                                                    0 0 0 2 0 2

                                                                    1 2 1 1 0 1

                                                                    x yf G G

                                                                    4242

                                                                    Edge direction and strengthEdge direction and strength

                                                                    Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                                    α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                                    The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                                    The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                                    2 2x yf G G

                                                                    4343

                                                                    Gradient MasksGradient Masks

                                                                    1 0 0 1

                                                                    0 1 1 0

                                                                    Roberts

                                                                    1 0 0 1

                                                                    0 1 1 0

                                                                    Roberts

                                                                    1 2 1 1 0 1

                                                                    0 0 0 2 0 2

                                                                    1 2 1 1 0 1

                                                                    Sobel

                                                                    1 2 1 1 0 1

                                                                    0 0 0 2 0 2

                                                                    1 2 1 1 0 1

                                                                    Sobel

                                                                    1 1 1 1 0 1

                                                                    0 0 0 1 0 1

                                                                    1 1 1 1 0 1

                                                                    Prewitt

                                                                    1 1 1 1 0 1

                                                                    0 0 0 1 0 1

                                                                    1 1 1 1 0 1

                                                                    Prewitt

                                                                    4444

                                                                    Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                    0 1 1 1 1 0

                                                                    1 0 1 1 0 1

                                                                    1 1 0 0 1 1

                                                                    Prewitt

                                                                    0 1 1 1 1 0

                                                                    1 0 1 1 0 1

                                                                    1 1 0 0 1 1

                                                                    Prewitt

                                                                    4545

                                                                    Review of Second DerivateReview of Second Derivate

                                                                    Laplacian OperatorLaplacian Operator

                                                                    21 1

                                                                    1 1 4

                                                                    f x y f x yf

                                                                    f x y f x y f x y

                                                                    0 1 0

                                                                    1 4 1

                                                                    0 1 0

                                                                    0 1 0

                                                                    1 4 1

                                                                    0 1 0

                                                                    LaplacianLaplacian

                                                                    MaskMask

                                                                    1 1 1

                                                                    1 8 1

                                                                    1 1 1

                                                                    1 1 1

                                                                    1 8 1

                                                                    1 1 1

                                                                    4646

                                                                    Example of edge detectionExample of edge detection

                                                                    See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                    Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                    derivative)derivative)

                                                                    produces double edgesproduces double edges

                                                                    unable to detect edge directionunable to detect edge direction

                                                                    4747

                                                                    Canny edge detectorCanny edge detector

                                                                    The most powerful edge-detection The most powerful edge-detection

                                                                    method method

                                                                    It differs from the other edge-It differs from the other edge-

                                                                    detection methods in that detection methods in that

                                                                    it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                    strong and weak edges) strong and weak edges)

                                                                    and includes the weak edges in the and includes the weak edges in the

                                                                    output only if they are connected to output only if they are connected to

                                                                    strong edges strong edges

                                                                    This method is therefore less likely This method is therefore less likely

                                                                    than the others to be fooled by than the others to be fooled by

                                                                    noise and more likely to detect true noise and more likely to detect true

                                                                    weak edgesweak edges

                                                                    4848

                                                                    Laplacian of GaussianLaplacian of Gaussian

                                                                    Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                    2 2 22

                                                                    4 2

                                                                    2 2 2

                                                                    2exp

                                                                    r rh

                                                                    r x y

                                                                    determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                    4949

                                                                    Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                    0 0 1 0 0

                                                                    0 1 2 1 0

                                                                    1 2 16 2 1

                                                                    0 1 2 1 0

                                                                    0 0 1 0 0

                                                                    0 0 1 0 0

                                                                    0 1 2 1 0

                                                                    1 2 16 2 1

                                                                    0 1 2 1 0

                                                                    0 0 1 0 0

                                                                    The coefficient must sum to The coefficient must sum to

                                                                    zerozero

                                                                    5050

                                                                    Edge Detection and Edge Detection and SegmentationSegmentation

                                                                    Image resulting from edge detection cannot be used as a segmentation result

                                                                    Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                    5151

                                                                    75 Region-based 75 Region-based SegmentationSegmentation

                                                                    GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                    Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                    5252

                                                                    Two methods of Region Two methods of Region SegmentationSegmentation

                                                                    Region GrowingRegion Growing

                                                                    Region SplittingRegion Splitting

                                                                    Region growing is the opposite of the Region growing is the opposite of the

                                                                    split and merge approachsplit and merge approach

                                                                    5353

                                                                    Region GrowingRegion Growing

                                                                    The objective of segmentation is to The objective of segmentation is to

                                                                    partition an image into regionspartition an image into regions

                                                                    A region is a connected component with A region is a connected component with

                                                                    some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                    texture)texture)

                                                                    In region growing we start with a set In region growing we start with a set

                                                                    of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                    appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                    pixels if they have pixels if they have similar propertiessimilar properties

                                                                    such as specific ranges of gray level such as specific ranges of gray level

                                                                    and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                    Need initialization Need initialization similarity similarity

                                                                    criterioncriterion

                                                                    5454

                                                                    Steps of Region GrowingSteps of Region Growing

                                                                    Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                    pixel andpixel and compare it with neighbor compare it with neighbor

                                                                    ppixelsixels

                                                                    When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                    is grown from the seed pixel by is grown from the seed pixel by

                                                                    addingadding neighboneighborr pixel pixelss

                                                                    When the growth of one region stopsWhen the growth of one region stops

                                                                    choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                    againagain

                                                                    5555

                                                                    Region Region growing growing

                                                                    An initial set of small An initial set of small

                                                                    areas are iterativelyareas are iteratively

                                                                    merged according to merged according to

                                                                    similarity constraintssimilarity constraints

                                                                    5656

                                                                    Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                    X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                    We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                    On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                    Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                    SeedSeed pointspoints

                                                                    5757

                                                                    CriterionCriterion

                                                                    There is a valley at around 190 in the There is a valley at around 190 in the

                                                                    histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                    to be considered as a part of region to the to be considered as a part of region to the

                                                                    seed pointseed point

                                                                    The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                    to at least one pixel in that regionto at least one pixel in that region

                                                                    Result of region growing and boundaries of Result of region growing and boundaries of

                                                                    defectsdefects

                                                                    5858

                                                                    Region SplittingRegion Splitting

                                                                    The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                    ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                    If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                    This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                    Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                    5959

                                                                    Region SplittingRegion Splitting

                                                                    DisadvantageDisadvantage

                                                                    they create regions that may be adjacent they create regions that may be adjacent

                                                                    and homogeneous but not mergedand homogeneous but not merged

                                                                    6060

                                                                    Region Splitting and MergingRegion Splitting and Merging

                                                                    ProcedureProcedure

                                                                    11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                    22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                    33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                    6161

                                                                    Region Splitting and Merging

                                                                    Quadtree

                                                                    (四叉树 )

                                                                    6262

                                                                    PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                    where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                    mmii is the mean gray level of that region is the mean gray level of that region

                                                                    σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                    ExampleExample

                                                                    Original Original

                                                                    imageimageThresholded imageThresholded image Result of Result of

                                                                    Splitting and Splitting and

                                                                    MergingMerging

                                                                    • Slide 1
                                                                    • Slide 2
                                                                    • Slide 3
                                                                    • Slide 4
                                                                    • Slide 5
                                                                    • Slide 6
                                                                    • Slide 7
                                                                    • Slide 8
                                                                    • Slide 9
                                                                    • Slide 10
                                                                    • Slide 11
                                                                    • Slide 12
                                                                    • Slide 13
                                                                    • Slide 14
                                                                    • Slide 15
                                                                    • Slide 16
                                                                    • Slide 17
                                                                    • Slide 18
                                                                    • Slide 19
                                                                    • Slide 20
                                                                    • Slide 21
                                                                    • Slide 22
                                                                    • Slide 23
                                                                    • Slide 24
                                                                    • Slide 25
                                                                    • Slide 26
                                                                    • Slide 27
                                                                    • Slide 28
                                                                    • Slide 29
                                                                    • Slide 30
                                                                    • Slide 31
                                                                    • Slide 32
                                                                    • Slide 33
                                                                    • Slide 34
                                                                    • Slide 35
                                                                    • Slide 36
                                                                    • Slide 37
                                                                    • Slide 38
                                                                    • Slide 39
                                                                    • Slide 40
                                                                    • Slide 41
                                                                    • Slide 42
                                                                    • Slide 43
                                                                    • Slide 44
                                                                    • Slide 45
                                                                    • Slide 46
                                                                    • Slide 47
                                                                    • Slide 48
                                                                    • Slide 49
                                                                    • Slide 50
                                                                    • Slide 51
                                                                    • Slide 52
                                                                    • Slide 53
                                                                    • Slide 54
                                                                    • Slide 55
                                                                    • Slide 56
                                                                    • Slide 57
                                                                    • Slide 58
                                                                    • Slide 59
                                                                    • Slide 60
                                                                    • Slide 61
                                                                    • Slide 62

                                                                      3535

                                                                      What is edgeWhat is edge

                                                                      In other wordsIn other words an edge is a set of an edge is a set of

                                                                      connected pixelsconnected pixels

                                                                      that lie on the boundary between two that lie on the boundary between two

                                                                      regions with relatively distinct gray-level regions with relatively distinct gray-level

                                                                      propertiesproperties

                                                                      Note edge vs boundaryNote edge vs boundary

                                                                      ―an edge is a ldquolocalrdquo conceptan edge is a ldquolocalrdquo concept

                                                                      ―whereas a region boundary owing to whereas a region boundary owing to

                                                                      the way it is defined is a more global the way it is defined is a more global

                                                                      ideaidea

                                                                      3636

                                                                      Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                                      because of because of

                                                                      optics optics

                                                                      sampling sampling

                                                                      image image

                                                                      acquisition acquisition

                                                                      imperfectionimperfection

                                                                      3737

                                                                      Thick and Thin EdgeThick and Thin Edge

                                                                      The slope of the ramp is inversely The slope of the ramp is inversely

                                                                      proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                                      edgeedge

                                                                      Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                                      pathpath

                                                                      Instead an edge point now is any point Instead an edge point now is any point

                                                                      contained in the ramp and contained in the ramp and an edge would an edge would

                                                                      then be a set of such points that are then be a set of such points that are

                                                                      connectedconnected

                                                                      The thickness is determined by the length of the The thickness is determined by the length of the

                                                                      rampramp

                                                                      The length is determined by the slope which is in The length is determined by the slope which is in

                                                                      turn determined by the degree of blurringturn determined by the degree of blurring

                                                                      Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                                      edges tend to be thinedges tend to be thin

                                                                      3838

                                                                      First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                                      the signs of the the signs of the

                                                                      derivatives would be derivatives would be

                                                                      reversed for an edge reversed for an edge

                                                                      that transitions from that transitions from

                                                                      light to darklight to dark

                                                                      First First derivatderivatee

                                                                      SeconSecond d derivatderivatee

                                                                      Gray-Gray-level level profileprofile

                                                                      3939

                                                                      Second derivativesSecond derivatives

                                                                      an undesirable featurean undesirable feature

                                                                      produces 2 values for every edge in an produces 2 values for every edge in an

                                                                      imageimage

                                                                      zero-crossing propertyzero-crossing property

                                                                      an imaginary straight line joining the an imaginary straight line joining the

                                                                      extreme positive and negative values of extreme positive and negative values of

                                                                      the second derivative would cross zero the second derivative would cross zero

                                                                      near the midpoint of the edgenear the midpoint of the edge

                                                                      quite useful for locating the centers of quite useful for locating the centers of

                                                                      thick edgesthick edges

                                                                      4040

                                                                      Basic idea of edge detectionBasic idea of edge detection

                                                                      A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                                      the edge direction and the results are the edge direction and the results are

                                                                      interpretedinterpreted

                                                                      The magnitude of the first derivative is The magnitude of the first derivative is

                                                                      used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                                      ramp)ramp)

                                                                      The sign of the second derivative can The sign of the second derivative can

                                                                      determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                                      dark or light side of an edgedark or light side of an edge

                                                                      4141

                                                                      Review of First DerivateReview of First Derivate

                                                                      Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                                      Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                                      Sobel operators 3Sobel operators 333

                                                                      6 5 8 5x yG z z G z z

                                                                      1 2 3

                                                                      4 5 6

                                                                      7 8 9

                                                                      z z z

                                                                      z z z

                                                                      z z z

                                                                      1 2 3

                                                                      4 5 6

                                                                      7 8 9

                                                                      z z z

                                                                      z z z

                                                                      z z z

                                                                      9 5 8 6x yG z z G z z 1 0 0 1

                                                                      0 1 1 0

                                                                      1 0 0 1

                                                                      0 1 1 0

                                                                      7 8 9 1 2 3

                                                                      3 6 9 1 4 7

                                                                      2 2

                                                                      2 2

                                                                      x

                                                                      y

                                                                      G z z z z z z

                                                                      G z z z z z z

                                                                      1 2 1 1 0 1

                                                                      0 0 0 2 0 2

                                                                      1 2 1 1 0 1

                                                                      1 2 1 1 0 1

                                                                      0 0 0 2 0 2

                                                                      1 2 1 1 0 1

                                                                      x yf G G

                                                                      4242

                                                                      Edge direction and strengthEdge direction and strength

                                                                      Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                                      α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                                      The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                                      The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                                      2 2x yf G G

                                                                      4343

                                                                      Gradient MasksGradient Masks

                                                                      1 0 0 1

                                                                      0 1 1 0

                                                                      Roberts

                                                                      1 0 0 1

                                                                      0 1 1 0

                                                                      Roberts

                                                                      1 2 1 1 0 1

                                                                      0 0 0 2 0 2

                                                                      1 2 1 1 0 1

                                                                      Sobel

                                                                      1 2 1 1 0 1

                                                                      0 0 0 2 0 2

                                                                      1 2 1 1 0 1

                                                                      Sobel

                                                                      1 1 1 1 0 1

                                                                      0 0 0 1 0 1

                                                                      1 1 1 1 0 1

                                                                      Prewitt

                                                                      1 1 1 1 0 1

                                                                      0 0 0 1 0 1

                                                                      1 1 1 1 0 1

                                                                      Prewitt

                                                                      4444

                                                                      Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                      0 1 1 1 1 0

                                                                      1 0 1 1 0 1

                                                                      1 1 0 0 1 1

                                                                      Prewitt

                                                                      0 1 1 1 1 0

                                                                      1 0 1 1 0 1

                                                                      1 1 0 0 1 1

                                                                      Prewitt

                                                                      4545

                                                                      Review of Second DerivateReview of Second Derivate

                                                                      Laplacian OperatorLaplacian Operator

                                                                      21 1

                                                                      1 1 4

                                                                      f x y f x yf

                                                                      f x y f x y f x y

                                                                      0 1 0

                                                                      1 4 1

                                                                      0 1 0

                                                                      0 1 0

                                                                      1 4 1

                                                                      0 1 0

                                                                      LaplacianLaplacian

                                                                      MaskMask

                                                                      1 1 1

                                                                      1 8 1

                                                                      1 1 1

                                                                      1 1 1

                                                                      1 8 1

                                                                      1 1 1

                                                                      4646

                                                                      Example of edge detectionExample of edge detection

                                                                      See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                      Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                      derivative)derivative)

                                                                      produces double edgesproduces double edges

                                                                      unable to detect edge directionunable to detect edge direction

                                                                      4747

                                                                      Canny edge detectorCanny edge detector

                                                                      The most powerful edge-detection The most powerful edge-detection

                                                                      method method

                                                                      It differs from the other edge-It differs from the other edge-

                                                                      detection methods in that detection methods in that

                                                                      it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                      strong and weak edges) strong and weak edges)

                                                                      and includes the weak edges in the and includes the weak edges in the

                                                                      output only if they are connected to output only if they are connected to

                                                                      strong edges strong edges

                                                                      This method is therefore less likely This method is therefore less likely

                                                                      than the others to be fooled by than the others to be fooled by

                                                                      noise and more likely to detect true noise and more likely to detect true

                                                                      weak edgesweak edges

                                                                      4848

                                                                      Laplacian of GaussianLaplacian of Gaussian

                                                                      Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                      2 2 22

                                                                      4 2

                                                                      2 2 2

                                                                      2exp

                                                                      r rh

                                                                      r x y

                                                                      determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                      4949

                                                                      Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                      0 0 1 0 0

                                                                      0 1 2 1 0

                                                                      1 2 16 2 1

                                                                      0 1 2 1 0

                                                                      0 0 1 0 0

                                                                      0 0 1 0 0

                                                                      0 1 2 1 0

                                                                      1 2 16 2 1

                                                                      0 1 2 1 0

                                                                      0 0 1 0 0

                                                                      The coefficient must sum to The coefficient must sum to

                                                                      zerozero

                                                                      5050

                                                                      Edge Detection and Edge Detection and SegmentationSegmentation

                                                                      Image resulting from edge detection cannot be used as a segmentation result

                                                                      Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                      5151

                                                                      75 Region-based 75 Region-based SegmentationSegmentation

                                                                      GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                      Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                      5252

                                                                      Two methods of Region Two methods of Region SegmentationSegmentation

                                                                      Region GrowingRegion Growing

                                                                      Region SplittingRegion Splitting

                                                                      Region growing is the opposite of the Region growing is the opposite of the

                                                                      split and merge approachsplit and merge approach

                                                                      5353

                                                                      Region GrowingRegion Growing

                                                                      The objective of segmentation is to The objective of segmentation is to

                                                                      partition an image into regionspartition an image into regions

                                                                      A region is a connected component with A region is a connected component with

                                                                      some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                      texture)texture)

                                                                      In region growing we start with a set In region growing we start with a set

                                                                      of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                      appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                      pixels if they have pixels if they have similar propertiessimilar properties

                                                                      such as specific ranges of gray level such as specific ranges of gray level

                                                                      and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                      Need initialization Need initialization similarity similarity

                                                                      criterioncriterion

                                                                      5454

                                                                      Steps of Region GrowingSteps of Region Growing

                                                                      Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                      pixel andpixel and compare it with neighbor compare it with neighbor

                                                                      ppixelsixels

                                                                      When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                      is grown from the seed pixel by is grown from the seed pixel by

                                                                      addingadding neighboneighborr pixel pixelss

                                                                      When the growth of one region stopsWhen the growth of one region stops

                                                                      choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                      againagain

                                                                      5555

                                                                      Region Region growing growing

                                                                      An initial set of small An initial set of small

                                                                      areas are iterativelyareas are iteratively

                                                                      merged according to merged according to

                                                                      similarity constraintssimilarity constraints

                                                                      5656

                                                                      Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                      X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                      We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                      On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                      Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                      SeedSeed pointspoints

                                                                      5757

                                                                      CriterionCriterion

                                                                      There is a valley at around 190 in the There is a valley at around 190 in the

                                                                      histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                      to be considered as a part of region to the to be considered as a part of region to the

                                                                      seed pointseed point

                                                                      The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                      to at least one pixel in that regionto at least one pixel in that region

                                                                      Result of region growing and boundaries of Result of region growing and boundaries of

                                                                      defectsdefects

                                                                      5858

                                                                      Region SplittingRegion Splitting

                                                                      The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                      ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                      If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                      This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                      Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                      5959

                                                                      Region SplittingRegion Splitting

                                                                      DisadvantageDisadvantage

                                                                      they create regions that may be adjacent they create regions that may be adjacent

                                                                      and homogeneous but not mergedand homogeneous but not merged

                                                                      6060

                                                                      Region Splitting and MergingRegion Splitting and Merging

                                                                      ProcedureProcedure

                                                                      11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                      22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                      33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                      6161

                                                                      Region Splitting and Merging

                                                                      Quadtree

                                                                      (四叉树 )

                                                                      6262

                                                                      PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                      where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                      mmii is the mean gray level of that region is the mean gray level of that region

                                                                      σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                      ExampleExample

                                                                      Original Original

                                                                      imageimageThresholded imageThresholded image Result of Result of

                                                                      Splitting and Splitting and

                                                                      MergingMerging

                                                                      • Slide 1
                                                                      • Slide 2
                                                                      • Slide 3
                                                                      • Slide 4
                                                                      • Slide 5
                                                                      • Slide 6
                                                                      • Slide 7
                                                                      • Slide 8
                                                                      • Slide 9
                                                                      • Slide 10
                                                                      • Slide 11
                                                                      • Slide 12
                                                                      • Slide 13
                                                                      • Slide 14
                                                                      • Slide 15
                                                                      • Slide 16
                                                                      • Slide 17
                                                                      • Slide 18
                                                                      • Slide 19
                                                                      • Slide 20
                                                                      • Slide 21
                                                                      • Slide 22
                                                                      • Slide 23
                                                                      • Slide 24
                                                                      • Slide 25
                                                                      • Slide 26
                                                                      • Slide 27
                                                                      • Slide 28
                                                                      • Slide 29
                                                                      • Slide 30
                                                                      • Slide 31
                                                                      • Slide 32
                                                                      • Slide 33
                                                                      • Slide 34
                                                                      • Slide 35
                                                                      • Slide 36
                                                                      • Slide 37
                                                                      • Slide 38
                                                                      • Slide 39
                                                                      • Slide 40
                                                                      • Slide 41
                                                                      • Slide 42
                                                                      • Slide 43
                                                                      • Slide 44
                                                                      • Slide 45
                                                                      • Slide 46
                                                                      • Slide 47
                                                                      • Slide 48
                                                                      • Slide 49
                                                                      • Slide 50
                                                                      • Slide 51
                                                                      • Slide 52
                                                                      • Slide 53
                                                                      • Slide 54
                                                                      • Slide 55
                                                                      • Slide 56
                                                                      • Slide 57
                                                                      • Slide 58
                                                                      • Slide 59
                                                                      • Slide 60
                                                                      • Slide 61
                                                                      • Slide 62

                                                                        3636

                                                                        Ideal and Ramp (Ideal and Ramp (斜坡斜坡 ) Edges) Edges

                                                                        because of because of

                                                                        optics optics

                                                                        sampling sampling

                                                                        image image

                                                                        acquisition acquisition

                                                                        imperfectionimperfection

                                                                        3737

                                                                        Thick and Thin EdgeThick and Thin Edge

                                                                        The slope of the ramp is inversely The slope of the ramp is inversely

                                                                        proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                                        edgeedge

                                                                        Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                                        pathpath

                                                                        Instead an edge point now is any point Instead an edge point now is any point

                                                                        contained in the ramp and contained in the ramp and an edge would an edge would

                                                                        then be a set of such points that are then be a set of such points that are

                                                                        connectedconnected

                                                                        The thickness is determined by the length of the The thickness is determined by the length of the

                                                                        rampramp

                                                                        The length is determined by the slope which is in The length is determined by the slope which is in

                                                                        turn determined by the degree of blurringturn determined by the degree of blurring

                                                                        Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                                        edges tend to be thinedges tend to be thin

                                                                        3838

                                                                        First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                                        the signs of the the signs of the

                                                                        derivatives would be derivatives would be

                                                                        reversed for an edge reversed for an edge

                                                                        that transitions from that transitions from

                                                                        light to darklight to dark

                                                                        First First derivatderivatee

                                                                        SeconSecond d derivatderivatee

                                                                        Gray-Gray-level level profileprofile

                                                                        3939

                                                                        Second derivativesSecond derivatives

                                                                        an undesirable featurean undesirable feature

                                                                        produces 2 values for every edge in an produces 2 values for every edge in an

                                                                        imageimage

                                                                        zero-crossing propertyzero-crossing property

                                                                        an imaginary straight line joining the an imaginary straight line joining the

                                                                        extreme positive and negative values of extreme positive and negative values of

                                                                        the second derivative would cross zero the second derivative would cross zero

                                                                        near the midpoint of the edgenear the midpoint of the edge

                                                                        quite useful for locating the centers of quite useful for locating the centers of

                                                                        thick edgesthick edges

                                                                        4040

                                                                        Basic idea of edge detectionBasic idea of edge detection

                                                                        A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                                        the edge direction and the results are the edge direction and the results are

                                                                        interpretedinterpreted

                                                                        The magnitude of the first derivative is The magnitude of the first derivative is

                                                                        used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                                        ramp)ramp)

                                                                        The sign of the second derivative can The sign of the second derivative can

                                                                        determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                                        dark or light side of an edgedark or light side of an edge

                                                                        4141

                                                                        Review of First DerivateReview of First Derivate

                                                                        Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                                        Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                                        Sobel operators 3Sobel operators 333

                                                                        6 5 8 5x yG z z G z z

                                                                        1 2 3

                                                                        4 5 6

                                                                        7 8 9

                                                                        z z z

                                                                        z z z

                                                                        z z z

                                                                        1 2 3

                                                                        4 5 6

                                                                        7 8 9

                                                                        z z z

                                                                        z z z

                                                                        z z z

                                                                        9 5 8 6x yG z z G z z 1 0 0 1

                                                                        0 1 1 0

                                                                        1 0 0 1

                                                                        0 1 1 0

                                                                        7 8 9 1 2 3

                                                                        3 6 9 1 4 7

                                                                        2 2

                                                                        2 2

                                                                        x

                                                                        y

                                                                        G z z z z z z

                                                                        G z z z z z z

                                                                        1 2 1 1 0 1

                                                                        0 0 0 2 0 2

                                                                        1 2 1 1 0 1

                                                                        1 2 1 1 0 1

                                                                        0 0 0 2 0 2

                                                                        1 2 1 1 0 1

                                                                        x yf G G

                                                                        4242

                                                                        Edge direction and strengthEdge direction and strength

                                                                        Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                                        α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                                        The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                                        The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                                        2 2x yf G G

                                                                        4343

                                                                        Gradient MasksGradient Masks

                                                                        1 0 0 1

                                                                        0 1 1 0

                                                                        Roberts

                                                                        1 0 0 1

                                                                        0 1 1 0

                                                                        Roberts

                                                                        1 2 1 1 0 1

                                                                        0 0 0 2 0 2

                                                                        1 2 1 1 0 1

                                                                        Sobel

                                                                        1 2 1 1 0 1

                                                                        0 0 0 2 0 2

                                                                        1 2 1 1 0 1

                                                                        Sobel

                                                                        1 1 1 1 0 1

                                                                        0 0 0 1 0 1

                                                                        1 1 1 1 0 1

                                                                        Prewitt

                                                                        1 1 1 1 0 1

                                                                        0 0 0 1 0 1

                                                                        1 1 1 1 0 1

                                                                        Prewitt

                                                                        4444

                                                                        Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                        0 1 1 1 1 0

                                                                        1 0 1 1 0 1

                                                                        1 1 0 0 1 1

                                                                        Prewitt

                                                                        0 1 1 1 1 0

                                                                        1 0 1 1 0 1

                                                                        1 1 0 0 1 1

                                                                        Prewitt

                                                                        4545

                                                                        Review of Second DerivateReview of Second Derivate

                                                                        Laplacian OperatorLaplacian Operator

                                                                        21 1

                                                                        1 1 4

                                                                        f x y f x yf

                                                                        f x y f x y f x y

                                                                        0 1 0

                                                                        1 4 1

                                                                        0 1 0

                                                                        0 1 0

                                                                        1 4 1

                                                                        0 1 0

                                                                        LaplacianLaplacian

                                                                        MaskMask

                                                                        1 1 1

                                                                        1 8 1

                                                                        1 1 1

                                                                        1 1 1

                                                                        1 8 1

                                                                        1 1 1

                                                                        4646

                                                                        Example of edge detectionExample of edge detection

                                                                        See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                        Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                        derivative)derivative)

                                                                        produces double edgesproduces double edges

                                                                        unable to detect edge directionunable to detect edge direction

                                                                        4747

                                                                        Canny edge detectorCanny edge detector

                                                                        The most powerful edge-detection The most powerful edge-detection

                                                                        method method

                                                                        It differs from the other edge-It differs from the other edge-

                                                                        detection methods in that detection methods in that

                                                                        it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                        strong and weak edges) strong and weak edges)

                                                                        and includes the weak edges in the and includes the weak edges in the

                                                                        output only if they are connected to output only if they are connected to

                                                                        strong edges strong edges

                                                                        This method is therefore less likely This method is therefore less likely

                                                                        than the others to be fooled by than the others to be fooled by

                                                                        noise and more likely to detect true noise and more likely to detect true

                                                                        weak edgesweak edges

                                                                        4848

                                                                        Laplacian of GaussianLaplacian of Gaussian

                                                                        Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                        2 2 22

                                                                        4 2

                                                                        2 2 2

                                                                        2exp

                                                                        r rh

                                                                        r x y

                                                                        determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                        4949

                                                                        Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                        0 0 1 0 0

                                                                        0 1 2 1 0

                                                                        1 2 16 2 1

                                                                        0 1 2 1 0

                                                                        0 0 1 0 0

                                                                        0 0 1 0 0

                                                                        0 1 2 1 0

                                                                        1 2 16 2 1

                                                                        0 1 2 1 0

                                                                        0 0 1 0 0

                                                                        The coefficient must sum to The coefficient must sum to

                                                                        zerozero

                                                                        5050

                                                                        Edge Detection and Edge Detection and SegmentationSegmentation

                                                                        Image resulting from edge detection cannot be used as a segmentation result

                                                                        Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                        5151

                                                                        75 Region-based 75 Region-based SegmentationSegmentation

                                                                        GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                        Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                        PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                        For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                        PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                        For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                        5252

                                                                        Two methods of Region Two methods of Region SegmentationSegmentation

                                                                        Region GrowingRegion Growing

                                                                        Region SplittingRegion Splitting

                                                                        Region growing is the opposite of the Region growing is the opposite of the

                                                                        split and merge approachsplit and merge approach

                                                                        5353

                                                                        Region GrowingRegion Growing

                                                                        The objective of segmentation is to The objective of segmentation is to

                                                                        partition an image into regionspartition an image into regions

                                                                        A region is a connected component with A region is a connected component with

                                                                        some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                        texture)texture)

                                                                        In region growing we start with a set In region growing we start with a set

                                                                        of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                        appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                        pixels if they have pixels if they have similar propertiessimilar properties

                                                                        such as specific ranges of gray level such as specific ranges of gray level

                                                                        and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                        Need initialization Need initialization similarity similarity

                                                                        criterioncriterion

                                                                        5454

                                                                        Steps of Region GrowingSteps of Region Growing

                                                                        Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                        pixel andpixel and compare it with neighbor compare it with neighbor

                                                                        ppixelsixels

                                                                        When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                        is grown from the seed pixel by is grown from the seed pixel by

                                                                        addingadding neighboneighborr pixel pixelss

                                                                        When the growth of one region stopsWhen the growth of one region stops

                                                                        choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                        againagain

                                                                        5555

                                                                        Region Region growing growing

                                                                        An initial set of small An initial set of small

                                                                        areas are iterativelyareas are iteratively

                                                                        merged according to merged according to

                                                                        similarity constraintssimilarity constraints

                                                                        5656

                                                                        Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                        X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                        We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                        On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                        Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                        SeedSeed pointspoints

                                                                        5757

                                                                        CriterionCriterion

                                                                        There is a valley at around 190 in the There is a valley at around 190 in the

                                                                        histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                        to be considered as a part of region to the to be considered as a part of region to the

                                                                        seed pointseed point

                                                                        The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                        to at least one pixel in that regionto at least one pixel in that region

                                                                        Result of region growing and boundaries of Result of region growing and boundaries of

                                                                        defectsdefects

                                                                        5858

                                                                        Region SplittingRegion Splitting

                                                                        The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                        ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                        If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                        This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                        Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                        5959

                                                                        Region SplittingRegion Splitting

                                                                        DisadvantageDisadvantage

                                                                        they create regions that may be adjacent they create regions that may be adjacent

                                                                        and homogeneous but not mergedand homogeneous but not merged

                                                                        6060

                                                                        Region Splitting and MergingRegion Splitting and Merging

                                                                        ProcedureProcedure

                                                                        11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                        22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                        33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                        6161

                                                                        Region Splitting and Merging

                                                                        Quadtree

                                                                        (四叉树 )

                                                                        6262

                                                                        PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                        where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                        mmii is the mean gray level of that region is the mean gray level of that region

                                                                        σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                        ExampleExample

                                                                        Original Original

                                                                        imageimageThresholded imageThresholded image Result of Result of

                                                                        Splitting and Splitting and

                                                                        MergingMerging

                                                                        • Slide 1
                                                                        • Slide 2
                                                                        • Slide 3
                                                                        • Slide 4
                                                                        • Slide 5
                                                                        • Slide 6
                                                                        • Slide 7
                                                                        • Slide 8
                                                                        • Slide 9
                                                                        • Slide 10
                                                                        • Slide 11
                                                                        • Slide 12
                                                                        • Slide 13
                                                                        • Slide 14
                                                                        • Slide 15
                                                                        • Slide 16
                                                                        • Slide 17
                                                                        • Slide 18
                                                                        • Slide 19
                                                                        • Slide 20
                                                                        • Slide 21
                                                                        • Slide 22
                                                                        • Slide 23
                                                                        • Slide 24
                                                                        • Slide 25
                                                                        • Slide 26
                                                                        • Slide 27
                                                                        • Slide 28
                                                                        • Slide 29
                                                                        • Slide 30
                                                                        • Slide 31
                                                                        • Slide 32
                                                                        • Slide 33
                                                                        • Slide 34
                                                                        • Slide 35
                                                                        • Slide 36
                                                                        • Slide 37
                                                                        • Slide 38
                                                                        • Slide 39
                                                                        • Slide 40
                                                                        • Slide 41
                                                                        • Slide 42
                                                                        • Slide 43
                                                                        • Slide 44
                                                                        • Slide 45
                                                                        • Slide 46
                                                                        • Slide 47
                                                                        • Slide 48
                                                                        • Slide 49
                                                                        • Slide 50
                                                                        • Slide 51
                                                                        • Slide 52
                                                                        • Slide 53
                                                                        • Slide 54
                                                                        • Slide 55
                                                                        • Slide 56
                                                                        • Slide 57
                                                                        • Slide 58
                                                                        • Slide 59
                                                                        • Slide 60
                                                                        • Slide 61
                                                                        • Slide 62

                                                                          3737

                                                                          Thick and Thin EdgeThick and Thin Edge

                                                                          The slope of the ramp is inversely The slope of the ramp is inversely

                                                                          proportional to the degree of blurring in the proportional to the degree of blurring in the

                                                                          edgeedge

                                                                          Namely we no longer have Namely we no longer have a thin (one pixel thick) a thin (one pixel thick)

                                                                          pathpath

                                                                          Instead an edge point now is any point Instead an edge point now is any point

                                                                          contained in the ramp and contained in the ramp and an edge would an edge would

                                                                          then be a set of such points that are then be a set of such points that are

                                                                          connectedconnected

                                                                          The thickness is determined by the length of the The thickness is determined by the length of the

                                                                          rampramp

                                                                          The length is determined by the slope which is in The length is determined by the slope which is in

                                                                          turn determined by the degree of blurringturn determined by the degree of blurring

                                                                          Blurred edges tend to be thick and sharp Blurred edges tend to be thick and sharp

                                                                          edges tend to be thinedges tend to be thin

                                                                          3838

                                                                          First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                                          the signs of the the signs of the

                                                                          derivatives would be derivatives would be

                                                                          reversed for an edge reversed for an edge

                                                                          that transitions from that transitions from

                                                                          light to darklight to dark

                                                                          First First derivatderivatee

                                                                          SeconSecond d derivatderivatee

                                                                          Gray-Gray-level level profileprofile

                                                                          3939

                                                                          Second derivativesSecond derivatives

                                                                          an undesirable featurean undesirable feature

                                                                          produces 2 values for every edge in an produces 2 values for every edge in an

                                                                          imageimage

                                                                          zero-crossing propertyzero-crossing property

                                                                          an imaginary straight line joining the an imaginary straight line joining the

                                                                          extreme positive and negative values of extreme positive and negative values of

                                                                          the second derivative would cross zero the second derivative would cross zero

                                                                          near the midpoint of the edgenear the midpoint of the edge

                                                                          quite useful for locating the centers of quite useful for locating the centers of

                                                                          thick edgesthick edges

                                                                          4040

                                                                          Basic idea of edge detectionBasic idea of edge detection

                                                                          A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                                          the edge direction and the results are the edge direction and the results are

                                                                          interpretedinterpreted

                                                                          The magnitude of the first derivative is The magnitude of the first derivative is

                                                                          used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                                          ramp)ramp)

                                                                          The sign of the second derivative can The sign of the second derivative can

                                                                          determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                                          dark or light side of an edgedark or light side of an edge

                                                                          4141

                                                                          Review of First DerivateReview of First Derivate

                                                                          Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                                          Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                                          Sobel operators 3Sobel operators 333

                                                                          6 5 8 5x yG z z G z z

                                                                          1 2 3

                                                                          4 5 6

                                                                          7 8 9

                                                                          z z z

                                                                          z z z

                                                                          z z z

                                                                          1 2 3

                                                                          4 5 6

                                                                          7 8 9

                                                                          z z z

                                                                          z z z

                                                                          z z z

                                                                          9 5 8 6x yG z z G z z 1 0 0 1

                                                                          0 1 1 0

                                                                          1 0 0 1

                                                                          0 1 1 0

                                                                          7 8 9 1 2 3

                                                                          3 6 9 1 4 7

                                                                          2 2

                                                                          2 2

                                                                          x

                                                                          y

                                                                          G z z z z z z

                                                                          G z z z z z z

                                                                          1 2 1 1 0 1

                                                                          0 0 0 2 0 2

                                                                          1 2 1 1 0 1

                                                                          1 2 1 1 0 1

                                                                          0 0 0 2 0 2

                                                                          1 2 1 1 0 1

                                                                          x yf G G

                                                                          4242

                                                                          Edge direction and strengthEdge direction and strength

                                                                          Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                                          α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                                          The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                                          The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                                          2 2x yf G G

                                                                          4343

                                                                          Gradient MasksGradient Masks

                                                                          1 0 0 1

                                                                          0 1 1 0

                                                                          Roberts

                                                                          1 0 0 1

                                                                          0 1 1 0

                                                                          Roberts

                                                                          1 2 1 1 0 1

                                                                          0 0 0 2 0 2

                                                                          1 2 1 1 0 1

                                                                          Sobel

                                                                          1 2 1 1 0 1

                                                                          0 0 0 2 0 2

                                                                          1 2 1 1 0 1

                                                                          Sobel

                                                                          1 1 1 1 0 1

                                                                          0 0 0 1 0 1

                                                                          1 1 1 1 0 1

                                                                          Prewitt

                                                                          1 1 1 1 0 1

                                                                          0 0 0 1 0 1

                                                                          1 1 1 1 0 1

                                                                          Prewitt

                                                                          4444

                                                                          Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                          0 1 1 1 1 0

                                                                          1 0 1 1 0 1

                                                                          1 1 0 0 1 1

                                                                          Prewitt

                                                                          0 1 1 1 1 0

                                                                          1 0 1 1 0 1

                                                                          1 1 0 0 1 1

                                                                          Prewitt

                                                                          4545

                                                                          Review of Second DerivateReview of Second Derivate

                                                                          Laplacian OperatorLaplacian Operator

                                                                          21 1

                                                                          1 1 4

                                                                          f x y f x yf

                                                                          f x y f x y f x y

                                                                          0 1 0

                                                                          1 4 1

                                                                          0 1 0

                                                                          0 1 0

                                                                          1 4 1

                                                                          0 1 0

                                                                          LaplacianLaplacian

                                                                          MaskMask

                                                                          1 1 1

                                                                          1 8 1

                                                                          1 1 1

                                                                          1 1 1

                                                                          1 8 1

                                                                          1 1 1

                                                                          4646

                                                                          Example of edge detectionExample of edge detection

                                                                          See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                          Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                          derivative)derivative)

                                                                          produces double edgesproduces double edges

                                                                          unable to detect edge directionunable to detect edge direction

                                                                          4747

                                                                          Canny edge detectorCanny edge detector

                                                                          The most powerful edge-detection The most powerful edge-detection

                                                                          method method

                                                                          It differs from the other edge-It differs from the other edge-

                                                                          detection methods in that detection methods in that

                                                                          it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                          strong and weak edges) strong and weak edges)

                                                                          and includes the weak edges in the and includes the weak edges in the

                                                                          output only if they are connected to output only if they are connected to

                                                                          strong edges strong edges

                                                                          This method is therefore less likely This method is therefore less likely

                                                                          than the others to be fooled by than the others to be fooled by

                                                                          noise and more likely to detect true noise and more likely to detect true

                                                                          weak edgesweak edges

                                                                          4848

                                                                          Laplacian of GaussianLaplacian of Gaussian

                                                                          Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                          2 2 22

                                                                          4 2

                                                                          2 2 2

                                                                          2exp

                                                                          r rh

                                                                          r x y

                                                                          determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                          4949

                                                                          Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                          0 0 1 0 0

                                                                          0 1 2 1 0

                                                                          1 2 16 2 1

                                                                          0 1 2 1 0

                                                                          0 0 1 0 0

                                                                          0 0 1 0 0

                                                                          0 1 2 1 0

                                                                          1 2 16 2 1

                                                                          0 1 2 1 0

                                                                          0 0 1 0 0

                                                                          The coefficient must sum to The coefficient must sum to

                                                                          zerozero

                                                                          5050

                                                                          Edge Detection and Edge Detection and SegmentationSegmentation

                                                                          Image resulting from edge detection cannot be used as a segmentation result

                                                                          Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                          5151

                                                                          75 Region-based 75 Region-based SegmentationSegmentation

                                                                          GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                          Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                          PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                          For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                          PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                          For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                          5252

                                                                          Two methods of Region Two methods of Region SegmentationSegmentation

                                                                          Region GrowingRegion Growing

                                                                          Region SplittingRegion Splitting

                                                                          Region growing is the opposite of the Region growing is the opposite of the

                                                                          split and merge approachsplit and merge approach

                                                                          5353

                                                                          Region GrowingRegion Growing

                                                                          The objective of segmentation is to The objective of segmentation is to

                                                                          partition an image into regionspartition an image into regions

                                                                          A region is a connected component with A region is a connected component with

                                                                          some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                          texture)texture)

                                                                          In region growing we start with a set In region growing we start with a set

                                                                          of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                          appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                          pixels if they have pixels if they have similar propertiessimilar properties

                                                                          such as specific ranges of gray level such as specific ranges of gray level

                                                                          and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                          Need initialization Need initialization similarity similarity

                                                                          criterioncriterion

                                                                          5454

                                                                          Steps of Region GrowingSteps of Region Growing

                                                                          Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                          pixel andpixel and compare it with neighbor compare it with neighbor

                                                                          ppixelsixels

                                                                          When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                          is grown from the seed pixel by is grown from the seed pixel by

                                                                          addingadding neighboneighborr pixel pixelss

                                                                          When the growth of one region stopsWhen the growth of one region stops

                                                                          choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                          againagain

                                                                          5555

                                                                          Region Region growing growing

                                                                          An initial set of small An initial set of small

                                                                          areas are iterativelyareas are iteratively

                                                                          merged according to merged according to

                                                                          similarity constraintssimilarity constraints

                                                                          5656

                                                                          Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                          X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                          We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                          On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                          Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                          SeedSeed pointspoints

                                                                          5757

                                                                          CriterionCriterion

                                                                          There is a valley at around 190 in the There is a valley at around 190 in the

                                                                          histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                          to be considered as a part of region to the to be considered as a part of region to the

                                                                          seed pointseed point

                                                                          The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                          to at least one pixel in that regionto at least one pixel in that region

                                                                          Result of region growing and boundaries of Result of region growing and boundaries of

                                                                          defectsdefects

                                                                          5858

                                                                          Region SplittingRegion Splitting

                                                                          The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                          ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                          If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                          This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                          Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                          5959

                                                                          Region SplittingRegion Splitting

                                                                          DisadvantageDisadvantage

                                                                          they create regions that may be adjacent they create regions that may be adjacent

                                                                          and homogeneous but not mergedand homogeneous but not merged

                                                                          6060

                                                                          Region Splitting and MergingRegion Splitting and Merging

                                                                          ProcedureProcedure

                                                                          11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                          22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                          33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                          6161

                                                                          Region Splitting and Merging

                                                                          Quadtree

                                                                          (四叉树 )

                                                                          6262

                                                                          PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                          where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                          mmii is the mean gray level of that region is the mean gray level of that region

                                                                          σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                          ExampleExample

                                                                          Original Original

                                                                          imageimageThresholded imageThresholded image Result of Result of

                                                                          Splitting and Splitting and

                                                                          MergingMerging

                                                                          • Slide 1
                                                                          • Slide 2
                                                                          • Slide 3
                                                                          • Slide 4
                                                                          • Slide 5
                                                                          • Slide 6
                                                                          • Slide 7
                                                                          • Slide 8
                                                                          • Slide 9
                                                                          • Slide 10
                                                                          • Slide 11
                                                                          • Slide 12
                                                                          • Slide 13
                                                                          • Slide 14
                                                                          • Slide 15
                                                                          • Slide 16
                                                                          • Slide 17
                                                                          • Slide 18
                                                                          • Slide 19
                                                                          • Slide 20
                                                                          • Slide 21
                                                                          • Slide 22
                                                                          • Slide 23
                                                                          • Slide 24
                                                                          • Slide 25
                                                                          • Slide 26
                                                                          • Slide 27
                                                                          • Slide 28
                                                                          • Slide 29
                                                                          • Slide 30
                                                                          • Slide 31
                                                                          • Slide 32
                                                                          • Slide 33
                                                                          • Slide 34
                                                                          • Slide 35
                                                                          • Slide 36
                                                                          • Slide 37
                                                                          • Slide 38
                                                                          • Slide 39
                                                                          • Slide 40
                                                                          • Slide 41
                                                                          • Slide 42
                                                                          • Slide 43
                                                                          • Slide 44
                                                                          • Slide 45
                                                                          • Slide 46
                                                                          • Slide 47
                                                                          • Slide 48
                                                                          • Slide 49
                                                                          • Slide 50
                                                                          • Slide 51
                                                                          • Slide 52
                                                                          • Slide 53
                                                                          • Slide 54
                                                                          • Slide 55
                                                                          • Slide 56
                                                                          • Slide 57
                                                                          • Slide 58
                                                                          • Slide 59
                                                                          • Slide 60
                                                                          • Slide 61
                                                                          • Slide 62

                                                                            3838

                                                                            First and Second derivatives (First and Second derivatives ( 导数导数 ))

                                                                            the signs of the the signs of the

                                                                            derivatives would be derivatives would be

                                                                            reversed for an edge reversed for an edge

                                                                            that transitions from that transitions from

                                                                            light to darklight to dark

                                                                            First First derivatderivatee

                                                                            SeconSecond d derivatderivatee

                                                                            Gray-Gray-level level profileprofile

                                                                            3939

                                                                            Second derivativesSecond derivatives

                                                                            an undesirable featurean undesirable feature

                                                                            produces 2 values for every edge in an produces 2 values for every edge in an

                                                                            imageimage

                                                                            zero-crossing propertyzero-crossing property

                                                                            an imaginary straight line joining the an imaginary straight line joining the

                                                                            extreme positive and negative values of extreme positive and negative values of

                                                                            the second derivative would cross zero the second derivative would cross zero

                                                                            near the midpoint of the edgenear the midpoint of the edge

                                                                            quite useful for locating the centers of quite useful for locating the centers of

                                                                            thick edgesthick edges

                                                                            4040

                                                                            Basic idea of edge detectionBasic idea of edge detection

                                                                            A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                                            the edge direction and the results are the edge direction and the results are

                                                                            interpretedinterpreted

                                                                            The magnitude of the first derivative is The magnitude of the first derivative is

                                                                            used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                                            ramp)ramp)

                                                                            The sign of the second derivative can The sign of the second derivative can

                                                                            determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                                            dark or light side of an edgedark or light side of an edge

                                                                            4141

                                                                            Review of First DerivateReview of First Derivate

                                                                            Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                                            Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                                            Sobel operators 3Sobel operators 333

                                                                            6 5 8 5x yG z z G z z

                                                                            1 2 3

                                                                            4 5 6

                                                                            7 8 9

                                                                            z z z

                                                                            z z z

                                                                            z z z

                                                                            1 2 3

                                                                            4 5 6

                                                                            7 8 9

                                                                            z z z

                                                                            z z z

                                                                            z z z

                                                                            9 5 8 6x yG z z G z z 1 0 0 1

                                                                            0 1 1 0

                                                                            1 0 0 1

                                                                            0 1 1 0

                                                                            7 8 9 1 2 3

                                                                            3 6 9 1 4 7

                                                                            2 2

                                                                            2 2

                                                                            x

                                                                            y

                                                                            G z z z z z z

                                                                            G z z z z z z

                                                                            1 2 1 1 0 1

                                                                            0 0 0 2 0 2

                                                                            1 2 1 1 0 1

                                                                            1 2 1 1 0 1

                                                                            0 0 0 2 0 2

                                                                            1 2 1 1 0 1

                                                                            x yf G G

                                                                            4242

                                                                            Edge direction and strengthEdge direction and strength

                                                                            Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                                            α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                                            The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                                            The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                                            2 2x yf G G

                                                                            4343

                                                                            Gradient MasksGradient Masks

                                                                            1 0 0 1

                                                                            0 1 1 0

                                                                            Roberts

                                                                            1 0 0 1

                                                                            0 1 1 0

                                                                            Roberts

                                                                            1 2 1 1 0 1

                                                                            0 0 0 2 0 2

                                                                            1 2 1 1 0 1

                                                                            Sobel

                                                                            1 2 1 1 0 1

                                                                            0 0 0 2 0 2

                                                                            1 2 1 1 0 1

                                                                            Sobel

                                                                            1 1 1 1 0 1

                                                                            0 0 0 1 0 1

                                                                            1 1 1 1 0 1

                                                                            Prewitt

                                                                            1 1 1 1 0 1

                                                                            0 0 0 1 0 1

                                                                            1 1 1 1 0 1

                                                                            Prewitt

                                                                            4444

                                                                            Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                            0 1 1 1 1 0

                                                                            1 0 1 1 0 1

                                                                            1 1 0 0 1 1

                                                                            Prewitt

                                                                            0 1 1 1 1 0

                                                                            1 0 1 1 0 1

                                                                            1 1 0 0 1 1

                                                                            Prewitt

                                                                            4545

                                                                            Review of Second DerivateReview of Second Derivate

                                                                            Laplacian OperatorLaplacian Operator

                                                                            21 1

                                                                            1 1 4

                                                                            f x y f x yf

                                                                            f x y f x y f x y

                                                                            0 1 0

                                                                            1 4 1

                                                                            0 1 0

                                                                            0 1 0

                                                                            1 4 1

                                                                            0 1 0

                                                                            LaplacianLaplacian

                                                                            MaskMask

                                                                            1 1 1

                                                                            1 8 1

                                                                            1 1 1

                                                                            1 1 1

                                                                            1 8 1

                                                                            1 1 1

                                                                            4646

                                                                            Example of edge detectionExample of edge detection

                                                                            See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                            Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                            derivative)derivative)

                                                                            produces double edgesproduces double edges

                                                                            unable to detect edge directionunable to detect edge direction

                                                                            4747

                                                                            Canny edge detectorCanny edge detector

                                                                            The most powerful edge-detection The most powerful edge-detection

                                                                            method method

                                                                            It differs from the other edge-It differs from the other edge-

                                                                            detection methods in that detection methods in that

                                                                            it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                            strong and weak edges) strong and weak edges)

                                                                            and includes the weak edges in the and includes the weak edges in the

                                                                            output only if they are connected to output only if they are connected to

                                                                            strong edges strong edges

                                                                            This method is therefore less likely This method is therefore less likely

                                                                            than the others to be fooled by than the others to be fooled by

                                                                            noise and more likely to detect true noise and more likely to detect true

                                                                            weak edgesweak edges

                                                                            4848

                                                                            Laplacian of GaussianLaplacian of Gaussian

                                                                            Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                            2 2 22

                                                                            4 2

                                                                            2 2 2

                                                                            2exp

                                                                            r rh

                                                                            r x y

                                                                            determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                            4949

                                                                            Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                            0 0 1 0 0

                                                                            0 1 2 1 0

                                                                            1 2 16 2 1

                                                                            0 1 2 1 0

                                                                            0 0 1 0 0

                                                                            0 0 1 0 0

                                                                            0 1 2 1 0

                                                                            1 2 16 2 1

                                                                            0 1 2 1 0

                                                                            0 0 1 0 0

                                                                            The coefficient must sum to The coefficient must sum to

                                                                            zerozero

                                                                            5050

                                                                            Edge Detection and Edge Detection and SegmentationSegmentation

                                                                            Image resulting from edge detection cannot be used as a segmentation result

                                                                            Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                            5151

                                                                            75 Region-based 75 Region-based SegmentationSegmentation

                                                                            GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                            Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                            PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                            For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                            PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                            For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                            5252

                                                                            Two methods of Region Two methods of Region SegmentationSegmentation

                                                                            Region GrowingRegion Growing

                                                                            Region SplittingRegion Splitting

                                                                            Region growing is the opposite of the Region growing is the opposite of the

                                                                            split and merge approachsplit and merge approach

                                                                            5353

                                                                            Region GrowingRegion Growing

                                                                            The objective of segmentation is to The objective of segmentation is to

                                                                            partition an image into regionspartition an image into regions

                                                                            A region is a connected component with A region is a connected component with

                                                                            some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                            texture)texture)

                                                                            In region growing we start with a set In region growing we start with a set

                                                                            of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                            appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                            pixels if they have pixels if they have similar propertiessimilar properties

                                                                            such as specific ranges of gray level such as specific ranges of gray level

                                                                            and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                            Need initialization Need initialization similarity similarity

                                                                            criterioncriterion

                                                                            5454

                                                                            Steps of Region GrowingSteps of Region Growing

                                                                            Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                            pixel andpixel and compare it with neighbor compare it with neighbor

                                                                            ppixelsixels

                                                                            When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                            is grown from the seed pixel by is grown from the seed pixel by

                                                                            addingadding neighboneighborr pixel pixelss

                                                                            When the growth of one region stopsWhen the growth of one region stops

                                                                            choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                            againagain

                                                                            5555

                                                                            Region Region growing growing

                                                                            An initial set of small An initial set of small

                                                                            areas are iterativelyareas are iteratively

                                                                            merged according to merged according to

                                                                            similarity constraintssimilarity constraints

                                                                            5656

                                                                            Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                            X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                            We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                            On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                            Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                            SeedSeed pointspoints

                                                                            5757

                                                                            CriterionCriterion

                                                                            There is a valley at around 190 in the There is a valley at around 190 in the

                                                                            histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                            to be considered as a part of region to the to be considered as a part of region to the

                                                                            seed pointseed point

                                                                            The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                            to at least one pixel in that regionto at least one pixel in that region

                                                                            Result of region growing and boundaries of Result of region growing and boundaries of

                                                                            defectsdefects

                                                                            5858

                                                                            Region SplittingRegion Splitting

                                                                            The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                            ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                            If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                            This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                            Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                            5959

                                                                            Region SplittingRegion Splitting

                                                                            DisadvantageDisadvantage

                                                                            they create regions that may be adjacent they create regions that may be adjacent

                                                                            and homogeneous but not mergedand homogeneous but not merged

                                                                            6060

                                                                            Region Splitting and MergingRegion Splitting and Merging

                                                                            ProcedureProcedure

                                                                            11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                            22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                            33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                            6161

                                                                            Region Splitting and Merging

                                                                            Quadtree

                                                                            (四叉树 )

                                                                            6262

                                                                            PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                            where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                            mmii is the mean gray level of that region is the mean gray level of that region

                                                                            σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                            ExampleExample

                                                                            Original Original

                                                                            imageimageThresholded imageThresholded image Result of Result of

                                                                            Splitting and Splitting and

                                                                            MergingMerging

                                                                            • Slide 1
                                                                            • Slide 2
                                                                            • Slide 3
                                                                            • Slide 4
                                                                            • Slide 5
                                                                            • Slide 6
                                                                            • Slide 7
                                                                            • Slide 8
                                                                            • Slide 9
                                                                            • Slide 10
                                                                            • Slide 11
                                                                            • Slide 12
                                                                            • Slide 13
                                                                            • Slide 14
                                                                            • Slide 15
                                                                            • Slide 16
                                                                            • Slide 17
                                                                            • Slide 18
                                                                            • Slide 19
                                                                            • Slide 20
                                                                            • Slide 21
                                                                            • Slide 22
                                                                            • Slide 23
                                                                            • Slide 24
                                                                            • Slide 25
                                                                            • Slide 26
                                                                            • Slide 27
                                                                            • Slide 28
                                                                            • Slide 29
                                                                            • Slide 30
                                                                            • Slide 31
                                                                            • Slide 32
                                                                            • Slide 33
                                                                            • Slide 34
                                                                            • Slide 35
                                                                            • Slide 36
                                                                            • Slide 37
                                                                            • Slide 38
                                                                            • Slide 39
                                                                            • Slide 40
                                                                            • Slide 41
                                                                            • Slide 42
                                                                            • Slide 43
                                                                            • Slide 44
                                                                            • Slide 45
                                                                            • Slide 46
                                                                            • Slide 47
                                                                            • Slide 48
                                                                            • Slide 49
                                                                            • Slide 50
                                                                            • Slide 51
                                                                            • Slide 52
                                                                            • Slide 53
                                                                            • Slide 54
                                                                            • Slide 55
                                                                            • Slide 56
                                                                            • Slide 57
                                                                            • Slide 58
                                                                            • Slide 59
                                                                            • Slide 60
                                                                            • Slide 61
                                                                            • Slide 62

                                                                              3939

                                                                              Second derivativesSecond derivatives

                                                                              an undesirable featurean undesirable feature

                                                                              produces 2 values for every edge in an produces 2 values for every edge in an

                                                                              imageimage

                                                                              zero-crossing propertyzero-crossing property

                                                                              an imaginary straight line joining the an imaginary straight line joining the

                                                                              extreme positive and negative values of extreme positive and negative values of

                                                                              the second derivative would cross zero the second derivative would cross zero

                                                                              near the midpoint of the edgenear the midpoint of the edge

                                                                              quite useful for locating the centers of quite useful for locating the centers of

                                                                              thick edgesthick edges

                                                                              4040

                                                                              Basic idea of edge detectionBasic idea of edge detection

                                                                              A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                                              the edge direction and the results are the edge direction and the results are

                                                                              interpretedinterpreted

                                                                              The magnitude of the first derivative is The magnitude of the first derivative is

                                                                              used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                                              ramp)ramp)

                                                                              The sign of the second derivative can The sign of the second derivative can

                                                                              determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                                              dark or light side of an edgedark or light side of an edge

                                                                              4141

                                                                              Review of First DerivateReview of First Derivate

                                                                              Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                                              Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                                              Sobel operators 3Sobel operators 333

                                                                              6 5 8 5x yG z z G z z

                                                                              1 2 3

                                                                              4 5 6

                                                                              7 8 9

                                                                              z z z

                                                                              z z z

                                                                              z z z

                                                                              1 2 3

                                                                              4 5 6

                                                                              7 8 9

                                                                              z z z

                                                                              z z z

                                                                              z z z

                                                                              9 5 8 6x yG z z G z z 1 0 0 1

                                                                              0 1 1 0

                                                                              1 0 0 1

                                                                              0 1 1 0

                                                                              7 8 9 1 2 3

                                                                              3 6 9 1 4 7

                                                                              2 2

                                                                              2 2

                                                                              x

                                                                              y

                                                                              G z z z z z z

                                                                              G z z z z z z

                                                                              1 2 1 1 0 1

                                                                              0 0 0 2 0 2

                                                                              1 2 1 1 0 1

                                                                              1 2 1 1 0 1

                                                                              0 0 0 2 0 2

                                                                              1 2 1 1 0 1

                                                                              x yf G G

                                                                              4242

                                                                              Edge direction and strengthEdge direction and strength

                                                                              Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                                              α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                                              The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                                              The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                                              2 2x yf G G

                                                                              4343

                                                                              Gradient MasksGradient Masks

                                                                              1 0 0 1

                                                                              0 1 1 0

                                                                              Roberts

                                                                              1 0 0 1

                                                                              0 1 1 0

                                                                              Roberts

                                                                              1 2 1 1 0 1

                                                                              0 0 0 2 0 2

                                                                              1 2 1 1 0 1

                                                                              Sobel

                                                                              1 2 1 1 0 1

                                                                              0 0 0 2 0 2

                                                                              1 2 1 1 0 1

                                                                              Sobel

                                                                              1 1 1 1 0 1

                                                                              0 0 0 1 0 1

                                                                              1 1 1 1 0 1

                                                                              Prewitt

                                                                              1 1 1 1 0 1

                                                                              0 0 0 1 0 1

                                                                              1 1 1 1 0 1

                                                                              Prewitt

                                                                              4444

                                                                              Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                              0 1 1 1 1 0

                                                                              1 0 1 1 0 1

                                                                              1 1 0 0 1 1

                                                                              Prewitt

                                                                              0 1 1 1 1 0

                                                                              1 0 1 1 0 1

                                                                              1 1 0 0 1 1

                                                                              Prewitt

                                                                              4545

                                                                              Review of Second DerivateReview of Second Derivate

                                                                              Laplacian OperatorLaplacian Operator

                                                                              21 1

                                                                              1 1 4

                                                                              f x y f x yf

                                                                              f x y f x y f x y

                                                                              0 1 0

                                                                              1 4 1

                                                                              0 1 0

                                                                              0 1 0

                                                                              1 4 1

                                                                              0 1 0

                                                                              LaplacianLaplacian

                                                                              MaskMask

                                                                              1 1 1

                                                                              1 8 1

                                                                              1 1 1

                                                                              1 1 1

                                                                              1 8 1

                                                                              1 1 1

                                                                              4646

                                                                              Example of edge detectionExample of edge detection

                                                                              See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                              Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                              derivative)derivative)

                                                                              produces double edgesproduces double edges

                                                                              unable to detect edge directionunable to detect edge direction

                                                                              4747

                                                                              Canny edge detectorCanny edge detector

                                                                              The most powerful edge-detection The most powerful edge-detection

                                                                              method method

                                                                              It differs from the other edge-It differs from the other edge-

                                                                              detection methods in that detection methods in that

                                                                              it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                              strong and weak edges) strong and weak edges)

                                                                              and includes the weak edges in the and includes the weak edges in the

                                                                              output only if they are connected to output only if they are connected to

                                                                              strong edges strong edges

                                                                              This method is therefore less likely This method is therefore less likely

                                                                              than the others to be fooled by than the others to be fooled by

                                                                              noise and more likely to detect true noise and more likely to detect true

                                                                              weak edgesweak edges

                                                                              4848

                                                                              Laplacian of GaussianLaplacian of Gaussian

                                                                              Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                              2 2 22

                                                                              4 2

                                                                              2 2 2

                                                                              2exp

                                                                              r rh

                                                                              r x y

                                                                              determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                              4949

                                                                              Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                              0 0 1 0 0

                                                                              0 1 2 1 0

                                                                              1 2 16 2 1

                                                                              0 1 2 1 0

                                                                              0 0 1 0 0

                                                                              0 0 1 0 0

                                                                              0 1 2 1 0

                                                                              1 2 16 2 1

                                                                              0 1 2 1 0

                                                                              0 0 1 0 0

                                                                              The coefficient must sum to The coefficient must sum to

                                                                              zerozero

                                                                              5050

                                                                              Edge Detection and Edge Detection and SegmentationSegmentation

                                                                              Image resulting from edge detection cannot be used as a segmentation result

                                                                              Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                              5151

                                                                              75 Region-based 75 Region-based SegmentationSegmentation

                                                                              GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                              Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                              PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                              For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                              PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                              For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                              5252

                                                                              Two methods of Region Two methods of Region SegmentationSegmentation

                                                                              Region GrowingRegion Growing

                                                                              Region SplittingRegion Splitting

                                                                              Region growing is the opposite of the Region growing is the opposite of the

                                                                              split and merge approachsplit and merge approach

                                                                              5353

                                                                              Region GrowingRegion Growing

                                                                              The objective of segmentation is to The objective of segmentation is to

                                                                              partition an image into regionspartition an image into regions

                                                                              A region is a connected component with A region is a connected component with

                                                                              some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                              texture)texture)

                                                                              In region growing we start with a set In region growing we start with a set

                                                                              of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                              appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                              pixels if they have pixels if they have similar propertiessimilar properties

                                                                              such as specific ranges of gray level such as specific ranges of gray level

                                                                              and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                              Need initialization Need initialization similarity similarity

                                                                              criterioncriterion

                                                                              5454

                                                                              Steps of Region GrowingSteps of Region Growing

                                                                              Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                              pixel andpixel and compare it with neighbor compare it with neighbor

                                                                              ppixelsixels

                                                                              When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                              is grown from the seed pixel by is grown from the seed pixel by

                                                                              addingadding neighboneighborr pixel pixelss

                                                                              When the growth of one region stopsWhen the growth of one region stops

                                                                              choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                              againagain

                                                                              5555

                                                                              Region Region growing growing

                                                                              An initial set of small An initial set of small

                                                                              areas are iterativelyareas are iteratively

                                                                              merged according to merged according to

                                                                              similarity constraintssimilarity constraints

                                                                              5656

                                                                              Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                              X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                              We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                              On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                              Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                              SeedSeed pointspoints

                                                                              5757

                                                                              CriterionCriterion

                                                                              There is a valley at around 190 in the There is a valley at around 190 in the

                                                                              histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                              to be considered as a part of region to the to be considered as a part of region to the

                                                                              seed pointseed point

                                                                              The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                              to at least one pixel in that regionto at least one pixel in that region

                                                                              Result of region growing and boundaries of Result of region growing and boundaries of

                                                                              defectsdefects

                                                                              5858

                                                                              Region SplittingRegion Splitting

                                                                              The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                              ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                              If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                              This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                              Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                              5959

                                                                              Region SplittingRegion Splitting

                                                                              DisadvantageDisadvantage

                                                                              they create regions that may be adjacent they create regions that may be adjacent

                                                                              and homogeneous but not mergedand homogeneous but not merged

                                                                              6060

                                                                              Region Splitting and MergingRegion Splitting and Merging

                                                                              ProcedureProcedure

                                                                              11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                              22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                              33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                              6161

                                                                              Region Splitting and Merging

                                                                              Quadtree

                                                                              (四叉树 )

                                                                              6262

                                                                              PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                              where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                              mmii is the mean gray level of that region is the mean gray level of that region

                                                                              σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                              ExampleExample

                                                                              Original Original

                                                                              imageimageThresholded imageThresholded image Result of Result of

                                                                              Splitting and Splitting and

                                                                              MergingMerging

                                                                              • Slide 1
                                                                              • Slide 2
                                                                              • Slide 3
                                                                              • Slide 4
                                                                              • Slide 5
                                                                              • Slide 6
                                                                              • Slide 7
                                                                              • Slide 8
                                                                              • Slide 9
                                                                              • Slide 10
                                                                              • Slide 11
                                                                              • Slide 12
                                                                              • Slide 13
                                                                              • Slide 14
                                                                              • Slide 15
                                                                              • Slide 16
                                                                              • Slide 17
                                                                              • Slide 18
                                                                              • Slide 19
                                                                              • Slide 20
                                                                              • Slide 21
                                                                              • Slide 22
                                                                              • Slide 23
                                                                              • Slide 24
                                                                              • Slide 25
                                                                              • Slide 26
                                                                              • Slide 27
                                                                              • Slide 28
                                                                              • Slide 29
                                                                              • Slide 30
                                                                              • Slide 31
                                                                              • Slide 32
                                                                              • Slide 33
                                                                              • Slide 34
                                                                              • Slide 35
                                                                              • Slide 36
                                                                              • Slide 37
                                                                              • Slide 38
                                                                              • Slide 39
                                                                              • Slide 40
                                                                              • Slide 41
                                                                              • Slide 42
                                                                              • Slide 43
                                                                              • Slide 44
                                                                              • Slide 45
                                                                              • Slide 46
                                                                              • Slide 47
                                                                              • Slide 48
                                                                              • Slide 49
                                                                              • Slide 50
                                                                              • Slide 51
                                                                              • Slide 52
                                                                              • Slide 53
                                                                              • Slide 54
                                                                              • Slide 55
                                                                              • Slide 56
                                                                              • Slide 57
                                                                              • Slide 58
                                                                              • Slide 59
                                                                              • Slide 60
                                                                              • Slide 61
                                                                              • Slide 62

                                                                                4040

                                                                                Basic idea of edge detectionBasic idea of edge detection

                                                                                A profile is defined perpendicularly to A profile is defined perpendicularly to

                                                                                the edge direction and the results are the edge direction and the results are

                                                                                interpretedinterpreted

                                                                                The magnitude of the first derivative is The magnitude of the first derivative is

                                                                                used to detect an edge (if a point is on a used to detect an edge (if a point is on a

                                                                                ramp)ramp)

                                                                                The sign of the second derivative can The sign of the second derivative can

                                                                                determine whether an edge pixel is on the determine whether an edge pixel is on the

                                                                                dark or light side of an edgedark or light side of an edge

                                                                                4141

                                                                                Review of First DerivateReview of First Derivate

                                                                                Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                                                Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                                                Sobel operators 3Sobel operators 333

                                                                                6 5 8 5x yG z z G z z

                                                                                1 2 3

                                                                                4 5 6

                                                                                7 8 9

                                                                                z z z

                                                                                z z z

                                                                                z z z

                                                                                1 2 3

                                                                                4 5 6

                                                                                7 8 9

                                                                                z z z

                                                                                z z z

                                                                                z z z

                                                                                9 5 8 6x yG z z G z z 1 0 0 1

                                                                                0 1 1 0

                                                                                1 0 0 1

                                                                                0 1 1 0

                                                                                7 8 9 1 2 3

                                                                                3 6 9 1 4 7

                                                                                2 2

                                                                                2 2

                                                                                x

                                                                                y

                                                                                G z z z z z z

                                                                                G z z z z z z

                                                                                1 2 1 1 0 1

                                                                                0 0 0 2 0 2

                                                                                1 2 1 1 0 1

                                                                                1 2 1 1 0 1

                                                                                0 0 0 2 0 2

                                                                                1 2 1 1 0 1

                                                                                x yf G G

                                                                                4242

                                                                                Edge direction and strengthEdge direction and strength

                                                                                Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                                                α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                                                The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                                                The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                                                2 2x yf G G

                                                                                4343

                                                                                Gradient MasksGradient Masks

                                                                                1 0 0 1

                                                                                0 1 1 0

                                                                                Roberts

                                                                                1 0 0 1

                                                                                0 1 1 0

                                                                                Roberts

                                                                                1 2 1 1 0 1

                                                                                0 0 0 2 0 2

                                                                                1 2 1 1 0 1

                                                                                Sobel

                                                                                1 2 1 1 0 1

                                                                                0 0 0 2 0 2

                                                                                1 2 1 1 0 1

                                                                                Sobel

                                                                                1 1 1 1 0 1

                                                                                0 0 0 1 0 1

                                                                                1 1 1 1 0 1

                                                                                Prewitt

                                                                                1 1 1 1 0 1

                                                                                0 0 0 1 0 1

                                                                                1 1 1 1 0 1

                                                                                Prewitt

                                                                                4444

                                                                                Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                                0 1 1 1 1 0

                                                                                1 0 1 1 0 1

                                                                                1 1 0 0 1 1

                                                                                Prewitt

                                                                                0 1 1 1 1 0

                                                                                1 0 1 1 0 1

                                                                                1 1 0 0 1 1

                                                                                Prewitt

                                                                                4545

                                                                                Review of Second DerivateReview of Second Derivate

                                                                                Laplacian OperatorLaplacian Operator

                                                                                21 1

                                                                                1 1 4

                                                                                f x y f x yf

                                                                                f x y f x y f x y

                                                                                0 1 0

                                                                                1 4 1

                                                                                0 1 0

                                                                                0 1 0

                                                                                1 4 1

                                                                                0 1 0

                                                                                LaplacianLaplacian

                                                                                MaskMask

                                                                                1 1 1

                                                                                1 8 1

                                                                                1 1 1

                                                                                1 1 1

                                                                                1 8 1

                                                                                1 1 1

                                                                                4646

                                                                                Example of edge detectionExample of edge detection

                                                                                See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                                Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                                derivative)derivative)

                                                                                produces double edgesproduces double edges

                                                                                unable to detect edge directionunable to detect edge direction

                                                                                4747

                                                                                Canny edge detectorCanny edge detector

                                                                                The most powerful edge-detection The most powerful edge-detection

                                                                                method method

                                                                                It differs from the other edge-It differs from the other edge-

                                                                                detection methods in that detection methods in that

                                                                                it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                                strong and weak edges) strong and weak edges)

                                                                                and includes the weak edges in the and includes the weak edges in the

                                                                                output only if they are connected to output only if they are connected to

                                                                                strong edges strong edges

                                                                                This method is therefore less likely This method is therefore less likely

                                                                                than the others to be fooled by than the others to be fooled by

                                                                                noise and more likely to detect true noise and more likely to detect true

                                                                                weak edgesweak edges

                                                                                4848

                                                                                Laplacian of GaussianLaplacian of Gaussian

                                                                                Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                                2 2 22

                                                                                4 2

                                                                                2 2 2

                                                                                2exp

                                                                                r rh

                                                                                r x y

                                                                                determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                                4949

                                                                                Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                                0 0 1 0 0

                                                                                0 1 2 1 0

                                                                                1 2 16 2 1

                                                                                0 1 2 1 0

                                                                                0 0 1 0 0

                                                                                0 0 1 0 0

                                                                                0 1 2 1 0

                                                                                1 2 16 2 1

                                                                                0 1 2 1 0

                                                                                0 0 1 0 0

                                                                                The coefficient must sum to The coefficient must sum to

                                                                                zerozero

                                                                                5050

                                                                                Edge Detection and Edge Detection and SegmentationSegmentation

                                                                                Image resulting from edge detection cannot be used as a segmentation result

                                                                                Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                                5151

                                                                                75 Region-based 75 Region-based SegmentationSegmentation

                                                                                GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                                Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                                PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                5252

                                                                                Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                Region GrowingRegion Growing

                                                                                Region SplittingRegion Splitting

                                                                                Region growing is the opposite of the Region growing is the opposite of the

                                                                                split and merge approachsplit and merge approach

                                                                                5353

                                                                                Region GrowingRegion Growing

                                                                                The objective of segmentation is to The objective of segmentation is to

                                                                                partition an image into regionspartition an image into regions

                                                                                A region is a connected component with A region is a connected component with

                                                                                some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                texture)texture)

                                                                                In region growing we start with a set In region growing we start with a set

                                                                                of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                pixels if they have pixels if they have similar propertiessimilar properties

                                                                                such as specific ranges of gray level such as specific ranges of gray level

                                                                                and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                Need initialization Need initialization similarity similarity

                                                                                criterioncriterion

                                                                                5454

                                                                                Steps of Region GrowingSteps of Region Growing

                                                                                Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                ppixelsixels

                                                                                When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                is grown from the seed pixel by is grown from the seed pixel by

                                                                                addingadding neighboneighborr pixel pixelss

                                                                                When the growth of one region stopsWhen the growth of one region stops

                                                                                choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                againagain

                                                                                5555

                                                                                Region Region growing growing

                                                                                An initial set of small An initial set of small

                                                                                areas are iterativelyareas are iteratively

                                                                                merged according to merged according to

                                                                                similarity constraintssimilarity constraints

                                                                                5656

                                                                                Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                SeedSeed pointspoints

                                                                                5757

                                                                                CriterionCriterion

                                                                                There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                to be considered as a part of region to the to be considered as a part of region to the

                                                                                seed pointseed point

                                                                                The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                to at least one pixel in that regionto at least one pixel in that region

                                                                                Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                defectsdefects

                                                                                5858

                                                                                Region SplittingRegion Splitting

                                                                                The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                5959

                                                                                Region SplittingRegion Splitting

                                                                                DisadvantageDisadvantage

                                                                                they create regions that may be adjacent they create regions that may be adjacent

                                                                                and homogeneous but not mergedand homogeneous but not merged

                                                                                6060

                                                                                Region Splitting and MergingRegion Splitting and Merging

                                                                                ProcedureProcedure

                                                                                11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                6161

                                                                                Region Splitting and Merging

                                                                                Quadtree

                                                                                (四叉树 )

                                                                                6262

                                                                                PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                mmii is the mean gray level of that region is the mean gray level of that region

                                                                                σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                ExampleExample

                                                                                Original Original

                                                                                imageimageThresholded imageThresholded image Result of Result of

                                                                                Splitting and Splitting and

                                                                                MergingMerging

                                                                                • Slide 1
                                                                                • Slide 2
                                                                                • Slide 3
                                                                                • Slide 4
                                                                                • Slide 5
                                                                                • Slide 6
                                                                                • Slide 7
                                                                                • Slide 8
                                                                                • Slide 9
                                                                                • Slide 10
                                                                                • Slide 11
                                                                                • Slide 12
                                                                                • Slide 13
                                                                                • Slide 14
                                                                                • Slide 15
                                                                                • Slide 16
                                                                                • Slide 17
                                                                                • Slide 18
                                                                                • Slide 19
                                                                                • Slide 20
                                                                                • Slide 21
                                                                                • Slide 22
                                                                                • Slide 23
                                                                                • Slide 24
                                                                                • Slide 25
                                                                                • Slide 26
                                                                                • Slide 27
                                                                                • Slide 28
                                                                                • Slide 29
                                                                                • Slide 30
                                                                                • Slide 31
                                                                                • Slide 32
                                                                                • Slide 33
                                                                                • Slide 34
                                                                                • Slide 35
                                                                                • Slide 36
                                                                                • Slide 37
                                                                                • Slide 38
                                                                                • Slide 39
                                                                                • Slide 40
                                                                                • Slide 41
                                                                                • Slide 42
                                                                                • Slide 43
                                                                                • Slide 44
                                                                                • Slide 45
                                                                                • Slide 46
                                                                                • Slide 47
                                                                                • Slide 48
                                                                                • Slide 49
                                                                                • Slide 50
                                                                                • Slide 51
                                                                                • Slide 52
                                                                                • Slide 53
                                                                                • Slide 54
                                                                                • Slide 55
                                                                                • Slide 56
                                                                                • Slide 57
                                                                                • Slide 58
                                                                                • Slide 59
                                                                                • Slide 60
                                                                                • Slide 61
                                                                                • Slide 62

                                                                                  4141

                                                                                  Review of First DerivateReview of First Derivate

                                                                                  Gradient OperatorGradient Operator simplest approximation 2simplest approximation 222

                                                                                  Roberts cross-gradientoperators 2Roberts cross-gradientoperators 222

                                                                                  Sobel operators 3Sobel operators 333

                                                                                  6 5 8 5x yG z z G z z

                                                                                  1 2 3

                                                                                  4 5 6

                                                                                  7 8 9

                                                                                  z z z

                                                                                  z z z

                                                                                  z z z

                                                                                  1 2 3

                                                                                  4 5 6

                                                                                  7 8 9

                                                                                  z z z

                                                                                  z z z

                                                                                  z z z

                                                                                  9 5 8 6x yG z z G z z 1 0 0 1

                                                                                  0 1 1 0

                                                                                  1 0 0 1

                                                                                  0 1 1 0

                                                                                  7 8 9 1 2 3

                                                                                  3 6 9 1 4 7

                                                                                  2 2

                                                                                  2 2

                                                                                  x

                                                                                  y

                                                                                  G z z z z z z

                                                                                  G z z z z z z

                                                                                  1 2 1 1 0 1

                                                                                  0 0 0 2 0 2

                                                                                  1 2 1 1 0 1

                                                                                  1 2 1 1 0 1

                                                                                  0 0 0 2 0 2

                                                                                  1 2 1 1 0 1

                                                                                  x yf G G

                                                                                  4242

                                                                                  Edge direction and strengthEdge direction and strength

                                                                                  Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                                                  α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                                                  The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                                                  The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                                                  2 2x yf G G

                                                                                  4343

                                                                                  Gradient MasksGradient Masks

                                                                                  1 0 0 1

                                                                                  0 1 1 0

                                                                                  Roberts

                                                                                  1 0 0 1

                                                                                  0 1 1 0

                                                                                  Roberts

                                                                                  1 2 1 1 0 1

                                                                                  0 0 0 2 0 2

                                                                                  1 2 1 1 0 1

                                                                                  Sobel

                                                                                  1 2 1 1 0 1

                                                                                  0 0 0 2 0 2

                                                                                  1 2 1 1 0 1

                                                                                  Sobel

                                                                                  1 1 1 1 0 1

                                                                                  0 0 0 1 0 1

                                                                                  1 1 1 1 0 1

                                                                                  Prewitt

                                                                                  1 1 1 1 0 1

                                                                                  0 0 0 1 0 1

                                                                                  1 1 1 1 0 1

                                                                                  Prewitt

                                                                                  4444

                                                                                  Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                                  0 1 1 1 1 0

                                                                                  1 0 1 1 0 1

                                                                                  1 1 0 0 1 1

                                                                                  Prewitt

                                                                                  0 1 1 1 1 0

                                                                                  1 0 1 1 0 1

                                                                                  1 1 0 0 1 1

                                                                                  Prewitt

                                                                                  4545

                                                                                  Review of Second DerivateReview of Second Derivate

                                                                                  Laplacian OperatorLaplacian Operator

                                                                                  21 1

                                                                                  1 1 4

                                                                                  f x y f x yf

                                                                                  f x y f x y f x y

                                                                                  0 1 0

                                                                                  1 4 1

                                                                                  0 1 0

                                                                                  0 1 0

                                                                                  1 4 1

                                                                                  0 1 0

                                                                                  LaplacianLaplacian

                                                                                  MaskMask

                                                                                  1 1 1

                                                                                  1 8 1

                                                                                  1 1 1

                                                                                  1 1 1

                                                                                  1 8 1

                                                                                  1 1 1

                                                                                  4646

                                                                                  Example of edge detectionExample of edge detection

                                                                                  See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                                  Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                                  derivative)derivative)

                                                                                  produces double edgesproduces double edges

                                                                                  unable to detect edge directionunable to detect edge direction

                                                                                  4747

                                                                                  Canny edge detectorCanny edge detector

                                                                                  The most powerful edge-detection The most powerful edge-detection

                                                                                  method method

                                                                                  It differs from the other edge-It differs from the other edge-

                                                                                  detection methods in that detection methods in that

                                                                                  it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                                  strong and weak edges) strong and weak edges)

                                                                                  and includes the weak edges in the and includes the weak edges in the

                                                                                  output only if they are connected to output only if they are connected to

                                                                                  strong edges strong edges

                                                                                  This method is therefore less likely This method is therefore less likely

                                                                                  than the others to be fooled by than the others to be fooled by

                                                                                  noise and more likely to detect true noise and more likely to detect true

                                                                                  weak edgesweak edges

                                                                                  4848

                                                                                  Laplacian of GaussianLaplacian of Gaussian

                                                                                  Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                                  2 2 22

                                                                                  4 2

                                                                                  2 2 2

                                                                                  2exp

                                                                                  r rh

                                                                                  r x y

                                                                                  determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                                  4949

                                                                                  Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                                  0 0 1 0 0

                                                                                  0 1 2 1 0

                                                                                  1 2 16 2 1

                                                                                  0 1 2 1 0

                                                                                  0 0 1 0 0

                                                                                  0 0 1 0 0

                                                                                  0 1 2 1 0

                                                                                  1 2 16 2 1

                                                                                  0 1 2 1 0

                                                                                  0 0 1 0 0

                                                                                  The coefficient must sum to The coefficient must sum to

                                                                                  zerozero

                                                                                  5050

                                                                                  Edge Detection and Edge Detection and SegmentationSegmentation

                                                                                  Image resulting from edge detection cannot be used as a segmentation result

                                                                                  Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                                  5151

                                                                                  75 Region-based 75 Region-based SegmentationSegmentation

                                                                                  GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                                  Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                                  PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                  For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                  PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                  For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                  5252

                                                                                  Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                  Region GrowingRegion Growing

                                                                                  Region SplittingRegion Splitting

                                                                                  Region growing is the opposite of the Region growing is the opposite of the

                                                                                  split and merge approachsplit and merge approach

                                                                                  5353

                                                                                  Region GrowingRegion Growing

                                                                                  The objective of segmentation is to The objective of segmentation is to

                                                                                  partition an image into regionspartition an image into regions

                                                                                  A region is a connected component with A region is a connected component with

                                                                                  some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                  texture)texture)

                                                                                  In region growing we start with a set In region growing we start with a set

                                                                                  of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                  appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                  pixels if they have pixels if they have similar propertiessimilar properties

                                                                                  such as specific ranges of gray level such as specific ranges of gray level

                                                                                  and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                  Need initialization Need initialization similarity similarity

                                                                                  criterioncriterion

                                                                                  5454

                                                                                  Steps of Region GrowingSteps of Region Growing

                                                                                  Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                  pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                  ppixelsixels

                                                                                  When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                  is grown from the seed pixel by is grown from the seed pixel by

                                                                                  addingadding neighboneighborr pixel pixelss

                                                                                  When the growth of one region stopsWhen the growth of one region stops

                                                                                  choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                  againagain

                                                                                  5555

                                                                                  Region Region growing growing

                                                                                  An initial set of small An initial set of small

                                                                                  areas are iterativelyareas are iteratively

                                                                                  merged according to merged according to

                                                                                  similarity constraintssimilarity constraints

                                                                                  5656

                                                                                  Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                  X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                  We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                  On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                  Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                  SeedSeed pointspoints

                                                                                  5757

                                                                                  CriterionCriterion

                                                                                  There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                  histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                  to be considered as a part of region to the to be considered as a part of region to the

                                                                                  seed pointseed point

                                                                                  The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                  to at least one pixel in that regionto at least one pixel in that region

                                                                                  Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                  defectsdefects

                                                                                  5858

                                                                                  Region SplittingRegion Splitting

                                                                                  The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                  ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                  If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                  This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                  Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                  5959

                                                                                  Region SplittingRegion Splitting

                                                                                  DisadvantageDisadvantage

                                                                                  they create regions that may be adjacent they create regions that may be adjacent

                                                                                  and homogeneous but not mergedand homogeneous but not merged

                                                                                  6060

                                                                                  Region Splitting and MergingRegion Splitting and Merging

                                                                                  ProcedureProcedure

                                                                                  11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                  22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                  33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                  6161

                                                                                  Region Splitting and Merging

                                                                                  Quadtree

                                                                                  (四叉树 )

                                                                                  6262

                                                                                  PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                  where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                  mmii is the mean gray level of that region is the mean gray level of that region

                                                                                  σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                  ExampleExample

                                                                                  Original Original

                                                                                  imageimageThresholded imageThresholded image Result of Result of

                                                                                  Splitting and Splitting and

                                                                                  MergingMerging

                                                                                  • Slide 1
                                                                                  • Slide 2
                                                                                  • Slide 3
                                                                                  • Slide 4
                                                                                  • Slide 5
                                                                                  • Slide 6
                                                                                  • Slide 7
                                                                                  • Slide 8
                                                                                  • Slide 9
                                                                                  • Slide 10
                                                                                  • Slide 11
                                                                                  • Slide 12
                                                                                  • Slide 13
                                                                                  • Slide 14
                                                                                  • Slide 15
                                                                                  • Slide 16
                                                                                  • Slide 17
                                                                                  • Slide 18
                                                                                  • Slide 19
                                                                                  • Slide 20
                                                                                  • Slide 21
                                                                                  • Slide 22
                                                                                  • Slide 23
                                                                                  • Slide 24
                                                                                  • Slide 25
                                                                                  • Slide 26
                                                                                  • Slide 27
                                                                                  • Slide 28
                                                                                  • Slide 29
                                                                                  • Slide 30
                                                                                  • Slide 31
                                                                                  • Slide 32
                                                                                  • Slide 33
                                                                                  • Slide 34
                                                                                  • Slide 35
                                                                                  • Slide 36
                                                                                  • Slide 37
                                                                                  • Slide 38
                                                                                  • Slide 39
                                                                                  • Slide 40
                                                                                  • Slide 41
                                                                                  • Slide 42
                                                                                  • Slide 43
                                                                                  • Slide 44
                                                                                  • Slide 45
                                                                                  • Slide 46
                                                                                  • Slide 47
                                                                                  • Slide 48
                                                                                  • Slide 49
                                                                                  • Slide 50
                                                                                  • Slide 51
                                                                                  • Slide 52
                                                                                  • Slide 53
                                                                                  • Slide 54
                                                                                  • Slide 55
                                                                                  • Slide 56
                                                                                  • Slide 57
                                                                                  • Slide 58
                                                                                  • Slide 59
                                                                                  • Slide 60
                                                                                  • Slide 61
                                                                                  • Slide 62

                                                                                    4242

                                                                                    Edge direction and strengthEdge direction and strength

                                                                                    Let α(xy) represents the direction angle of tLet α(xy) represents the direction angle of the vector f at (xy)he vector f at (xy)

                                                                                    α(xy)=tanα(xy)=tan-1-1(GyGx)(GyGx)

                                                                                    The direction of an edge at (xy) perpendiculThe direction of an edge at (xy) perpendicular (ar ( 垂直垂直 ) to the direction of the gradient vec) to the direction of the gradient vector at that pointtor at that point

                                                                                    The edge strength is given by the gradient mThe edge strength is given by the gradient magnitudeagnitude

                                                                                    2 2x yf G G

                                                                                    4343

                                                                                    Gradient MasksGradient Masks

                                                                                    1 0 0 1

                                                                                    0 1 1 0

                                                                                    Roberts

                                                                                    1 0 0 1

                                                                                    0 1 1 0

                                                                                    Roberts

                                                                                    1 2 1 1 0 1

                                                                                    0 0 0 2 0 2

                                                                                    1 2 1 1 0 1

                                                                                    Sobel

                                                                                    1 2 1 1 0 1

                                                                                    0 0 0 2 0 2

                                                                                    1 2 1 1 0 1

                                                                                    Sobel

                                                                                    1 1 1 1 0 1

                                                                                    0 0 0 1 0 1

                                                                                    1 1 1 1 0 1

                                                                                    Prewitt

                                                                                    1 1 1 1 0 1

                                                                                    0 0 0 1 0 1

                                                                                    1 1 1 1 0 1

                                                                                    Prewitt

                                                                                    4444

                                                                                    Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                                    0 1 1 1 1 0

                                                                                    1 0 1 1 0 1

                                                                                    1 1 0 0 1 1

                                                                                    Prewitt

                                                                                    0 1 1 1 1 0

                                                                                    1 0 1 1 0 1

                                                                                    1 1 0 0 1 1

                                                                                    Prewitt

                                                                                    4545

                                                                                    Review of Second DerivateReview of Second Derivate

                                                                                    Laplacian OperatorLaplacian Operator

                                                                                    21 1

                                                                                    1 1 4

                                                                                    f x y f x yf

                                                                                    f x y f x y f x y

                                                                                    0 1 0

                                                                                    1 4 1

                                                                                    0 1 0

                                                                                    0 1 0

                                                                                    1 4 1

                                                                                    0 1 0

                                                                                    LaplacianLaplacian

                                                                                    MaskMask

                                                                                    1 1 1

                                                                                    1 8 1

                                                                                    1 1 1

                                                                                    1 1 1

                                                                                    1 8 1

                                                                                    1 1 1

                                                                                    4646

                                                                                    Example of edge detectionExample of edge detection

                                                                                    See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                                    Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                                    derivative)derivative)

                                                                                    produces double edgesproduces double edges

                                                                                    unable to detect edge directionunable to detect edge direction

                                                                                    4747

                                                                                    Canny edge detectorCanny edge detector

                                                                                    The most powerful edge-detection The most powerful edge-detection

                                                                                    method method

                                                                                    It differs from the other edge-It differs from the other edge-

                                                                                    detection methods in that detection methods in that

                                                                                    it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                                    strong and weak edges) strong and weak edges)

                                                                                    and includes the weak edges in the and includes the weak edges in the

                                                                                    output only if they are connected to output only if they are connected to

                                                                                    strong edges strong edges

                                                                                    This method is therefore less likely This method is therefore less likely

                                                                                    than the others to be fooled by than the others to be fooled by

                                                                                    noise and more likely to detect true noise and more likely to detect true

                                                                                    weak edgesweak edges

                                                                                    4848

                                                                                    Laplacian of GaussianLaplacian of Gaussian

                                                                                    Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                                    2 2 22

                                                                                    4 2

                                                                                    2 2 2

                                                                                    2exp

                                                                                    r rh

                                                                                    r x y

                                                                                    determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                                    4949

                                                                                    Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                                    0 0 1 0 0

                                                                                    0 1 2 1 0

                                                                                    1 2 16 2 1

                                                                                    0 1 2 1 0

                                                                                    0 0 1 0 0

                                                                                    0 0 1 0 0

                                                                                    0 1 2 1 0

                                                                                    1 2 16 2 1

                                                                                    0 1 2 1 0

                                                                                    0 0 1 0 0

                                                                                    The coefficient must sum to The coefficient must sum to

                                                                                    zerozero

                                                                                    5050

                                                                                    Edge Detection and Edge Detection and SegmentationSegmentation

                                                                                    Image resulting from edge detection cannot be used as a segmentation result

                                                                                    Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                                    5151

                                                                                    75 Region-based 75 Region-based SegmentationSegmentation

                                                                                    GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                                    Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                                    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                    5252

                                                                                    Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                    Region GrowingRegion Growing

                                                                                    Region SplittingRegion Splitting

                                                                                    Region growing is the opposite of the Region growing is the opposite of the

                                                                                    split and merge approachsplit and merge approach

                                                                                    5353

                                                                                    Region GrowingRegion Growing

                                                                                    The objective of segmentation is to The objective of segmentation is to

                                                                                    partition an image into regionspartition an image into regions

                                                                                    A region is a connected component with A region is a connected component with

                                                                                    some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                    texture)texture)

                                                                                    In region growing we start with a set In region growing we start with a set

                                                                                    of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                    appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                    pixels if they have pixels if they have similar propertiessimilar properties

                                                                                    such as specific ranges of gray level such as specific ranges of gray level

                                                                                    and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                    Need initialization Need initialization similarity similarity

                                                                                    criterioncriterion

                                                                                    5454

                                                                                    Steps of Region GrowingSteps of Region Growing

                                                                                    Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                    pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                    ppixelsixels

                                                                                    When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                    is grown from the seed pixel by is grown from the seed pixel by

                                                                                    addingadding neighboneighborr pixel pixelss

                                                                                    When the growth of one region stopsWhen the growth of one region stops

                                                                                    choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                    againagain

                                                                                    5555

                                                                                    Region Region growing growing

                                                                                    An initial set of small An initial set of small

                                                                                    areas are iterativelyareas are iteratively

                                                                                    merged according to merged according to

                                                                                    similarity constraintssimilarity constraints

                                                                                    5656

                                                                                    Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                    X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                    We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                    On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                    Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                    SeedSeed pointspoints

                                                                                    5757

                                                                                    CriterionCriterion

                                                                                    There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                    histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                    to be considered as a part of region to the to be considered as a part of region to the

                                                                                    seed pointseed point

                                                                                    The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                    to at least one pixel in that regionto at least one pixel in that region

                                                                                    Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                    defectsdefects

                                                                                    5858

                                                                                    Region SplittingRegion Splitting

                                                                                    The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                    ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                    If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                    This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                    Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                    5959

                                                                                    Region SplittingRegion Splitting

                                                                                    DisadvantageDisadvantage

                                                                                    they create regions that may be adjacent they create regions that may be adjacent

                                                                                    and homogeneous but not mergedand homogeneous but not merged

                                                                                    6060

                                                                                    Region Splitting and MergingRegion Splitting and Merging

                                                                                    ProcedureProcedure

                                                                                    11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                    22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                    33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                    6161

                                                                                    Region Splitting and Merging

                                                                                    Quadtree

                                                                                    (四叉树 )

                                                                                    6262

                                                                                    PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                    where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                    mmii is the mean gray level of that region is the mean gray level of that region

                                                                                    σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                    ExampleExample

                                                                                    Original Original

                                                                                    imageimageThresholded imageThresholded image Result of Result of

                                                                                    Splitting and Splitting and

                                                                                    MergingMerging

                                                                                    • Slide 1
                                                                                    • Slide 2
                                                                                    • Slide 3
                                                                                    • Slide 4
                                                                                    • Slide 5
                                                                                    • Slide 6
                                                                                    • Slide 7
                                                                                    • Slide 8
                                                                                    • Slide 9
                                                                                    • Slide 10
                                                                                    • Slide 11
                                                                                    • Slide 12
                                                                                    • Slide 13
                                                                                    • Slide 14
                                                                                    • Slide 15
                                                                                    • Slide 16
                                                                                    • Slide 17
                                                                                    • Slide 18
                                                                                    • Slide 19
                                                                                    • Slide 20
                                                                                    • Slide 21
                                                                                    • Slide 22
                                                                                    • Slide 23
                                                                                    • Slide 24
                                                                                    • Slide 25
                                                                                    • Slide 26
                                                                                    • Slide 27
                                                                                    • Slide 28
                                                                                    • Slide 29
                                                                                    • Slide 30
                                                                                    • Slide 31
                                                                                    • Slide 32
                                                                                    • Slide 33
                                                                                    • Slide 34
                                                                                    • Slide 35
                                                                                    • Slide 36
                                                                                    • Slide 37
                                                                                    • Slide 38
                                                                                    • Slide 39
                                                                                    • Slide 40
                                                                                    • Slide 41
                                                                                    • Slide 42
                                                                                    • Slide 43
                                                                                    • Slide 44
                                                                                    • Slide 45
                                                                                    • Slide 46
                                                                                    • Slide 47
                                                                                    • Slide 48
                                                                                    • Slide 49
                                                                                    • Slide 50
                                                                                    • Slide 51
                                                                                    • Slide 52
                                                                                    • Slide 53
                                                                                    • Slide 54
                                                                                    • Slide 55
                                                                                    • Slide 56
                                                                                    • Slide 57
                                                                                    • Slide 58
                                                                                    • Slide 59
                                                                                    • Slide 60
                                                                                    • Slide 61
                                                                                    • Slide 62

                                                                                      4343

                                                                                      Gradient MasksGradient Masks

                                                                                      1 0 0 1

                                                                                      0 1 1 0

                                                                                      Roberts

                                                                                      1 0 0 1

                                                                                      0 1 1 0

                                                                                      Roberts

                                                                                      1 2 1 1 0 1

                                                                                      0 0 0 2 0 2

                                                                                      1 2 1 1 0 1

                                                                                      Sobel

                                                                                      1 2 1 1 0 1

                                                                                      0 0 0 2 0 2

                                                                                      1 2 1 1 0 1

                                                                                      Sobel

                                                                                      1 1 1 1 0 1

                                                                                      0 0 0 1 0 1

                                                                                      1 1 1 1 0 1

                                                                                      Prewitt

                                                                                      1 1 1 1 0 1

                                                                                      0 0 0 1 0 1

                                                                                      1 1 1 1 0 1

                                                                                      Prewitt

                                                                                      4444

                                                                                      Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                                      0 1 1 1 1 0

                                                                                      1 0 1 1 0 1

                                                                                      1 1 0 0 1 1

                                                                                      Prewitt

                                                                                      0 1 1 1 1 0

                                                                                      1 0 1 1 0 1

                                                                                      1 1 0 0 1 1

                                                                                      Prewitt

                                                                                      4545

                                                                                      Review of Second DerivateReview of Second Derivate

                                                                                      Laplacian OperatorLaplacian Operator

                                                                                      21 1

                                                                                      1 1 4

                                                                                      f x y f x yf

                                                                                      f x y f x y f x y

                                                                                      0 1 0

                                                                                      1 4 1

                                                                                      0 1 0

                                                                                      0 1 0

                                                                                      1 4 1

                                                                                      0 1 0

                                                                                      LaplacianLaplacian

                                                                                      MaskMask

                                                                                      1 1 1

                                                                                      1 8 1

                                                                                      1 1 1

                                                                                      1 1 1

                                                                                      1 8 1

                                                                                      1 1 1

                                                                                      4646

                                                                                      Example of edge detectionExample of edge detection

                                                                                      See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                                      Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                                      derivative)derivative)

                                                                                      produces double edgesproduces double edges

                                                                                      unable to detect edge directionunable to detect edge direction

                                                                                      4747

                                                                                      Canny edge detectorCanny edge detector

                                                                                      The most powerful edge-detection The most powerful edge-detection

                                                                                      method method

                                                                                      It differs from the other edge-It differs from the other edge-

                                                                                      detection methods in that detection methods in that

                                                                                      it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                                      strong and weak edges) strong and weak edges)

                                                                                      and includes the weak edges in the and includes the weak edges in the

                                                                                      output only if they are connected to output only if they are connected to

                                                                                      strong edges strong edges

                                                                                      This method is therefore less likely This method is therefore less likely

                                                                                      than the others to be fooled by than the others to be fooled by

                                                                                      noise and more likely to detect true noise and more likely to detect true

                                                                                      weak edgesweak edges

                                                                                      4848

                                                                                      Laplacian of GaussianLaplacian of Gaussian

                                                                                      Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                                      2 2 22

                                                                                      4 2

                                                                                      2 2 2

                                                                                      2exp

                                                                                      r rh

                                                                                      r x y

                                                                                      determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                                      4949

                                                                                      Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                                      0 0 1 0 0

                                                                                      0 1 2 1 0

                                                                                      1 2 16 2 1

                                                                                      0 1 2 1 0

                                                                                      0 0 1 0 0

                                                                                      0 0 1 0 0

                                                                                      0 1 2 1 0

                                                                                      1 2 16 2 1

                                                                                      0 1 2 1 0

                                                                                      0 0 1 0 0

                                                                                      The coefficient must sum to The coefficient must sum to

                                                                                      zerozero

                                                                                      5050

                                                                                      Edge Detection and Edge Detection and SegmentationSegmentation

                                                                                      Image resulting from edge detection cannot be used as a segmentation result

                                                                                      Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                                      5151

                                                                                      75 Region-based 75 Region-based SegmentationSegmentation

                                                                                      GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                                      Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                                      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                      5252

                                                                                      Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                      Region GrowingRegion Growing

                                                                                      Region SplittingRegion Splitting

                                                                                      Region growing is the opposite of the Region growing is the opposite of the

                                                                                      split and merge approachsplit and merge approach

                                                                                      5353

                                                                                      Region GrowingRegion Growing

                                                                                      The objective of segmentation is to The objective of segmentation is to

                                                                                      partition an image into regionspartition an image into regions

                                                                                      A region is a connected component with A region is a connected component with

                                                                                      some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                      texture)texture)

                                                                                      In region growing we start with a set In region growing we start with a set

                                                                                      of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                      appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                      pixels if they have pixels if they have similar propertiessimilar properties

                                                                                      such as specific ranges of gray level such as specific ranges of gray level

                                                                                      and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                      Need initialization Need initialization similarity similarity

                                                                                      criterioncriterion

                                                                                      5454

                                                                                      Steps of Region GrowingSteps of Region Growing

                                                                                      Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                      pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                      ppixelsixels

                                                                                      When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                      is grown from the seed pixel by is grown from the seed pixel by

                                                                                      addingadding neighboneighborr pixel pixelss

                                                                                      When the growth of one region stopsWhen the growth of one region stops

                                                                                      choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                      againagain

                                                                                      5555

                                                                                      Region Region growing growing

                                                                                      An initial set of small An initial set of small

                                                                                      areas are iterativelyareas are iteratively

                                                                                      merged according to merged according to

                                                                                      similarity constraintssimilarity constraints

                                                                                      5656

                                                                                      Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                      X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                      We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                      On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                      Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                      SeedSeed pointspoints

                                                                                      5757

                                                                                      CriterionCriterion

                                                                                      There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                      histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                      to be considered as a part of region to the to be considered as a part of region to the

                                                                                      seed pointseed point

                                                                                      The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                      to at least one pixel in that regionto at least one pixel in that region

                                                                                      Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                      defectsdefects

                                                                                      5858

                                                                                      Region SplittingRegion Splitting

                                                                                      The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                      ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                      If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                      This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                      Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                      5959

                                                                                      Region SplittingRegion Splitting

                                                                                      DisadvantageDisadvantage

                                                                                      they create regions that may be adjacent they create regions that may be adjacent

                                                                                      and homogeneous but not mergedand homogeneous but not merged

                                                                                      6060

                                                                                      Region Splitting and MergingRegion Splitting and Merging

                                                                                      ProcedureProcedure

                                                                                      11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                      22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                      33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                      6161

                                                                                      Region Splitting and Merging

                                                                                      Quadtree

                                                                                      (四叉树 )

                                                                                      6262

                                                                                      PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                      where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                      mmii is the mean gray level of that region is the mean gray level of that region

                                                                                      σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                      ExampleExample

                                                                                      Original Original

                                                                                      imageimageThresholded imageThresholded image Result of Result of

                                                                                      Splitting and Splitting and

                                                                                      MergingMerging

                                                                                      • Slide 1
                                                                                      • Slide 2
                                                                                      • Slide 3
                                                                                      • Slide 4
                                                                                      • Slide 5
                                                                                      • Slide 6
                                                                                      • Slide 7
                                                                                      • Slide 8
                                                                                      • Slide 9
                                                                                      • Slide 10
                                                                                      • Slide 11
                                                                                      • Slide 12
                                                                                      • Slide 13
                                                                                      • Slide 14
                                                                                      • Slide 15
                                                                                      • Slide 16
                                                                                      • Slide 17
                                                                                      • Slide 18
                                                                                      • Slide 19
                                                                                      • Slide 20
                                                                                      • Slide 21
                                                                                      • Slide 22
                                                                                      • Slide 23
                                                                                      • Slide 24
                                                                                      • Slide 25
                                                                                      • Slide 26
                                                                                      • Slide 27
                                                                                      • Slide 28
                                                                                      • Slide 29
                                                                                      • Slide 30
                                                                                      • Slide 31
                                                                                      • Slide 32
                                                                                      • Slide 33
                                                                                      • Slide 34
                                                                                      • Slide 35
                                                                                      • Slide 36
                                                                                      • Slide 37
                                                                                      • Slide 38
                                                                                      • Slide 39
                                                                                      • Slide 40
                                                                                      • Slide 41
                                                                                      • Slide 42
                                                                                      • Slide 43
                                                                                      • Slide 44
                                                                                      • Slide 45
                                                                                      • Slide 46
                                                                                      • Slide 47
                                                                                      • Slide 48
                                                                                      • Slide 49
                                                                                      • Slide 50
                                                                                      • Slide 51
                                                                                      • Slide 52
                                                                                      • Slide 53
                                                                                      • Slide 54
                                                                                      • Slide 55
                                                                                      • Slide 56
                                                                                      • Slide 57
                                                                                      • Slide 58
                                                                                      • Slide 59
                                                                                      • Slide 60
                                                                                      • Slide 61
                                                                                      • Slide 62

                                                                                        4444

                                                                                        Diagonal edges with PrewittDiagonal edges with Prewittand Sobel masksand Sobel masks

                                                                                        0 1 1 1 1 0

                                                                                        1 0 1 1 0 1

                                                                                        1 1 0 0 1 1

                                                                                        Prewitt

                                                                                        0 1 1 1 1 0

                                                                                        1 0 1 1 0 1

                                                                                        1 1 0 0 1 1

                                                                                        Prewitt

                                                                                        4545

                                                                                        Review of Second DerivateReview of Second Derivate

                                                                                        Laplacian OperatorLaplacian Operator

                                                                                        21 1

                                                                                        1 1 4

                                                                                        f x y f x yf

                                                                                        f x y f x y f x y

                                                                                        0 1 0

                                                                                        1 4 1

                                                                                        0 1 0

                                                                                        0 1 0

                                                                                        1 4 1

                                                                                        0 1 0

                                                                                        LaplacianLaplacian

                                                                                        MaskMask

                                                                                        1 1 1

                                                                                        1 8 1

                                                                                        1 1 1

                                                                                        1 1 1

                                                                                        1 8 1

                                                                                        1 1 1

                                                                                        4646

                                                                                        Example of edge detectionExample of edge detection

                                                                                        See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                                        Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                                        derivative)derivative)

                                                                                        produces double edgesproduces double edges

                                                                                        unable to detect edge directionunable to detect edge direction

                                                                                        4747

                                                                                        Canny edge detectorCanny edge detector

                                                                                        The most powerful edge-detection The most powerful edge-detection

                                                                                        method method

                                                                                        It differs from the other edge-It differs from the other edge-

                                                                                        detection methods in that detection methods in that

                                                                                        it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                                        strong and weak edges) strong and weak edges)

                                                                                        and includes the weak edges in the and includes the weak edges in the

                                                                                        output only if they are connected to output only if they are connected to

                                                                                        strong edges strong edges

                                                                                        This method is therefore less likely This method is therefore less likely

                                                                                        than the others to be fooled by than the others to be fooled by

                                                                                        noise and more likely to detect true noise and more likely to detect true

                                                                                        weak edgesweak edges

                                                                                        4848

                                                                                        Laplacian of GaussianLaplacian of Gaussian

                                                                                        Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                                        2 2 22

                                                                                        4 2

                                                                                        2 2 2

                                                                                        2exp

                                                                                        r rh

                                                                                        r x y

                                                                                        determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                                        4949

                                                                                        Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                                        0 0 1 0 0

                                                                                        0 1 2 1 0

                                                                                        1 2 16 2 1

                                                                                        0 1 2 1 0

                                                                                        0 0 1 0 0

                                                                                        0 0 1 0 0

                                                                                        0 1 2 1 0

                                                                                        1 2 16 2 1

                                                                                        0 1 2 1 0

                                                                                        0 0 1 0 0

                                                                                        The coefficient must sum to The coefficient must sum to

                                                                                        zerozero

                                                                                        5050

                                                                                        Edge Detection and Edge Detection and SegmentationSegmentation

                                                                                        Image resulting from edge detection cannot be used as a segmentation result

                                                                                        Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                                        5151

                                                                                        75 Region-based 75 Region-based SegmentationSegmentation

                                                                                        GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                                        Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                                        PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                        For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                        PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                        For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                        5252

                                                                                        Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                        Region GrowingRegion Growing

                                                                                        Region SplittingRegion Splitting

                                                                                        Region growing is the opposite of the Region growing is the opposite of the

                                                                                        split and merge approachsplit and merge approach

                                                                                        5353

                                                                                        Region GrowingRegion Growing

                                                                                        The objective of segmentation is to The objective of segmentation is to

                                                                                        partition an image into regionspartition an image into regions

                                                                                        A region is a connected component with A region is a connected component with

                                                                                        some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                        texture)texture)

                                                                                        In region growing we start with a set In region growing we start with a set

                                                                                        of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                        appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                        pixels if they have pixels if they have similar propertiessimilar properties

                                                                                        such as specific ranges of gray level such as specific ranges of gray level

                                                                                        and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                        Need initialization Need initialization similarity similarity

                                                                                        criterioncriterion

                                                                                        5454

                                                                                        Steps of Region GrowingSteps of Region Growing

                                                                                        Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                        pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                        ppixelsixels

                                                                                        When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                        is grown from the seed pixel by is grown from the seed pixel by

                                                                                        addingadding neighboneighborr pixel pixelss

                                                                                        When the growth of one region stopsWhen the growth of one region stops

                                                                                        choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                        againagain

                                                                                        5555

                                                                                        Region Region growing growing

                                                                                        An initial set of small An initial set of small

                                                                                        areas are iterativelyareas are iteratively

                                                                                        merged according to merged according to

                                                                                        similarity constraintssimilarity constraints

                                                                                        5656

                                                                                        Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                        X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                        We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                        On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                        Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                        SeedSeed pointspoints

                                                                                        5757

                                                                                        CriterionCriterion

                                                                                        There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                        histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                        to be considered as a part of region to the to be considered as a part of region to the

                                                                                        seed pointseed point

                                                                                        The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                        to at least one pixel in that regionto at least one pixel in that region

                                                                                        Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                        defectsdefects

                                                                                        5858

                                                                                        Region SplittingRegion Splitting

                                                                                        The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                        ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                        If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                        This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                        Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                        5959

                                                                                        Region SplittingRegion Splitting

                                                                                        DisadvantageDisadvantage

                                                                                        they create regions that may be adjacent they create regions that may be adjacent

                                                                                        and homogeneous but not mergedand homogeneous but not merged

                                                                                        6060

                                                                                        Region Splitting and MergingRegion Splitting and Merging

                                                                                        ProcedureProcedure

                                                                                        11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                        22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                        33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                        6161

                                                                                        Region Splitting and Merging

                                                                                        Quadtree

                                                                                        (四叉树 )

                                                                                        6262

                                                                                        PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                        where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                        mmii is the mean gray level of that region is the mean gray level of that region

                                                                                        σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                        ExampleExample

                                                                                        Original Original

                                                                                        imageimageThresholded imageThresholded image Result of Result of

                                                                                        Splitting and Splitting and

                                                                                        MergingMerging

                                                                                        • Slide 1
                                                                                        • Slide 2
                                                                                        • Slide 3
                                                                                        • Slide 4
                                                                                        • Slide 5
                                                                                        • Slide 6
                                                                                        • Slide 7
                                                                                        • Slide 8
                                                                                        • Slide 9
                                                                                        • Slide 10
                                                                                        • Slide 11
                                                                                        • Slide 12
                                                                                        • Slide 13
                                                                                        • Slide 14
                                                                                        • Slide 15
                                                                                        • Slide 16
                                                                                        • Slide 17
                                                                                        • Slide 18
                                                                                        • Slide 19
                                                                                        • Slide 20
                                                                                        • Slide 21
                                                                                        • Slide 22
                                                                                        • Slide 23
                                                                                        • Slide 24
                                                                                        • Slide 25
                                                                                        • Slide 26
                                                                                        • Slide 27
                                                                                        • Slide 28
                                                                                        • Slide 29
                                                                                        • Slide 30
                                                                                        • Slide 31
                                                                                        • Slide 32
                                                                                        • Slide 33
                                                                                        • Slide 34
                                                                                        • Slide 35
                                                                                        • Slide 36
                                                                                        • Slide 37
                                                                                        • Slide 38
                                                                                        • Slide 39
                                                                                        • Slide 40
                                                                                        • Slide 41
                                                                                        • Slide 42
                                                                                        • Slide 43
                                                                                        • Slide 44
                                                                                        • Slide 45
                                                                                        • Slide 46
                                                                                        • Slide 47
                                                                                        • Slide 48
                                                                                        • Slide 49
                                                                                        • Slide 50
                                                                                        • Slide 51
                                                                                        • Slide 52
                                                                                        • Slide 53
                                                                                        • Slide 54
                                                                                        • Slide 55
                                                                                        • Slide 56
                                                                                        • Slide 57
                                                                                        • Slide 58
                                                                                        • Slide 59
                                                                                        • Slide 60
                                                                                        • Slide 61
                                                                                        • Slide 62

                                                                                          4545

                                                                                          Review of Second DerivateReview of Second Derivate

                                                                                          Laplacian OperatorLaplacian Operator

                                                                                          21 1

                                                                                          1 1 4

                                                                                          f x y f x yf

                                                                                          f x y f x y f x y

                                                                                          0 1 0

                                                                                          1 4 1

                                                                                          0 1 0

                                                                                          0 1 0

                                                                                          1 4 1

                                                                                          0 1 0

                                                                                          LaplacianLaplacian

                                                                                          MaskMask

                                                                                          1 1 1

                                                                                          1 8 1

                                                                                          1 1 1

                                                                                          1 1 1

                                                                                          1 8 1

                                                                                          1 1 1

                                                                                          4646

                                                                                          Example of edge detectionExample of edge detection

                                                                                          See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                                          Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                                          derivative)derivative)

                                                                                          produces double edgesproduces double edges

                                                                                          unable to detect edge directionunable to detect edge direction

                                                                                          4747

                                                                                          Canny edge detectorCanny edge detector

                                                                                          The most powerful edge-detection The most powerful edge-detection

                                                                                          method method

                                                                                          It differs from the other edge-It differs from the other edge-

                                                                                          detection methods in that detection methods in that

                                                                                          it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                                          strong and weak edges) strong and weak edges)

                                                                                          and includes the weak edges in the and includes the weak edges in the

                                                                                          output only if they are connected to output only if they are connected to

                                                                                          strong edges strong edges

                                                                                          This method is therefore less likely This method is therefore less likely

                                                                                          than the others to be fooled by than the others to be fooled by

                                                                                          noise and more likely to detect true noise and more likely to detect true

                                                                                          weak edgesweak edges

                                                                                          4848

                                                                                          Laplacian of GaussianLaplacian of Gaussian

                                                                                          Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                                          2 2 22

                                                                                          4 2

                                                                                          2 2 2

                                                                                          2exp

                                                                                          r rh

                                                                                          r x y

                                                                                          determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                                          4949

                                                                                          Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                                          0 0 1 0 0

                                                                                          0 1 2 1 0

                                                                                          1 2 16 2 1

                                                                                          0 1 2 1 0

                                                                                          0 0 1 0 0

                                                                                          0 0 1 0 0

                                                                                          0 1 2 1 0

                                                                                          1 2 16 2 1

                                                                                          0 1 2 1 0

                                                                                          0 0 1 0 0

                                                                                          The coefficient must sum to The coefficient must sum to

                                                                                          zerozero

                                                                                          5050

                                                                                          Edge Detection and Edge Detection and SegmentationSegmentation

                                                                                          Image resulting from edge detection cannot be used as a segmentation result

                                                                                          Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                                          5151

                                                                                          75 Region-based 75 Region-based SegmentationSegmentation

                                                                                          GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                                          Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                                          PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                          For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                          PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                          For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                          5252

                                                                                          Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                          Region GrowingRegion Growing

                                                                                          Region SplittingRegion Splitting

                                                                                          Region growing is the opposite of the Region growing is the opposite of the

                                                                                          split and merge approachsplit and merge approach

                                                                                          5353

                                                                                          Region GrowingRegion Growing

                                                                                          The objective of segmentation is to The objective of segmentation is to

                                                                                          partition an image into regionspartition an image into regions

                                                                                          A region is a connected component with A region is a connected component with

                                                                                          some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                          texture)texture)

                                                                                          In region growing we start with a set In region growing we start with a set

                                                                                          of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                          appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                          pixels if they have pixels if they have similar propertiessimilar properties

                                                                                          such as specific ranges of gray level such as specific ranges of gray level

                                                                                          and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                          Need initialization Need initialization similarity similarity

                                                                                          criterioncriterion

                                                                                          5454

                                                                                          Steps of Region GrowingSteps of Region Growing

                                                                                          Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                          pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                          ppixelsixels

                                                                                          When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                          is grown from the seed pixel by is grown from the seed pixel by

                                                                                          addingadding neighboneighborr pixel pixelss

                                                                                          When the growth of one region stopsWhen the growth of one region stops

                                                                                          choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                          againagain

                                                                                          5555

                                                                                          Region Region growing growing

                                                                                          An initial set of small An initial set of small

                                                                                          areas are iterativelyareas are iteratively

                                                                                          merged according to merged according to

                                                                                          similarity constraintssimilarity constraints

                                                                                          5656

                                                                                          Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                          X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                          We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                          On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                          Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                          SeedSeed pointspoints

                                                                                          5757

                                                                                          CriterionCriterion

                                                                                          There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                          histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                          to be considered as a part of region to the to be considered as a part of region to the

                                                                                          seed pointseed point

                                                                                          The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                          to at least one pixel in that regionto at least one pixel in that region

                                                                                          Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                          defectsdefects

                                                                                          5858

                                                                                          Region SplittingRegion Splitting

                                                                                          The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                          ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                          If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                          This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                          Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                          5959

                                                                                          Region SplittingRegion Splitting

                                                                                          DisadvantageDisadvantage

                                                                                          they create regions that may be adjacent they create regions that may be adjacent

                                                                                          and homogeneous but not mergedand homogeneous but not merged

                                                                                          6060

                                                                                          Region Splitting and MergingRegion Splitting and Merging

                                                                                          ProcedureProcedure

                                                                                          11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                          22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                          33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                          6161

                                                                                          Region Splitting and Merging

                                                                                          Quadtree

                                                                                          (四叉树 )

                                                                                          6262

                                                                                          PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                          where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                          mmii is the mean gray level of that region is the mean gray level of that region

                                                                                          σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                          ExampleExample

                                                                                          Original Original

                                                                                          imageimageThresholded imageThresholded image Result of Result of

                                                                                          Splitting and Splitting and

                                                                                          MergingMerging

                                                                                          • Slide 1
                                                                                          • Slide 2
                                                                                          • Slide 3
                                                                                          • Slide 4
                                                                                          • Slide 5
                                                                                          • Slide 6
                                                                                          • Slide 7
                                                                                          • Slide 8
                                                                                          • Slide 9
                                                                                          • Slide 10
                                                                                          • Slide 11
                                                                                          • Slide 12
                                                                                          • Slide 13
                                                                                          • Slide 14
                                                                                          • Slide 15
                                                                                          • Slide 16
                                                                                          • Slide 17
                                                                                          • Slide 18
                                                                                          • Slide 19
                                                                                          • Slide 20
                                                                                          • Slide 21
                                                                                          • Slide 22
                                                                                          • Slide 23
                                                                                          • Slide 24
                                                                                          • Slide 25
                                                                                          • Slide 26
                                                                                          • Slide 27
                                                                                          • Slide 28
                                                                                          • Slide 29
                                                                                          • Slide 30
                                                                                          • Slide 31
                                                                                          • Slide 32
                                                                                          • Slide 33
                                                                                          • Slide 34
                                                                                          • Slide 35
                                                                                          • Slide 36
                                                                                          • Slide 37
                                                                                          • Slide 38
                                                                                          • Slide 39
                                                                                          • Slide 40
                                                                                          • Slide 41
                                                                                          • Slide 42
                                                                                          • Slide 43
                                                                                          • Slide 44
                                                                                          • Slide 45
                                                                                          • Slide 46
                                                                                          • Slide 47
                                                                                          • Slide 48
                                                                                          • Slide 49
                                                                                          • Slide 50
                                                                                          • Slide 51
                                                                                          • Slide 52
                                                                                          • Slide 53
                                                                                          • Slide 54
                                                                                          • Slide 55
                                                                                          • Slide 56
                                                                                          • Slide 57
                                                                                          • Slide 58
                                                                                          • Slide 59
                                                                                          • Slide 60
                                                                                          • Slide 61
                                                                                          • Slide 62

                                                                                            4646

                                                                                            Example of edge detectionExample of edge detection

                                                                                            See Matlab Help--demo--toolbox--image proSee Matlab Help--demo--toolbox--image processing--analysishellip--Edge detectioncessing--analysishellip--Edge detection

                                                                                            Note The Laplacian is seldom used in practiNote The Laplacian is seldom used in practice becausece because unacceptably sensitive to noise (as second-order unacceptably sensitive to noise (as second-order

                                                                                            derivative)derivative)

                                                                                            produces double edgesproduces double edges

                                                                                            unable to detect edge directionunable to detect edge direction

                                                                                            4747

                                                                                            Canny edge detectorCanny edge detector

                                                                                            The most powerful edge-detection The most powerful edge-detection

                                                                                            method method

                                                                                            It differs from the other edge-It differs from the other edge-

                                                                                            detection methods in that detection methods in that

                                                                                            it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                                            strong and weak edges) strong and weak edges)

                                                                                            and includes the weak edges in the and includes the weak edges in the

                                                                                            output only if they are connected to output only if they are connected to

                                                                                            strong edges strong edges

                                                                                            This method is therefore less likely This method is therefore less likely

                                                                                            than the others to be fooled by than the others to be fooled by

                                                                                            noise and more likely to detect true noise and more likely to detect true

                                                                                            weak edgesweak edges

                                                                                            4848

                                                                                            Laplacian of GaussianLaplacian of Gaussian

                                                                                            Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                                            2 2 22

                                                                                            4 2

                                                                                            2 2 2

                                                                                            2exp

                                                                                            r rh

                                                                                            r x y

                                                                                            determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                                            4949

                                                                                            Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                                            0 0 1 0 0

                                                                                            0 1 2 1 0

                                                                                            1 2 16 2 1

                                                                                            0 1 2 1 0

                                                                                            0 0 1 0 0

                                                                                            0 0 1 0 0

                                                                                            0 1 2 1 0

                                                                                            1 2 16 2 1

                                                                                            0 1 2 1 0

                                                                                            0 0 1 0 0

                                                                                            The coefficient must sum to The coefficient must sum to

                                                                                            zerozero

                                                                                            5050

                                                                                            Edge Detection and Edge Detection and SegmentationSegmentation

                                                                                            Image resulting from edge detection cannot be used as a segmentation result

                                                                                            Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                                            5151

                                                                                            75 Region-based 75 Region-based SegmentationSegmentation

                                                                                            GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                                            Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                                            PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                            For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                            PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                            For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                            5252

                                                                                            Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                            Region GrowingRegion Growing

                                                                                            Region SplittingRegion Splitting

                                                                                            Region growing is the opposite of the Region growing is the opposite of the

                                                                                            split and merge approachsplit and merge approach

                                                                                            5353

                                                                                            Region GrowingRegion Growing

                                                                                            The objective of segmentation is to The objective of segmentation is to

                                                                                            partition an image into regionspartition an image into regions

                                                                                            A region is a connected component with A region is a connected component with

                                                                                            some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                            texture)texture)

                                                                                            In region growing we start with a set In region growing we start with a set

                                                                                            of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                            appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                            pixels if they have pixels if they have similar propertiessimilar properties

                                                                                            such as specific ranges of gray level such as specific ranges of gray level

                                                                                            and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                            Need initialization Need initialization similarity similarity

                                                                                            criterioncriterion

                                                                                            5454

                                                                                            Steps of Region GrowingSteps of Region Growing

                                                                                            Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                            pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                            ppixelsixels

                                                                                            When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                            is grown from the seed pixel by is grown from the seed pixel by

                                                                                            addingadding neighboneighborr pixel pixelss

                                                                                            When the growth of one region stopsWhen the growth of one region stops

                                                                                            choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                            againagain

                                                                                            5555

                                                                                            Region Region growing growing

                                                                                            An initial set of small An initial set of small

                                                                                            areas are iterativelyareas are iteratively

                                                                                            merged according to merged according to

                                                                                            similarity constraintssimilarity constraints

                                                                                            5656

                                                                                            Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                            X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                            We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                            On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                            Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                            SeedSeed pointspoints

                                                                                            5757

                                                                                            CriterionCriterion

                                                                                            There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                            histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                            to be considered as a part of region to the to be considered as a part of region to the

                                                                                            seed pointseed point

                                                                                            The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                            to at least one pixel in that regionto at least one pixel in that region

                                                                                            Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                            defectsdefects

                                                                                            5858

                                                                                            Region SplittingRegion Splitting

                                                                                            The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                            ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                            If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                            This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                            Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                            5959

                                                                                            Region SplittingRegion Splitting

                                                                                            DisadvantageDisadvantage

                                                                                            they create regions that may be adjacent they create regions that may be adjacent

                                                                                            and homogeneous but not mergedand homogeneous but not merged

                                                                                            6060

                                                                                            Region Splitting and MergingRegion Splitting and Merging

                                                                                            ProcedureProcedure

                                                                                            11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                            22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                            33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                            6161

                                                                                            Region Splitting and Merging

                                                                                            Quadtree

                                                                                            (四叉树 )

                                                                                            6262

                                                                                            PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                            where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                            mmii is the mean gray level of that region is the mean gray level of that region

                                                                                            σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                            ExampleExample

                                                                                            Original Original

                                                                                            imageimageThresholded imageThresholded image Result of Result of

                                                                                            Splitting and Splitting and

                                                                                            MergingMerging

                                                                                            • Slide 1
                                                                                            • Slide 2
                                                                                            • Slide 3
                                                                                            • Slide 4
                                                                                            • Slide 5
                                                                                            • Slide 6
                                                                                            • Slide 7
                                                                                            • Slide 8
                                                                                            • Slide 9
                                                                                            • Slide 10
                                                                                            • Slide 11
                                                                                            • Slide 12
                                                                                            • Slide 13
                                                                                            • Slide 14
                                                                                            • Slide 15
                                                                                            • Slide 16
                                                                                            • Slide 17
                                                                                            • Slide 18
                                                                                            • Slide 19
                                                                                            • Slide 20
                                                                                            • Slide 21
                                                                                            • Slide 22
                                                                                            • Slide 23
                                                                                            • Slide 24
                                                                                            • Slide 25
                                                                                            • Slide 26
                                                                                            • Slide 27
                                                                                            • Slide 28
                                                                                            • Slide 29
                                                                                            • Slide 30
                                                                                            • Slide 31
                                                                                            • Slide 32
                                                                                            • Slide 33
                                                                                            • Slide 34
                                                                                            • Slide 35
                                                                                            • Slide 36
                                                                                            • Slide 37
                                                                                            • Slide 38
                                                                                            • Slide 39
                                                                                            • Slide 40
                                                                                            • Slide 41
                                                                                            • Slide 42
                                                                                            • Slide 43
                                                                                            • Slide 44
                                                                                            • Slide 45
                                                                                            • Slide 46
                                                                                            • Slide 47
                                                                                            • Slide 48
                                                                                            • Slide 49
                                                                                            • Slide 50
                                                                                            • Slide 51
                                                                                            • Slide 52
                                                                                            • Slide 53
                                                                                            • Slide 54
                                                                                            • Slide 55
                                                                                            • Slide 56
                                                                                            • Slide 57
                                                                                            • Slide 58
                                                                                            • Slide 59
                                                                                            • Slide 60
                                                                                            • Slide 61
                                                                                            • Slide 62

                                                                                              4747

                                                                                              Canny edge detectorCanny edge detector

                                                                                              The most powerful edge-detection The most powerful edge-detection

                                                                                              method method

                                                                                              It differs from the other edge-It differs from the other edge-

                                                                                              detection methods in that detection methods in that

                                                                                              it uses two different thresholds (to detect it uses two different thresholds (to detect

                                                                                              strong and weak edges) strong and weak edges)

                                                                                              and includes the weak edges in the and includes the weak edges in the

                                                                                              output only if they are connected to output only if they are connected to

                                                                                              strong edges strong edges

                                                                                              This method is therefore less likely This method is therefore less likely

                                                                                              than the others to be fooled by than the others to be fooled by

                                                                                              noise and more likely to detect true noise and more likely to detect true

                                                                                              weak edgesweak edges

                                                                                              4848

                                                                                              Laplacian of GaussianLaplacian of Gaussian

                                                                                              Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                                              2 2 22

                                                                                              4 2

                                                                                              2 2 2

                                                                                              2exp

                                                                                              r rh

                                                                                              r x y

                                                                                              determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                                              4949

                                                                                              Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                                              0 0 1 0 0

                                                                                              0 1 2 1 0

                                                                                              1 2 16 2 1

                                                                                              0 1 2 1 0

                                                                                              0 0 1 0 0

                                                                                              0 0 1 0 0

                                                                                              0 1 2 1 0

                                                                                              1 2 16 2 1

                                                                                              0 1 2 1 0

                                                                                              0 0 1 0 0

                                                                                              The coefficient must sum to The coefficient must sum to

                                                                                              zerozero

                                                                                              5050

                                                                                              Edge Detection and Edge Detection and SegmentationSegmentation

                                                                                              Image resulting from edge detection cannot be used as a segmentation result

                                                                                              Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                                              5151

                                                                                              75 Region-based 75 Region-based SegmentationSegmentation

                                                                                              GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                                              Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                                              PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                              For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                              PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                              For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                              5252

                                                                                              Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                              Region GrowingRegion Growing

                                                                                              Region SplittingRegion Splitting

                                                                                              Region growing is the opposite of the Region growing is the opposite of the

                                                                                              split and merge approachsplit and merge approach

                                                                                              5353

                                                                                              Region GrowingRegion Growing

                                                                                              The objective of segmentation is to The objective of segmentation is to

                                                                                              partition an image into regionspartition an image into regions

                                                                                              A region is a connected component with A region is a connected component with

                                                                                              some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                              texture)texture)

                                                                                              In region growing we start with a set In region growing we start with a set

                                                                                              of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                              appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                              pixels if they have pixels if they have similar propertiessimilar properties

                                                                                              such as specific ranges of gray level such as specific ranges of gray level

                                                                                              and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                              Need initialization Need initialization similarity similarity

                                                                                              criterioncriterion

                                                                                              5454

                                                                                              Steps of Region GrowingSteps of Region Growing

                                                                                              Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                              pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                              ppixelsixels

                                                                                              When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                              is grown from the seed pixel by is grown from the seed pixel by

                                                                                              addingadding neighboneighborr pixel pixelss

                                                                                              When the growth of one region stopsWhen the growth of one region stops

                                                                                              choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                              againagain

                                                                                              5555

                                                                                              Region Region growing growing

                                                                                              An initial set of small An initial set of small

                                                                                              areas are iterativelyareas are iteratively

                                                                                              merged according to merged according to

                                                                                              similarity constraintssimilarity constraints

                                                                                              5656

                                                                                              Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                              X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                              We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                              On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                              Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                              SeedSeed pointspoints

                                                                                              5757

                                                                                              CriterionCriterion

                                                                                              There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                              histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                              to be considered as a part of region to the to be considered as a part of region to the

                                                                                              seed pointseed point

                                                                                              The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                              to at least one pixel in that regionto at least one pixel in that region

                                                                                              Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                              defectsdefects

                                                                                              5858

                                                                                              Region SplittingRegion Splitting

                                                                                              The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                              ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                              If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                              This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                              Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                              5959

                                                                                              Region SplittingRegion Splitting

                                                                                              DisadvantageDisadvantage

                                                                                              they create regions that may be adjacent they create regions that may be adjacent

                                                                                              and homogeneous but not mergedand homogeneous but not merged

                                                                                              6060

                                                                                              Region Splitting and MergingRegion Splitting and Merging

                                                                                              ProcedureProcedure

                                                                                              11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                              22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                              33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                              6161

                                                                                              Region Splitting and Merging

                                                                                              Quadtree

                                                                                              (四叉树 )

                                                                                              6262

                                                                                              PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                              where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                              mmii is the mean gray level of that region is the mean gray level of that region

                                                                                              σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                              ExampleExample

                                                                                              Original Original

                                                                                              imageimageThresholded imageThresholded image Result of Result of

                                                                                              Splitting and Splitting and

                                                                                              MergingMerging

                                                                                              • Slide 1
                                                                                              • Slide 2
                                                                                              • Slide 3
                                                                                              • Slide 4
                                                                                              • Slide 5
                                                                                              • Slide 6
                                                                                              • Slide 7
                                                                                              • Slide 8
                                                                                              • Slide 9
                                                                                              • Slide 10
                                                                                              • Slide 11
                                                                                              • Slide 12
                                                                                              • Slide 13
                                                                                              • Slide 14
                                                                                              • Slide 15
                                                                                              • Slide 16
                                                                                              • Slide 17
                                                                                              • Slide 18
                                                                                              • Slide 19
                                                                                              • Slide 20
                                                                                              • Slide 21
                                                                                              • Slide 22
                                                                                              • Slide 23
                                                                                              • Slide 24
                                                                                              • Slide 25
                                                                                              • Slide 26
                                                                                              • Slide 27
                                                                                              • Slide 28
                                                                                              • Slide 29
                                                                                              • Slide 30
                                                                                              • Slide 31
                                                                                              • Slide 32
                                                                                              • Slide 33
                                                                                              • Slide 34
                                                                                              • Slide 35
                                                                                              • Slide 36
                                                                                              • Slide 37
                                                                                              • Slide 38
                                                                                              • Slide 39
                                                                                              • Slide 40
                                                                                              • Slide 41
                                                                                              • Slide 42
                                                                                              • Slide 43
                                                                                              • Slide 44
                                                                                              • Slide 45
                                                                                              • Slide 46
                                                                                              • Slide 47
                                                                                              • Slide 48
                                                                                              • Slide 49
                                                                                              • Slide 50
                                                                                              • Slide 51
                                                                                              • Slide 52
                                                                                              • Slide 53
                                                                                              • Slide 54
                                                                                              • Slide 55
                                                                                              • Slide 56
                                                                                              • Slide 57
                                                                                              • Slide 58
                                                                                              • Slide 59
                                                                                              • Slide 60
                                                                                              • Slide 61
                                                                                              • Slide 62

                                                                                                4848

                                                                                                Laplacian of GaussianLaplacian of Gaussian

                                                                                                Laplacian combined with smoothing to find Laplacian combined with smoothing to find edges via zero-crossingedges via zero-crossing

                                                                                                2 2 22

                                                                                                4 2

                                                                                                2 2 2

                                                                                                2exp

                                                                                                r rh

                                                                                                r x y

                                                                                                determines the degrdetermines the degree of blurring that occee of blurring that occursurs

                                                                                                4949

                                                                                                Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                                                0 0 1 0 0

                                                                                                0 1 2 1 0

                                                                                                1 2 16 2 1

                                                                                                0 1 2 1 0

                                                                                                0 0 1 0 0

                                                                                                0 0 1 0 0

                                                                                                0 1 2 1 0

                                                                                                1 2 16 2 1

                                                                                                0 1 2 1 0

                                                                                                0 0 1 0 0

                                                                                                The coefficient must sum to The coefficient must sum to

                                                                                                zerozero

                                                                                                5050

                                                                                                Edge Detection and Edge Detection and SegmentationSegmentation

                                                                                                Image resulting from edge detection cannot be used as a segmentation result

                                                                                                Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                                                5151

                                                                                                75 Region-based 75 Region-based SegmentationSegmentation

                                                                                                GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                                                Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                                                PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                                For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                                PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                                For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                                5252

                                                                                                Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                                Region GrowingRegion Growing

                                                                                                Region SplittingRegion Splitting

                                                                                                Region growing is the opposite of the Region growing is the opposite of the

                                                                                                split and merge approachsplit and merge approach

                                                                                                5353

                                                                                                Region GrowingRegion Growing

                                                                                                The objective of segmentation is to The objective of segmentation is to

                                                                                                partition an image into regionspartition an image into regions

                                                                                                A region is a connected component with A region is a connected component with

                                                                                                some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                                texture)texture)

                                                                                                In region growing we start with a set In region growing we start with a set

                                                                                                of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                                appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                                pixels if they have pixels if they have similar propertiessimilar properties

                                                                                                such as specific ranges of gray level such as specific ranges of gray level

                                                                                                and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                                Need initialization Need initialization similarity similarity

                                                                                                criterioncriterion

                                                                                                5454

                                                                                                Steps of Region GrowingSteps of Region Growing

                                                                                                Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                                pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                                ppixelsixels

                                                                                                When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                                is grown from the seed pixel by is grown from the seed pixel by

                                                                                                addingadding neighboneighborr pixel pixelss

                                                                                                When the growth of one region stopsWhen the growth of one region stops

                                                                                                choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                                againagain

                                                                                                5555

                                                                                                Region Region growing growing

                                                                                                An initial set of small An initial set of small

                                                                                                areas are iterativelyareas are iteratively

                                                                                                merged according to merged according to

                                                                                                similarity constraintssimilarity constraints

                                                                                                5656

                                                                                                Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                                X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                                We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                                On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                                Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                                SeedSeed pointspoints

                                                                                                5757

                                                                                                CriterionCriterion

                                                                                                There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                                histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                                to be considered as a part of region to the to be considered as a part of region to the

                                                                                                seed pointseed point

                                                                                                The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                                to at least one pixel in that regionto at least one pixel in that region

                                                                                                Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                                defectsdefects

                                                                                                5858

                                                                                                Region SplittingRegion Splitting

                                                                                                The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                                ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                                If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                                This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                                Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                                5959

                                                                                                Region SplittingRegion Splitting

                                                                                                DisadvantageDisadvantage

                                                                                                they create regions that may be adjacent they create regions that may be adjacent

                                                                                                and homogeneous but not mergedand homogeneous but not merged

                                                                                                6060

                                                                                                Region Splitting and MergingRegion Splitting and Merging

                                                                                                ProcedureProcedure

                                                                                                11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                6161

                                                                                                Region Splitting and Merging

                                                                                                Quadtree

                                                                                                (四叉树 )

                                                                                                6262

                                                                                                PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                ExampleExample

                                                                                                Original Original

                                                                                                imageimageThresholded imageThresholded image Result of Result of

                                                                                                Splitting and Splitting and

                                                                                                MergingMerging

                                                                                                • Slide 1
                                                                                                • Slide 2
                                                                                                • Slide 3
                                                                                                • Slide 4
                                                                                                • Slide 5
                                                                                                • Slide 6
                                                                                                • Slide 7
                                                                                                • Slide 8
                                                                                                • Slide 9
                                                                                                • Slide 10
                                                                                                • Slide 11
                                                                                                • Slide 12
                                                                                                • Slide 13
                                                                                                • Slide 14
                                                                                                • Slide 15
                                                                                                • Slide 16
                                                                                                • Slide 17
                                                                                                • Slide 18
                                                                                                • Slide 19
                                                                                                • Slide 20
                                                                                                • Slide 21
                                                                                                • Slide 22
                                                                                                • Slide 23
                                                                                                • Slide 24
                                                                                                • Slide 25
                                                                                                • Slide 26
                                                                                                • Slide 27
                                                                                                • Slide 28
                                                                                                • Slide 29
                                                                                                • Slide 30
                                                                                                • Slide 31
                                                                                                • Slide 32
                                                                                                • Slide 33
                                                                                                • Slide 34
                                                                                                • Slide 35
                                                                                                • Slide 36
                                                                                                • Slide 37
                                                                                                • Slide 38
                                                                                                • Slide 39
                                                                                                • Slide 40
                                                                                                • Slide 41
                                                                                                • Slide 42
                                                                                                • Slide 43
                                                                                                • Slide 44
                                                                                                • Slide 45
                                                                                                • Slide 46
                                                                                                • Slide 47
                                                                                                • Slide 48
                                                                                                • Slide 49
                                                                                                • Slide 50
                                                                                                • Slide 51
                                                                                                • Slide 52
                                                                                                • Slide 53
                                                                                                • Slide 54
                                                                                                • Slide 55
                                                                                                • Slide 56
                                                                                                • Slide 57
                                                                                                • Slide 58
                                                                                                • Slide 59
                                                                                                • Slide 60
                                                                                                • Slide 61
                                                                                                • Slide 62

                                                                                                  4949

                                                                                                  Laplacian of Gaussian (Laplacian of Gaussian (Mexican hatMexican hat))

                                                                                                  0 0 1 0 0

                                                                                                  0 1 2 1 0

                                                                                                  1 2 16 2 1

                                                                                                  0 1 2 1 0

                                                                                                  0 0 1 0 0

                                                                                                  0 0 1 0 0

                                                                                                  0 1 2 1 0

                                                                                                  1 2 16 2 1

                                                                                                  0 1 2 1 0

                                                                                                  0 0 1 0 0

                                                                                                  The coefficient must sum to The coefficient must sum to

                                                                                                  zerozero

                                                                                                  5050

                                                                                                  Edge Detection and Edge Detection and SegmentationSegmentation

                                                                                                  Image resulting from edge detection cannot be used as a segmentation result

                                                                                                  Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                                                  5151

                                                                                                  75 Region-based 75 Region-based SegmentationSegmentation

                                                                                                  GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                                                  Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                                                  PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                                  For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                                  PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                                  For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                                  5252

                                                                                                  Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                                  Region GrowingRegion Growing

                                                                                                  Region SplittingRegion Splitting

                                                                                                  Region growing is the opposite of the Region growing is the opposite of the

                                                                                                  split and merge approachsplit and merge approach

                                                                                                  5353

                                                                                                  Region GrowingRegion Growing

                                                                                                  The objective of segmentation is to The objective of segmentation is to

                                                                                                  partition an image into regionspartition an image into regions

                                                                                                  A region is a connected component with A region is a connected component with

                                                                                                  some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                                  texture)texture)

                                                                                                  In region growing we start with a set In region growing we start with a set

                                                                                                  of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                                  appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                                  pixels if they have pixels if they have similar propertiessimilar properties

                                                                                                  such as specific ranges of gray level such as specific ranges of gray level

                                                                                                  and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                                  Need initialization Need initialization similarity similarity

                                                                                                  criterioncriterion

                                                                                                  5454

                                                                                                  Steps of Region GrowingSteps of Region Growing

                                                                                                  Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                                  pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                                  ppixelsixels

                                                                                                  When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                                  is grown from the seed pixel by is grown from the seed pixel by

                                                                                                  addingadding neighboneighborr pixel pixelss

                                                                                                  When the growth of one region stopsWhen the growth of one region stops

                                                                                                  choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                                  againagain

                                                                                                  5555

                                                                                                  Region Region growing growing

                                                                                                  An initial set of small An initial set of small

                                                                                                  areas are iterativelyareas are iteratively

                                                                                                  merged according to merged according to

                                                                                                  similarity constraintssimilarity constraints

                                                                                                  5656

                                                                                                  Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                                  X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                                  We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                                  On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                                  Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                                  SeedSeed pointspoints

                                                                                                  5757

                                                                                                  CriterionCriterion

                                                                                                  There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                                  histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                                  to be considered as a part of region to the to be considered as a part of region to the

                                                                                                  seed pointseed point

                                                                                                  The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                                  to at least one pixel in that regionto at least one pixel in that region

                                                                                                  Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                                  defectsdefects

                                                                                                  5858

                                                                                                  Region SplittingRegion Splitting

                                                                                                  The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                                  ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                                  If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                                  This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                                  Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                                  5959

                                                                                                  Region SplittingRegion Splitting

                                                                                                  DisadvantageDisadvantage

                                                                                                  they create regions that may be adjacent they create regions that may be adjacent

                                                                                                  and homogeneous but not mergedand homogeneous but not merged

                                                                                                  6060

                                                                                                  Region Splitting and MergingRegion Splitting and Merging

                                                                                                  ProcedureProcedure

                                                                                                  11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                  22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                  33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                  6161

                                                                                                  Region Splitting and Merging

                                                                                                  Quadtree

                                                                                                  (四叉树 )

                                                                                                  6262

                                                                                                  PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                  where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                  mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                  σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                  ExampleExample

                                                                                                  Original Original

                                                                                                  imageimageThresholded imageThresholded image Result of Result of

                                                                                                  Splitting and Splitting and

                                                                                                  MergingMerging

                                                                                                  • Slide 1
                                                                                                  • Slide 2
                                                                                                  • Slide 3
                                                                                                  • Slide 4
                                                                                                  • Slide 5
                                                                                                  • Slide 6
                                                                                                  • Slide 7
                                                                                                  • Slide 8
                                                                                                  • Slide 9
                                                                                                  • Slide 10
                                                                                                  • Slide 11
                                                                                                  • Slide 12
                                                                                                  • Slide 13
                                                                                                  • Slide 14
                                                                                                  • Slide 15
                                                                                                  • Slide 16
                                                                                                  • Slide 17
                                                                                                  • Slide 18
                                                                                                  • Slide 19
                                                                                                  • Slide 20
                                                                                                  • Slide 21
                                                                                                  • Slide 22
                                                                                                  • Slide 23
                                                                                                  • Slide 24
                                                                                                  • Slide 25
                                                                                                  • Slide 26
                                                                                                  • Slide 27
                                                                                                  • Slide 28
                                                                                                  • Slide 29
                                                                                                  • Slide 30
                                                                                                  • Slide 31
                                                                                                  • Slide 32
                                                                                                  • Slide 33
                                                                                                  • Slide 34
                                                                                                  • Slide 35
                                                                                                  • Slide 36
                                                                                                  • Slide 37
                                                                                                  • Slide 38
                                                                                                  • Slide 39
                                                                                                  • Slide 40
                                                                                                  • Slide 41
                                                                                                  • Slide 42
                                                                                                  • Slide 43
                                                                                                  • Slide 44
                                                                                                  • Slide 45
                                                                                                  • Slide 46
                                                                                                  • Slide 47
                                                                                                  • Slide 48
                                                                                                  • Slide 49
                                                                                                  • Slide 50
                                                                                                  • Slide 51
                                                                                                  • Slide 52
                                                                                                  • Slide 53
                                                                                                  • Slide 54
                                                                                                  • Slide 55
                                                                                                  • Slide 56
                                                                                                  • Slide 57
                                                                                                  • Slide 58
                                                                                                  • Slide 59
                                                                                                  • Slide 60
                                                                                                  • Slide 61
                                                                                                  • Slide 62

                                                                                                    5050

                                                                                                    Edge Detection and Edge Detection and SegmentationSegmentation

                                                                                                    Image resulting from edge detection cannot be used as a segmentation result

                                                                                                    Supplementary processing steps must follow to combine edges into edge chains that correspond better with borders (boundary) in the image

                                                                                                    5151

                                                                                                    75 Region-based 75 Region-based SegmentationSegmentation

                                                                                                    GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                                                    Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                                                    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                                    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                                    PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                                    For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                                    5252

                                                                                                    Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                                    Region GrowingRegion Growing

                                                                                                    Region SplittingRegion Splitting

                                                                                                    Region growing is the opposite of the Region growing is the opposite of the

                                                                                                    split and merge approachsplit and merge approach

                                                                                                    5353

                                                                                                    Region GrowingRegion Growing

                                                                                                    The objective of segmentation is to The objective of segmentation is to

                                                                                                    partition an image into regionspartition an image into regions

                                                                                                    A region is a connected component with A region is a connected component with

                                                                                                    some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                                    texture)texture)

                                                                                                    In region growing we start with a set In region growing we start with a set

                                                                                                    of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                                    appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                                    pixels if they have pixels if they have similar propertiessimilar properties

                                                                                                    such as specific ranges of gray level such as specific ranges of gray level

                                                                                                    and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                                    Need initialization Need initialization similarity similarity

                                                                                                    criterioncriterion

                                                                                                    5454

                                                                                                    Steps of Region GrowingSteps of Region Growing

                                                                                                    Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                                    pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                                    ppixelsixels

                                                                                                    When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                                    is grown from the seed pixel by is grown from the seed pixel by

                                                                                                    addingadding neighboneighborr pixel pixelss

                                                                                                    When the growth of one region stopsWhen the growth of one region stops

                                                                                                    choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                                    againagain

                                                                                                    5555

                                                                                                    Region Region growing growing

                                                                                                    An initial set of small An initial set of small

                                                                                                    areas are iterativelyareas are iteratively

                                                                                                    merged according to merged according to

                                                                                                    similarity constraintssimilarity constraints

                                                                                                    5656

                                                                                                    Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                                    X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                                    We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                                    On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                                    Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                                    SeedSeed pointspoints

                                                                                                    5757

                                                                                                    CriterionCriterion

                                                                                                    There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                                    histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                                    to be considered as a part of region to the to be considered as a part of region to the

                                                                                                    seed pointseed point

                                                                                                    The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                                    to at least one pixel in that regionto at least one pixel in that region

                                                                                                    Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                                    defectsdefects

                                                                                                    5858

                                                                                                    Region SplittingRegion Splitting

                                                                                                    The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                                    ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                                    If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                                    This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                                    Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                                    5959

                                                                                                    Region SplittingRegion Splitting

                                                                                                    DisadvantageDisadvantage

                                                                                                    they create regions that may be adjacent they create regions that may be adjacent

                                                                                                    and homogeneous but not mergedand homogeneous but not merged

                                                                                                    6060

                                                                                                    Region Splitting and MergingRegion Splitting and Merging

                                                                                                    ProcedureProcedure

                                                                                                    11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                    22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                    33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                    6161

                                                                                                    Region Splitting and Merging

                                                                                                    Quadtree

                                                                                                    (四叉树 )

                                                                                                    6262

                                                                                                    PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                    where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                    mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                    σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                    ExampleExample

                                                                                                    Original Original

                                                                                                    imageimageThresholded imageThresholded image Result of Result of

                                                                                                    Splitting and Splitting and

                                                                                                    MergingMerging

                                                                                                    • Slide 1
                                                                                                    • Slide 2
                                                                                                    • Slide 3
                                                                                                    • Slide 4
                                                                                                    • Slide 5
                                                                                                    • Slide 6
                                                                                                    • Slide 7
                                                                                                    • Slide 8
                                                                                                    • Slide 9
                                                                                                    • Slide 10
                                                                                                    • Slide 11
                                                                                                    • Slide 12
                                                                                                    • Slide 13
                                                                                                    • Slide 14
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                                                                                                    • Slide 48
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                                                                                                    • Slide 52
                                                                                                    • Slide 53
                                                                                                    • Slide 54
                                                                                                    • Slide 55
                                                                                                    • Slide 56
                                                                                                    • Slide 57
                                                                                                    • Slide 58
                                                                                                    • Slide 59
                                                                                                    • Slide 60
                                                                                                    • Slide 61
                                                                                                    • Slide 62

                                                                                                      5151

                                                                                                      75 Region-based 75 Region-based SegmentationSegmentation

                                                                                                      GoalGoal find regions that are ldquohomogeneou find regions that are ldquohomogeneousrdquo by some criterionsrdquo by some criterion

                                                                                                      Segmentation is a process that partitions Segmentation is a process that partitions R R iinto n subregions nto n subregions RR11 RR22 hellip hellip RRnn such thatsuch that

                                                                                                      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                                      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                                      PP((RRii) is a logical predicate ) is a logical predicate property defined over the property defined over the points in set points in set RRii

                                                                                                      For example For example PP((RRii) = TRUE i) = TRUE if all pixel in f all pixel in RRii have the sa have the same gray levelme gray level

                                                                                                      5252

                                                                                                      Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                                      Region GrowingRegion Growing

                                                                                                      Region SplittingRegion Splitting

                                                                                                      Region growing is the opposite of the Region growing is the opposite of the

                                                                                                      split and merge approachsplit and merge approach

                                                                                                      5353

                                                                                                      Region GrowingRegion Growing

                                                                                                      The objective of segmentation is to The objective of segmentation is to

                                                                                                      partition an image into regionspartition an image into regions

                                                                                                      A region is a connected component with A region is a connected component with

                                                                                                      some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                                      texture)texture)

                                                                                                      In region growing we start with a set In region growing we start with a set

                                                                                                      of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                                      appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                                      pixels if they have pixels if they have similar propertiessimilar properties

                                                                                                      such as specific ranges of gray level such as specific ranges of gray level

                                                                                                      and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                                      Need initialization Need initialization similarity similarity

                                                                                                      criterioncriterion

                                                                                                      5454

                                                                                                      Steps of Region GrowingSteps of Region Growing

                                                                                                      Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                                      pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                                      ppixelsixels

                                                                                                      When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                                      is grown from the seed pixel by is grown from the seed pixel by

                                                                                                      addingadding neighboneighborr pixel pixelss

                                                                                                      When the growth of one region stopsWhen the growth of one region stops

                                                                                                      choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                                      againagain

                                                                                                      5555

                                                                                                      Region Region growing growing

                                                                                                      An initial set of small An initial set of small

                                                                                                      areas are iterativelyareas are iteratively

                                                                                                      merged according to merged according to

                                                                                                      similarity constraintssimilarity constraints

                                                                                                      5656

                                                                                                      Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                                      X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                                      We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                                      On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                                      Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                                      SeedSeed pointspoints

                                                                                                      5757

                                                                                                      CriterionCriterion

                                                                                                      There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                                      histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                                      to be considered as a part of region to the to be considered as a part of region to the

                                                                                                      seed pointseed point

                                                                                                      The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                                      to at least one pixel in that regionto at least one pixel in that region

                                                                                                      Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                                      defectsdefects

                                                                                                      5858

                                                                                                      Region SplittingRegion Splitting

                                                                                                      The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                                      ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                                      If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                                      This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                                      Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                                      5959

                                                                                                      Region SplittingRegion Splitting

                                                                                                      DisadvantageDisadvantage

                                                                                                      they create regions that may be adjacent they create regions that may be adjacent

                                                                                                      and homogeneous but not mergedand homogeneous but not merged

                                                                                                      6060

                                                                                                      Region Splitting and MergingRegion Splitting and Merging

                                                                                                      ProcedureProcedure

                                                                                                      11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                      22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                      33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                      6161

                                                                                                      Region Splitting and Merging

                                                                                                      Quadtree

                                                                                                      (四叉树 )

                                                                                                      6262

                                                                                                      PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                      where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                      mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                      σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                      ExampleExample

                                                                                                      Original Original

                                                                                                      imageimageThresholded imageThresholded image Result of Result of

                                                                                                      Splitting and Splitting and

                                                                                                      MergingMerging

                                                                                                      • Slide 1
                                                                                                      • Slide 2
                                                                                                      • Slide 3
                                                                                                      • Slide 4
                                                                                                      • Slide 5
                                                                                                      • Slide 6
                                                                                                      • Slide 7
                                                                                                      • Slide 8
                                                                                                      • Slide 9
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                                                                                                      • Slide 56
                                                                                                      • Slide 57
                                                                                                      • Slide 58
                                                                                                      • Slide 59
                                                                                                      • Slide 60
                                                                                                      • Slide 61
                                                                                                      • Slide 62

                                                                                                        5252

                                                                                                        Two methods of Region Two methods of Region SegmentationSegmentation

                                                                                                        Region GrowingRegion Growing

                                                                                                        Region SplittingRegion Splitting

                                                                                                        Region growing is the opposite of the Region growing is the opposite of the

                                                                                                        split and merge approachsplit and merge approach

                                                                                                        5353

                                                                                                        Region GrowingRegion Growing

                                                                                                        The objective of segmentation is to The objective of segmentation is to

                                                                                                        partition an image into regionspartition an image into regions

                                                                                                        A region is a connected component with A region is a connected component with

                                                                                                        some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                                        texture)texture)

                                                                                                        In region growing we start with a set In region growing we start with a set

                                                                                                        of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                                        appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                                        pixels if they have pixels if they have similar propertiessimilar properties

                                                                                                        such as specific ranges of gray level such as specific ranges of gray level

                                                                                                        and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                                        Need initialization Need initialization similarity similarity

                                                                                                        criterioncriterion

                                                                                                        5454

                                                                                                        Steps of Region GrowingSteps of Region Growing

                                                                                                        Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                                        pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                                        ppixelsixels

                                                                                                        When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                                        is grown from the seed pixel by is grown from the seed pixel by

                                                                                                        addingadding neighboneighborr pixel pixelss

                                                                                                        When the growth of one region stopsWhen the growth of one region stops

                                                                                                        choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                                        againagain

                                                                                                        5555

                                                                                                        Region Region growing growing

                                                                                                        An initial set of small An initial set of small

                                                                                                        areas are iterativelyareas are iteratively

                                                                                                        merged according to merged according to

                                                                                                        similarity constraintssimilarity constraints

                                                                                                        5656

                                                                                                        Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                                        X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                                        We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                                        On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                                        Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                                        SeedSeed pointspoints

                                                                                                        5757

                                                                                                        CriterionCriterion

                                                                                                        There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                                        histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                                        to be considered as a part of region to the to be considered as a part of region to the

                                                                                                        seed pointseed point

                                                                                                        The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                                        to at least one pixel in that regionto at least one pixel in that region

                                                                                                        Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                                        defectsdefects

                                                                                                        5858

                                                                                                        Region SplittingRegion Splitting

                                                                                                        The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                                        ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                                        If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                                        This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                                        Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                                        5959

                                                                                                        Region SplittingRegion Splitting

                                                                                                        DisadvantageDisadvantage

                                                                                                        they create regions that may be adjacent they create regions that may be adjacent

                                                                                                        and homogeneous but not mergedand homogeneous but not merged

                                                                                                        6060

                                                                                                        Region Splitting and MergingRegion Splitting and Merging

                                                                                                        ProcedureProcedure

                                                                                                        11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                        22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                        33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                        6161

                                                                                                        Region Splitting and Merging

                                                                                                        Quadtree

                                                                                                        (四叉树 )

                                                                                                        6262

                                                                                                        PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                        where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                        mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                        σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                        ExampleExample

                                                                                                        Original Original

                                                                                                        imageimageThresholded imageThresholded image Result of Result of

                                                                                                        Splitting and Splitting and

                                                                                                        MergingMerging

                                                                                                        • Slide 1
                                                                                                        • Slide 2
                                                                                                        • Slide 3
                                                                                                        • Slide 4
                                                                                                        • Slide 5
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                                                                                                        • Slide 56
                                                                                                        • Slide 57
                                                                                                        • Slide 58
                                                                                                        • Slide 59
                                                                                                        • Slide 60
                                                                                                        • Slide 61
                                                                                                        • Slide 62

                                                                                                          5353

                                                                                                          Region GrowingRegion Growing

                                                                                                          The objective of segmentation is to The objective of segmentation is to

                                                                                                          partition an image into regionspartition an image into regions

                                                                                                          A region is a connected component with A region is a connected component with

                                                                                                          some uniformity (say gray-levels or some uniformity (say gray-levels or

                                                                                                          texture)texture)

                                                                                                          In region growing we start with a set In region growing we start with a set

                                                                                                          of of ldquoseedrdquoldquoseedrdquo points growing by points growing by

                                                                                                          appending to each seedrsquos neighbor appending to each seedrsquos neighbor

                                                                                                          pixels if they have pixels if they have similar propertiessimilar properties

                                                                                                          such as specific ranges of gray level such as specific ranges of gray level

                                                                                                          and and lsquo8-connected neighborrsquolsquo8-connected neighborrsquo

                                                                                                          Need initialization Need initialization similarity similarity

                                                                                                          criterioncriterion

                                                                                                          5454

                                                                                                          Steps of Region GrowingSteps of Region Growing

                                                                                                          Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                                          pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                                          ppixelsixels

                                                                                                          When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                                          is grown from the seed pixel by is grown from the seed pixel by

                                                                                                          addingadding neighboneighborr pixel pixelss

                                                                                                          When the growth of one region stopsWhen the growth of one region stops

                                                                                                          choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                                          againagain

                                                                                                          5555

                                                                                                          Region Region growing growing

                                                                                                          An initial set of small An initial set of small

                                                                                                          areas are iterativelyareas are iteratively

                                                                                                          merged according to merged according to

                                                                                                          similarity constraintssimilarity constraints

                                                                                                          5656

                                                                                                          Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                                          X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                                          We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                                          On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                                          Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                                          SeedSeed pointspoints

                                                                                                          5757

                                                                                                          CriterionCriterion

                                                                                                          There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                                          histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                                          to be considered as a part of region to the to be considered as a part of region to the

                                                                                                          seed pointseed point

                                                                                                          The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                                          to at least one pixel in that regionto at least one pixel in that region

                                                                                                          Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                                          defectsdefects

                                                                                                          5858

                                                                                                          Region SplittingRegion Splitting

                                                                                                          The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                                          ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                                          If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                                          This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                                          Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                                          5959

                                                                                                          Region SplittingRegion Splitting

                                                                                                          DisadvantageDisadvantage

                                                                                                          they create regions that may be adjacent they create regions that may be adjacent

                                                                                                          and homogeneous but not mergedand homogeneous but not merged

                                                                                                          6060

                                                                                                          Region Splitting and MergingRegion Splitting and Merging

                                                                                                          ProcedureProcedure

                                                                                                          11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                          22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                          33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                          6161

                                                                                                          Region Splitting and Merging

                                                                                                          Quadtree

                                                                                                          (四叉树 )

                                                                                                          6262

                                                                                                          PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                          where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                          mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                          σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                          ExampleExample

                                                                                                          Original Original

                                                                                                          imageimageThresholded imageThresholded image Result of Result of

                                                                                                          Splitting and Splitting and

                                                                                                          MergingMerging

                                                                                                          • Slide 1
                                                                                                          • Slide 2
                                                                                                          • Slide 3
                                                                                                          • Slide 4
                                                                                                          • Slide 5
                                                                                                          • Slide 6
                                                                                                          • Slide 7
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                                                                                                            5454

                                                                                                            Steps of Region GrowingSteps of Region Growing

                                                                                                            Start by choosing an arbitrary seed Start by choosing an arbitrary seed

                                                                                                            pixel andpixel and compare it with neighbor compare it with neighbor

                                                                                                            ppixelsixels

                                                                                                            When When seedseed and and neighbor neighbor ppixelsixels are are similarsimilar and 8-connected neighbor r and 8-connected neighbor region egion

                                                                                                            is grown from the seed pixel by is grown from the seed pixel by

                                                                                                            addingadding neighboneighborr pixel pixelss

                                                                                                            When the growth of one region stopsWhen the growth of one region stops

                                                                                                            choose another seed pixel which doeschoose another seed pixel which does not yet belong to any region and start not yet belong to any region and start

                                                                                                            againagain

                                                                                                            5555

                                                                                                            Region Region growing growing

                                                                                                            An initial set of small An initial set of small

                                                                                                            areas are iterativelyareas are iteratively

                                                                                                            merged according to merged according to

                                                                                                            similarity constraintssimilarity constraints

                                                                                                            5656

                                                                                                            Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                                            X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                                            We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                                            On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                                            Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                                            SeedSeed pointspoints

                                                                                                            5757

                                                                                                            CriterionCriterion

                                                                                                            There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                                            histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                                            to be considered as a part of region to the to be considered as a part of region to the

                                                                                                            seed pointseed point

                                                                                                            The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                                            to at least one pixel in that regionto at least one pixel in that region

                                                                                                            Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                                            defectsdefects

                                                                                                            5858

                                                                                                            Region SplittingRegion Splitting

                                                                                                            The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                                            ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                                            If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                                            This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                                            Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                                            5959

                                                                                                            Region SplittingRegion Splitting

                                                                                                            DisadvantageDisadvantage

                                                                                                            they create regions that may be adjacent they create regions that may be adjacent

                                                                                                            and homogeneous but not mergedand homogeneous but not merged

                                                                                                            6060

                                                                                                            Region Splitting and MergingRegion Splitting and Merging

                                                                                                            ProcedureProcedure

                                                                                                            11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                            22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                            33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                            6161

                                                                                                            Region Splitting and Merging

                                                                                                            Quadtree

                                                                                                            (四叉树 )

                                                                                                            6262

                                                                                                            PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                            where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                            mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                            σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                            ExampleExample

                                                                                                            Original Original

                                                                                                            imageimageThresholded imageThresholded image Result of Result of

                                                                                                            Splitting and Splitting and

                                                                                                            MergingMerging

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                                                                                                              5555

                                                                                                              Region Region growing growing

                                                                                                              An initial set of small An initial set of small

                                                                                                              areas are iterativelyareas are iteratively

                                                                                                              merged according to merged according to

                                                                                                              similarity constraintssimilarity constraints

                                                                                                              5656

                                                                                                              Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                                              X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                                              We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                                              On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                                              Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                                              SeedSeed pointspoints

                                                                                                              5757

                                                                                                              CriterionCriterion

                                                                                                              There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                                              histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                                              to be considered as a part of region to the to be considered as a part of region to the

                                                                                                              seed pointseed point

                                                                                                              The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                                              to at least one pixel in that regionto at least one pixel in that region

                                                                                                              Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                                              defectsdefects

                                                                                                              5858

                                                                                                              Region SplittingRegion Splitting

                                                                                                              The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                                              ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                                              If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                                              This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                                              Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                                              5959

                                                                                                              Region SplittingRegion Splitting

                                                                                                              DisadvantageDisadvantage

                                                                                                              they create regions that may be adjacent they create regions that may be adjacent

                                                                                                              and homogeneous but not mergedand homogeneous but not merged

                                                                                                              6060

                                                                                                              Region Splitting and MergingRegion Splitting and Merging

                                                                                                              ProcedureProcedure

                                                                                                              11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                              22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                              33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                              6161

                                                                                                              Region Splitting and Merging

                                                                                                              Quadtree

                                                                                                              (四叉树 )

                                                                                                              6262

                                                                                                              PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                              where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                              mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                              σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                              ExampleExample

                                                                                                              Original Original

                                                                                                              imageimageThresholded imageThresholded image Result of Result of

                                                                                                              Splitting and Splitting and

                                                                                                              MergingMerging

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                                                                                                                5656

                                                                                                                Example defective welds (Example defective welds ( 焊缝焊缝 ) detecting) detecting

                                                                                                                X ray of weld (the horizontal dark X ray of weld (the horizontal dark region) containing several cracks region) containing several cracks and porosities (and porosities ( 孔孔 the bright w the bright white streaks running horizontally hite streaks running horizontally through the image)through the image)

                                                                                                                We need initial seed points to groWe need initial seed points to grow into regionsw into regions

                                                                                                                On looking at the histogram aOn looking at the histogram and image cracks are brightnd image cracks are bright

                                                                                                                Select all pixels having value oSelect all pixels having value of 255 as seedsf 255 as seeds

                                                                                                                SeedSeed pointspoints

                                                                                                                5757

                                                                                                                CriterionCriterion

                                                                                                                There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                                                histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                                                to be considered as a part of region to the to be considered as a part of region to the

                                                                                                                seed pointseed point

                                                                                                                The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                                                to at least one pixel in that regionto at least one pixel in that region

                                                                                                                Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                                                defectsdefects

                                                                                                                5858

                                                                                                                Region SplittingRegion Splitting

                                                                                                                The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                                                ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                                                If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                                                This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                                                Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                                                5959

                                                                                                                Region SplittingRegion Splitting

                                                                                                                DisadvantageDisadvantage

                                                                                                                they create regions that may be adjacent they create regions that may be adjacent

                                                                                                                and homogeneous but not mergedand homogeneous but not merged

                                                                                                                6060

                                                                                                                Region Splitting and MergingRegion Splitting and Merging

                                                                                                                ProcedureProcedure

                                                                                                                11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                                22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                                33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                                6161

                                                                                                                Region Splitting and Merging

                                                                                                                Quadtree

                                                                                                                (四叉树 )

                                                                                                                6262

                                                                                                                PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                                where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                                mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                                σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                                ExampleExample

                                                                                                                Original Original

                                                                                                                imageimageThresholded imageThresholded image Result of Result of

                                                                                                                Splitting and Splitting and

                                                                                                                MergingMerging

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                                                                                                                  5757

                                                                                                                  CriterionCriterion

                                                                                                                  There is a valley at around 190 in the There is a valley at around 190 in the

                                                                                                                  histogram A pixel should have a valuehistogram A pixel should have a valuegtgt190 190

                                                                                                                  to be considered as a part of region to the to be considered as a part of region to the

                                                                                                                  seed pointseed point

                                                                                                                  The pixel should be a 8-connected neighbor The pixel should be a 8-connected neighbor

                                                                                                                  to at least one pixel in that regionto at least one pixel in that region

                                                                                                                  Result of region growing and boundaries of Result of region growing and boundaries of

                                                                                                                  defectsdefects

                                                                                                                  5858

                                                                                                                  Region SplittingRegion Splitting

                                                                                                                  The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                                                  ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                                                  If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                                                  This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                                                  Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                                                  5959

                                                                                                                  Region SplittingRegion Splitting

                                                                                                                  DisadvantageDisadvantage

                                                                                                                  they create regions that may be adjacent they create regions that may be adjacent

                                                                                                                  and homogeneous but not mergedand homogeneous but not merged

                                                                                                                  6060

                                                                                                                  Region Splitting and MergingRegion Splitting and Merging

                                                                                                                  ProcedureProcedure

                                                                                                                  11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                                  22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                                  33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                                  6161

                                                                                                                  Region Splitting and Merging

                                                                                                                  Quadtree

                                                                                                                  (四叉树 )

                                                                                                                  6262

                                                                                                                  PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                                  where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                                  mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                                  σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                                  ExampleExample

                                                                                                                  Original Original

                                                                                                                  imageimageThresholded imageThresholded image Result of Result of

                                                                                                                  Splitting and Splitting and

                                                                                                                  MergingMerging

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                                                                                                                    5858

                                                                                                                    Region SplittingRegion Splitting

                                                                                                                    The opposite approach to region mergingThe opposite approach to region merging A top-down approach and starts with the assumA top-down approach and starts with the assum

                                                                                                                    ption that the entire image is homogeneousption that the entire image is homogeneous

                                                                                                                    If this is not true the image is split into four sub iIf this is not true the image is split into four sub imagesmages

                                                                                                                    This splitting procedure is repeated recursivelyThis splitting procedure is repeated recursively(( 递归的递归的 ) until the image is split into homogeneo) until the image is split into homogeneous regionsus regions

                                                                                                                    Need homogeneity criterion split ruleNeed homogeneity criterion split rule

                                                                                                                    5959

                                                                                                                    Region SplittingRegion Splitting

                                                                                                                    DisadvantageDisadvantage

                                                                                                                    they create regions that may be adjacent they create regions that may be adjacent

                                                                                                                    and homogeneous but not mergedand homogeneous but not merged

                                                                                                                    6060

                                                                                                                    Region Splitting and MergingRegion Splitting and Merging

                                                                                                                    ProcedureProcedure

                                                                                                                    11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                                    22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                                    33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                                    6161

                                                                                                                    Region Splitting and Merging

                                                                                                                    Quadtree

                                                                                                                    (四叉树 )

                                                                                                                    6262

                                                                                                                    PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                                    where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                                    mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                                    σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                                    ExampleExample

                                                                                                                    Original Original

                                                                                                                    imageimageThresholded imageThresholded image Result of Result of

                                                                                                                    Splitting and Splitting and

                                                                                                                    MergingMerging

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                                                                                                                      5959

                                                                                                                      Region SplittingRegion Splitting

                                                                                                                      DisadvantageDisadvantage

                                                                                                                      they create regions that may be adjacent they create regions that may be adjacent

                                                                                                                      and homogeneous but not mergedand homogeneous but not merged

                                                                                                                      6060

                                                                                                                      Region Splitting and MergingRegion Splitting and Merging

                                                                                                                      ProcedureProcedure

                                                                                                                      11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                                      22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                                      33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                                      6161

                                                                                                                      Region Splitting and Merging

                                                                                                                      Quadtree

                                                                                                                      (四叉树 )

                                                                                                                      6262

                                                                                                                      PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                                      where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                                      mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                                      σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                                      ExampleExample

                                                                                                                      Original Original

                                                                                                                      imageimageThresholded imageThresholded image Result of Result of

                                                                                                                      Splitting and Splitting and

                                                                                                                      MergingMerging

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                                                                                                                        6060

                                                                                                                        Region Splitting and MergingRegion Splitting and Merging

                                                                                                                        ProcedureProcedure

                                                                                                                        11 Split image into 4 disjointed quadrants for Split image into 4 disjointed quadrants for any region Rany region Rii P(R P(Rii)=FALSE)=FALSE

                                                                                                                        22 Merge any adjacent regions RMerge any adjacent regions Rjj and R and Rkk for w for which P(Rhich P(RjjcupRcupRkk)=TRUE)=TRUE

                                                                                                                        33 Stop when no further splitting or merging iStop when no further splitting or merging is possibles possible

                                                                                                                        6161

                                                                                                                        Region Splitting and Merging

                                                                                                                        Quadtree

                                                                                                                        (四叉树 )

                                                                                                                        6262

                                                                                                                        PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                                        where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                                        mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                                        σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                                        ExampleExample

                                                                                                                        Original Original

                                                                                                                        imageimageThresholded imageThresholded image Result of Result of

                                                                                                                        Splitting and Splitting and

                                                                                                                        MergingMerging

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                                                                                                                          6161

                                                                                                                          Region Splitting and Merging

                                                                                                                          Quadtree

                                                                                                                          (四叉树 )

                                                                                                                          6262

                                                                                                                          PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                                          where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                                          mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                                          σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                                          ExampleExample

                                                                                                                          Original Original

                                                                                                                          imageimageThresholded imageThresholded image Result of Result of

                                                                                                                          Splitting and Splitting and

                                                                                                                          MergingMerging

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                                                                                                                            6262

                                                                                                                            PP((RRii) = TRUE if at least 80 of the pixels in ) = TRUE if at least 80 of the pixels in RRii have t have the property |he property |zzjj--mmii|le2σ|le2σii

                                                                                                                            where zwhere zjj is the gray level of the is the gray level of the jjth pixel in th pixel in RRii

                                                                                                                            mmii is the mean gray level of that region is the mean gray level of that region

                                                                                                                            σσii is the standard deviation of the gray levels in is the standard deviation of the gray levels in RRii

                                                                                                                            ExampleExample

                                                                                                                            Original Original

                                                                                                                            imageimageThresholded imageThresholded image Result of Result of

                                                                                                                            Splitting and Splitting and

                                                                                                                            MergingMerging

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