3D Computer Vision and Video Computing Introducti on Part I Feature Extraction (2) Edge Detection CSc I6716 Spring 2013 Zhigang Zhu, City College of New York [email protected]
Feb 16, 2016
3D Computer Visionand Video Computing Introduction
Part IFeature Extraction (2)
Edge Detection
CSc I6716Spring 2013
Zhigang Zhu, City College of New York [email protected]
3D Computer Visionand Video Computing Edge Detection
n What’s an edge?l “He was sitting on the Edge of his seat.”l “She paints with a hard Edge.”l “I almost ran off the Edge of the road.”l “She was standing by the Edge of the woods.”l “Film negatives should only be handled by their Edges.”l “We are on the Edge of tomorrow.”l “He likes to live life on the Edge.”l “She is feeling rather Edgy.”
n The definition of Edge is not always clear.n In Computer Vision, Edge is usually related to a
discontinuity within a local set of pixels.
3D Computer Visionand Video Computing Discontinuities
n A: Depth discontinuity: abrupt depth change in the worldn B: Surface normal discontinuity: change in surface orientationn C: Illumination discontinuity: shadows, lighting changesn D: Reflectance discontinuity: surface properties, markings
A
C
B
D
3D Computer Visionand Video Computing Illusory Edges
n Illusory edges will not be detectable by the algorithms that we will discuss
n No change in image irradiance - no image processing algorithm can directly address these situations
n Computer vision can deal with these sorts of things by drawing on information external to the image (perceptual grouping techniques)
Kanizsa Triangles
3D Computer Visionand Video Computing Another One
3D Computer Visionand Video Computing Goal
n Devise computational algorithms for the extraction of significant edges from the image.
n What is meant by significant is unclear.l Partly defined by the context in which the edge detector
is being applied
3D Computer Visionand Video Computing Edgels
n Define a local edge or edgel to be a rapid change in the image function over a small areal implies that edgels should be detectable over a local
neighborhoodn Edgels are NOT contours, boundaries, or lines
l edgels may lend support to the existence of those structuresl these structures are typically constructed from edgels
n Edgels have propertiesl Orientationl Magnitudel Position
3D Computer Visionand Video Computing Outline
n First order edge detectors (lecture - required)l Mathematicsl 1x2, Roberts, Sobel, Prewitt
n Canny edge detector (after-class reading)n Second order edge detector (after-class reading)
l Laplacian, LOG / DOGn Hough Transform – detect by voting
l Linesl Circlesl Other shapes
3D Computer Visionand Video Computing Locating Edgels
Rapid change in image => high local gradient => differentiation
f(x) = step edge
1st Derivative f ’(x)
2nd Derivative -f ’’(x)
maximum
zero crossing
3D Computer Visionand Video Computing Reality
3D Computer Visionand Video Computing Properties of an Edge
OrientationOriginal
Orientation
Magnitude
Position
3D Computer Visionand Video Computing Quantitative Edge Descriptors
n Edge Orientationl Edge Normal - unit vector in the direction of
maximum intensity change (maximum intensity gradient)
l Edge Direction - unit vector perpendicular to the edge normal
n Edge Position or Centerl image position at which edge is located
(usually saved as binary image)n Edge Strength / Magnitude
l related to local contrast or gradient - how rapid is the intensity variation across the edge along the edge normal.
3D Computer Visionand Video Computing Edge Degradation in Noise
Ideal step edge Step edge + noise
Increasing noise
3D Computer Visionand Video Computing Real Image
3D Computer Visionand Video Computing Edge Detection: Typical
n Noise Smoothingl Suppress as much noise as possible while retaining
‘true’ edgesl In the absence of other information, assume ‘white’
noise with a Gaussian distributionn Edge Enhancement
l Design a filter that responds to edges; filter output high are edge pixels and low elsewhere
n Edge Localizationl Determine which edge pixels should be discarded as
noise and which should be retainedu thin wide edges to 1-pixel width (nonmaximum
suppression)u establish minimum value to declare a local maximum from
edge filter to be an edge (thresholding)
3D Computer Visionand Video Computing Edge Detection Methods
n 1st Derivative Estimatel Gradient edge detectionl Compass edge detectionl Canny edge detector (*)
n 2nd Derivative Estimatel Laplacianl Difference of Gaussians
n Parametric Edge Models (*)
3D Computer Visionand Video Computing Gradient Methods
F(x)
xF’(x)
x
Edge= sharp variation
Large first derivative
3D Computer Visionand Video Computing Gradient of a Function
n Assume f is a continuous function in (x,y). Then
n are the rates of change of the function f in the x and y directions, respectively.
n The vector (Dx, Dy) is called the gradient of f.n This vector has a magnitude: and an orientation:
n q is the direction of the maximum change in f.n S is the size of that change.
s = Dx2+Dy
2
Dx
Dyq = tan-1 ( )
yf
xf
yx
D
D ,
3D Computer Visionand Video Computing Geometric Interpretation
n Butl I(i,j) is not a continuous function.
n Thereforel look for discrete approximations to the gradient.
y
x
f(x,y) S
f
Dy
qDx
3D Computer Visionand Video Computing Discrete Approximations
df(x)dx = lim
Dx 0
f(x + Dx) - f(x)Dx
df(x)dx
f(x) - f(x-1)1@ xx-1
f(x)
Convolve with -1 1
3D Computer Visionand Video Computing In Two Dimensions
n Discrete image function I
n Derivatives Differences
i
j
Image
row i-1
row i
row i+1
col j-1 col j col j+1
I(i-1,j-1)
I(i,j-1)
I(i+1,j-1)
I(i-1,j)
I(i,j)
I(i+1,j)
I(i-1,j+1)
I(i,j+1)
I(i+1,j+1)
-1 11
-1DjI = DiI =
3D Computer Visionand Video Computing 1x2 Example
1x2 Vertical
1x2 HorizontalCombined
3D Computer Visionand Video Computing Smoothing and Edge Detection
n Derivatives are 'noisy' operationsl edges are a high spatial frequency phenomenonl edge detectors are sensitive to and accent noise
n Averaging reduces noisel spatial averages can be computed using masks
n Combine smoothing with edge detection.
1 1 1
1 1 1
1 1 1
1 / 9 x
1 1 1
1 1 1
1 1 1
1 1 1
1 0 1
1 1 1
1 / 8 x
3D Computer Visionand Video Computing Effect of Blurring
Original Orig+1 Iter Orig+2 Iter
Image
Edges
ThresholdedEdges
3D Computer Visionand Video Computing Combining the Two
n Applying this mask is equivalent to taking the difference of averages on either side of the central pixel.
-1 -1 -1
11 1
0 0
Average
Average
3D Computer Visionand Video Computing Many Different Kernels
n Variablesl Size of kernell Pattern of weights
n 1x2 Operator (we’ve already seen this one
-1 11
-1DjI = DiI =
3D Computer Visionand Video Computing Roberts Cross Operator
n Does not return any information about the orientation of the edge 22 ),1()1,()1,1(),( yxIyxIyxIyxI
or
1 00 -1
0 1-1 0+
| I(x, y) - I(x+1, y+1) | + | I(x, y+1) - I(x+1, y) |
[ I(x, y) - I(x+1, y+1) ]2 + [ I(x, y+1) - I(x+1, y) ]2S =
S =
3D Computer Visionand Video Computing Sobel Operator
-1 -2 -1 0 0 0 1 2 1
-1 0 1-2 0 2 -1 0 1
S1= S2 =
Edge Magnitude =
Edge Direction =
S1 + S22 2
tan-1S1
S2
3D Computer Visionand Video Computing Anatomy of the Sobel
1 2 1 0 0 0
-1 -2 -1
-1 0 1 -2 0 2-1 0 1
= 1/4 * [ 1 2 1] 1 0 -1
Sobel kernel is separable!
1
-1
-2 1
1
2 Averaging done parallel to edge
1/4
1/4
= 1/4 * [-1 0 -1] 1 2 1
+
3D Computer Visionand Video Computing Prewitt Operator
-1 -1 -1 0 0 0 1 1 1
-1 0 1-1 0 1 -1 0 1
P1= P2 =
Edge Magnitude =
Edge Direction =
P1 + P22 2
tan-1P1
P2
3D Computer Visionand Video Computing Large Masks
1 x 2
1 x 5
1 x 9
1 x 9 uniform weights
-1 0 0 0 0 0 0 0 1
-1 -1 -1 -1 0 1 1 1 1
-1 0 0 0 1
-1 1
What happens as the mask size increases?
3D Computer Visionand Video Computing Large Kernels
7x7 Horizontal Edges only
13x13 Horizontal Edges only
3D Computer Visionand Video Computing Compass Masks
n Use eight masks aligned with the usual compass directions
n Select largest response (magnitude)n Orientation is the direction associated with the largest
response
SE
E
NENNW
W
SE S(-)
(+)
3D Computer Visionand Video Computing Many Different Kernels
1
111
1
-1 -1
Prewitt 1
-2
-1
-3
555
-3
-3 -3 -3
Kirsch
0
0
111
0
-1 -1 -1
Prewitt 2
0 0
121
0
-1 -2 -1
Sobel
0
-1 -1
11
0 0 0
- 2
2
Frei & Chen
3D Computer Visionand Video Computing Robinson Compass Masks
-1 0 1-2 0 2 -1 0 1
0 1 2-1 0 1 -2 -1 0
1 2 1 0 0 0 -1 -2 -1
2 1 0 1 0 -1 0 -1 -2
1 0 -1 2 0 -2 1 0 -1
0 -1 -2 1 0 -1 2 1 0
-1 -2 -1 0 0 0 1 2 1
-2 -1 0-1 0 1 0 1 2
3D Computer Visionand Video Computing Analysis of Edge Kernels
n Analysis based on a step edge inclined at an angle q (relative to y-axis) through center of window.
n Robinson/Sobel: true edge contrast less than 1.6% different from that computed by the operator.
n Error in edge directionl Robinson/Sobel: less than 1.5 degrees errorl Prewitt: less than 7.5 degrees error
n Summaryl Typically, 3 x 3 gradient operators perform better than 2 x 2.l Prewitt2 and Sobel perform better than any of the other 3x3 gradient
estimation operators.l In low signal to noise ratio situations, gradient estimation operators of
size larger than 3 x 3 have improved performance.l In large masks, weighting by distance from the central pixel is
beneficial.
3D Computer Visionand Video Computing Prewitt Example
Santa Fe Mission Prewitt Horizontal and Vertical Edges Combined
3D Computer Visionand Video Computing Edge Thresholding
n Global approach
See Haralick paper for thresholding based on statistical significance tests.
0
1000
2000
3000
4000
5000
Num
ber o
f Pix
els
Edge Gradient Magnitude
64 128
T=64T=128
Edge Histogram
3D Computer Visionand Video Computing Reading and Homework
- Go through slides 40-71 after class
- Reading: Lecture notes on feature extraction
- Homework 2: Due after two weeks
You may try different filters in Matlab, but do your homework by your programming … …
3D Computer Visionand Video Computing Canny Edge Detector
n Probably most widely usedn LF. Canny, "A computational approach to edge detection",
IEEE Trans. Pattern Anal. Machine Intelligence (PAMI), vol. PAMI vii-g, pp. 679-697, 1986.
n Based on a set of criteria that should be satisfied by an edge detector:l Good detection. There should be a minimum number of false
negatives and false positives. l Good localization. The edge location must be reported as
close as possible to the correct position. l Only one response to a single edge.
Cost function which could be optimized using variational methods
3D Computer Visionand Video Computing Canny Results
I = imread(‘image file name’);BW1 = edge(I,'sobel');BW2 = edge(I,'canny');imshow(BW1)figure, imshow(BW2)
s=1, T2=255, T1=1
‘Y’ or ‘T’ junction problem with Canny operator
3D Computer Visionand Video Computing Canny Results
s=1, T2=255, T1=220 s=1, T2=128, T1=1 s=2, T2=128, T1=1
M. Heath, S. Sarkar, T. Sanocki, and K.W. Bowyer, "A Robust Visual Method for Assessing the Relative Performance of Edge-Detection Algorithms" IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 19, No. 12, December 1997, pp. 1338-1359.
http://marathon.csee.usf.edu/edge/edge_detection.html
3D Computer Visionand Video Computing
n Second derivatives…
3D Computer Visionand Video Computing Edges from Second Derivatives
n Digital gradient operators estimate the first derivative of the image function in two or more directions.
f(x) = step edge
1st Derivative f’(x)
2nd Derivative f’’(x)
maximum
zero crossing
GR
AD
IEN
TM
ETH
OD
S
3D Computer Visionand Video Computing Second Derivatives
n Second derivative = rate of change of first derivative.n Maxima of first derivative = zero crossings of second
derivative.n For a discrete function, derivatives can be approximated by
differencing.n Consider the one dimensional case:
..... f(i-2) f(i-1) f(i) f(i+1) f(i+2) .....
Df(i-1) Df(i) Df(i+1) Df(i+2)
D f(i-1)2 2D f(i+1)D f(i)2
D f(i) = D f(i+1) - D f(i)2
= f(i+1) - 2 f(i) + f(i-1)
-2 11Mask:
3D Computer Visionand Video Computing Laplacian Operator
n Now consider a two-dimensional function f(x,y).n The second partials of f(x,y) are not isotropic.n Can be shown that the smallest possible isotropic
second derivative operator is the Laplacian:
n Two-dimensional discrete approximation is:
1
1 1
1
-4
2
2
2
22
yf
xff
3D Computer Visionand Video Computing
-1 -1 24 -1 -1-1 -1 -1 -1 -1-1 -1 -1 -1 -1
-1 -1 -1 -1 -1-1 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 +8 +8 +8 -1 -1 -1 -1 -1 -1 +8 +8 +8 -1 -1 -1 -1 -1 -1 +8 +8 +8 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
Example Laplacian Kernels
n Note that these are not the optimal approximations to the Laplacian of the sizes shown.
5X5
9X9
3D Computer Visionand Video Computing Example Application
5x5 Laplacian Filter 9x9 Laplacian Filter
3D Computer Visionand Video Computing Detailed View of Results
3D Computer Visionand Video Computing Interpretation of the Laplacian
n Consider the definition of the discrete Laplacian:
n Rewrite as:
n Factor out -5 to get:
n Laplacian can be obtained, up to the constant -5, by subtracting the average value around a point (i,j) from the image value at the point (i,j)!l What window and what averaging function?
I = I(i+1,j)+I(i-1,j)+I(i,j+1)+I(i,j-1) - 4I(i,j)
looks like a window sum
I = I(i+1,j)+I(i-1,j)+I(i,j+1)+I(i,j-1)+I(i,j) - 5I(i,j)
I = -5 {I(i,j) - window average}
3D Computer Visionand Video ComputingEnhancement using the Laplacian
n The Laplacian can be used to enhance images:
n If (i,j) is in the middle of a flat region or long ramp: I-2I = In If (i,j) is at low end of ramp or edge: I-2I < In If (i,j) is at high end of ramp or edge: I-2I > I
n Effect is one of deblurring the image
I(i,j) - I(i,j) =5 I(i,j) -[I(i+1,j) + I(i-1,j) + I(i,j+1) + I(i,j-1)]
3D Computer Visionand Video Computing Laplacian Enhancement
Blurred Original 3x3 Laplacian Enhanced
3D Computer Visionand Video Computing Noise
n Second derivative, like first derivative, enhances noise
n Combine second derivative operator with a smoothing operator.
n Questions:l Nature of optimal smoothing filter.l How to detect intensity changes at a given
scale.l How to combine information across multiple
scales.n Smoothing operator should be
l 'tunable' in what it leaves behindl smooth and localized in image space.
n One operator which satisfies these two constraints is the Gaussian:
3D Computer Visionand Video Computing 2D Gaussian Distribution
n The two-dimensional Gaussian distribution is defined by:
n From this distribution, can generate smoothing masks whose width depends upon s:
e(x + y )2 2
2 s 2s 2
1G(x,y) =
x
y
3D Computer Visionand Video Computing s Defines Kernel ‘Width’
s2 = .25 s2 = 1.0 s2 = 4.0
3D Computer Visionand Video Computing Creating Gaussian Kernels
n The mask weights are evaluated from the Gaussian distribution:
n This can be rewritten as:
n This can now be evaluated over a window of size nxn to obtain a kernel in which the (0,0) value is 1.
n k is a scaling constant
W(i,j) = k * exp (- )i2 + j22 s2
= exp (- )i2 + j22 s2
W(i,j)k
3D Computer Visionand Video Computing Example
n Choose s = 2. and n = 7, then:2
.011 .039 .082 .105 .082 .039 .011
.011 .039 .082 .105 .082 .039 .011
.039 .135 .287 .368 .287 .135 .039
.105 .039 .779 1.000 .779 .368 .105.082 .287 .606 .779 .606 .287 .082
.082 .287 .606 .779 .606 .287 .082
.039 .135 .287 .368 .287 .135 .039
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
i
j
W(1,2)k = exp(-12+ 22
2*2 ) To make this value 1, choose k = 91.
3D Computer Visionand Video Computing Example
1 4 7 10 7 4 1
7 26 55 71 55 26 7
4 12 26 33 26 12 4
1 4 7 10 7 4 1
7 26 55 71 55 26 7
4 12 26 33 26 12 4
10 33 71 91 71 33 10
W(i,j) = 1,115j = -3
3
i = -3
3
7x7 Gaussian Filter
Plot of Weight Values
3D Computer Visionand Video Computing Kernel Application
7x7 Gaussian Kernel 15x15 Gaussian Kernel
3D Computer Visionand Video Computing Why Gaussian for Smoothing
n Gaussian is not the only choice, but it has a number of important propertiesl If we convolve a Gaussian with another Gaussian, the
result is a Gaussianu This is called linear scale space
l Efficiency: separablel Central limit theorem
3D Computer Visionand Video Computing Why Gaussian for Smoothing
n Gaussian is separable
3D Computer Visionand Video Computing Why Gaussian for Smoothing – cont.
n Gaussian is the solution to the diffusion equation
n We can extend it to non-linear smoothing
3D Computer Visionand Video Computing 2G Filter
n Marr and Hildreth approach:1. Apply Gaussian smoothing using s's of increasing size:
2. Take the Laplacian of the resulting images:
3. Look for zero crossings.
n Second expression can be written as:
n Thus, can take Laplacian of the Gaussian and use that as the operator.
G I*
(G I)*
(2G ) * I
3D Computer Visionand Video Computing Mexican Hat Filter
n Laplacian of the Gaussian
n 2G is a circularly symmetric operator.n Also called the hat or Mexican-hat operator.
-1s 4
G (x,y) =2s2
(x + y )2 21 - e 2s2
(x + y )2 22
3D Computer Visionand Video Computing s2 Controls Size
s2 = 0.5 s2 = 1.0 s2 = 2.0
3D Computer Visionand Video Computing Kernels
n Remember the center surround cells in the human system?
0 0 -1 0 0
-1 -2 16 -2 -1 0 -1 -2 -1 0
0 -1 -2 -1 0 0 0 -1 0 0
0 0 0 0 0 0 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0
0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0
0 0 -1 -1 -1 -2 -3 -3 -3 -3 -3 -2 -1 -1 -1 0 0
0 0 -1 -1 -1 -2 -3 -3 -3 -3 -3 -2 -1 -1 -1 0 0
0 0 -1 -1 -2 -3 -3 -3 -3 -3 -3 -3 -2 -1 -1 0 0
0 0 -1 -1 -2 -3 -3 -3 -3 -3 -3 -3 -2 -1 -1 0 0
0 -1 -1 -2 -3 -3 -3 -2 -3 -2 -3 -3 -3 -2 -1 -1 0 0 -1 -2 -3 -3 -3 0 2 4 2 0 -3 -3 -3 -2 -1 0
0 -1 -1 -2 -3 -3 -3 -3 -3 -3 -3 -3 -3 -2 -1 -1 0 0 -1 -2 -3 -3 -3 0 2 4 2 0 -3 -3 -3 -2 -1 0
-1 -1 -3 -3 -3 0 4 10 12 10 4 0 -3 -3 -3 -1 -1
-1 -1 -3 -3 -3 0 4 10 12 10 4 0 -3 -3 -3 -1 -1
-1 -1 -3 -3 -2 2 10 18 21 18 10 2 -2 -3 -3 -1 -1
-1 -1 -3 -3 -2 2 10 18 21 18 10 2 -2 -3 -3 -1 -1 -1 -1 -3 -3 -3 4 12 21 24 21 112 4 -3 -3 -3 -1 -1
5x5
17 x 17
3D Computer Visionand Video Computing Example
13x13 Kernel
3D Computer Visionand Video Computing Example
13 x 13 Hat Filter Thesholded Positive
Thesholded Negative Zero Crossings
3D Computer Visionand Video Computing Scale Space
Thresholded Negative
17x17 LoG Filter Thresholded Positive
Zero Crossings
3D Computer Visionand Video Computing Scale Space
s2 = 2 s2 = 2
s2 =2 2 s2 = 4
3D Computer Visionand Video Computing Multi-Resolution Scale Space
n Observations:l For sufficiently different s 's, the zero crossings will be
unrelated unless there is 'something going on' in the image.l If there are coincident zero crossings in two or more
successive zero crossing images, then there is sufficient evidence for an edge in the image.
l If the coincident zero crossings disappear as s becomes larger, then either:
u two or more local intensity changes are being averaged together, oru two independent phenomena are operating to produce intensity changes
in the same region of the image but at different scales.
n Use these ideas to produce a 'first-pass' approach to edge detection using multi-resolution zero crossing data.
n Never completely worked outn See Tony Lindbergh’s thesis and papers
3D Computer Visionand Video Computing Color Edge Detection
n Typical Approachesl Fusion of results on R, G, B separately
l Multi-dimensional gradient methods
l Vector methodsl Color signatures: Stanford (Rubner and Thomasi)
3D Computer Visionand Video Computing Hierarchical Feature Extraction
n Most features are extracted by combining a small set of primitive features (edges, corners, regions)
l Grouping: which edges/corners/curves form a group?u perceptual organization at the intermediate-level of vision
l Model Fitting: what structure best describes the group?
n Consider a slightly simpler problem…..
3D Computer Visionand Video Computing From Edgels to Lines
n Given local edge elements:
n Can we organize these into more 'complete' structures, such as straight lines?
n Group edge points into lines?n Consider a fairly simple technique...
3D Computer Visionand Video Computing Edgels to Lines
n Given a set of local edge elementsl With or without orientation information
n How can we extract longer straight lines?n General idea:
l Find an alternative space in which lines map to pointsl Each edge element 'votes' for the straight line which it
may be a part of.l Points receiving a high number of votes might
correspond to actual straight lines in the image.n The idea behind the Hough transform is that a change
in representation converts a point grouping problem into a peak detection problem
3D Computer Visionand Video Computing Edgels to Lines
n Consider two (edge) points, P(x,y) and P’(x’,y’) in image space:
n The set of all lines through P=(x,y) is y=mx + b, for appropriate choices of m and b.l Similarly for P’
n But this is also the equation of a line in (m,b) space, or parameter space.
P
P 'L
x
y
3D Computer Visionand Video Computing Parameter Space
b
m
b = -mx+y
b’ = -m’x'+y'
x,y; x',y' are fixed
(m,b)
L 1
L2
n The intersection represents the parameters of the equation of a line y=mx+b going through both (x,y) and (x',y').
n The more colinear edgels there are in the image, the more lines will intersect in parameter space
n Leads directly to an algorithm
3D Computer Visionand Video Computing General Idea
n General Idea: l The Hough space (m,b) is a representation of every
possible line segment in the planel Make the Hough space (m and b) discretel Let every edge point in the image plane ‘vote for’ any
line it might belong to.
3D Computer Visionand Video Computing Hough Transform
n Line Detection Algorithm: Hough Transforml Quantize b and m into appropriate 'buckets'.
u Need to decide what’s ‘appropriate’l Create accumulator array H(m,b), all of whose
elements are initially zero.l For each point (i,j) in the edge image for which the edge
magnitude is above a specific threshold, increment all points in H(m,b) for all discrete values of m and b satisfying b = -mj+i.
u Note that H is a two dimensional histograml Local maxima in H corresponds to colinear edge points
in the edge image.
3D Computer Visionand Video Computing Quantized Parameter Space
n Quantization
b single votes two votes
m
The problem of line detection in image space has been transformed into the problem of clusterdetection in parameter space
3D Computer Visionand Video Computing Example
n The problem of line detection in image space has been transformed into the problem of cluster detection in parameter space
Image Edges
AccumulatorArray
Result
3D Computer Visionand Video Computing Problems
n Vertical lines have infinite slopesl difficult to quantize m to take this into account.
n Use alternative parameterization of a linel polar coordinate representation
r1
q1
q2
r2
y
x
r 1 x 1 cos q y 1 sin q+=
r = x cos q + y sin q
3D Computer Visionand Video Computing Why?
n (r,q) is an efficient representation:l Small: only two parameters (like y=mx+b)l Finite: 0 £ r £ Ö(row2+col2), 0 £ q £ 2l Unique: only one representation per line
3D Computer Visionand Video Computing Alternate Representation
n Curve in (r,q) space is now a sinusoid l but the algorithm remains valid.
r
2 q
r1 x1 cos q y 1 sin q+=
r 2 x 2 cos q y2 sin q+=
3D Computer Visionand Video Computing Example
x
y
P = (4, 4)
P = (-3, 5)
1
2
P2 P1
r
q
r 4 c 4 s+=r 3 c 5 s+=
s 2 c 2+ 1=
s 750
50=
c 150
50=
q 1.4289=
r 4.5255=
Solve for r and q
r 3 cos q 5 sin q+=
r 4 cos q 4 sin q+=
Two Constraints
(r, q )(r, q ) Space
3D Computer Visionand Video Computing Real Example
Image Edges
AccumulatorArray
Result
3D Computer Visionand Video Computing Modifications
n Note that this technique only uses the fact that an edge exists at point (i,j).
n What about the orientation of the edge?l More constraints!
n Use estimate of edge orientation as q.n Each edgel now maps to a point in Hough space.
Image
Origin is arbitrary
The three edgels have same (r, q)
3D Computer Visionand Video Computing Gradient Data
n Colinear edges in Cartesian coordinate space now form point clusters in (m,b) parameter space.
E1
E2
E3
L1L2
L3
b
L1
L2 L3
mm
3D Computer Visionand Video Computing Gradient Data
n ‘Average’ point in Hough Space:
n Leads to an ‘average’ line in image space:
b
L1
L2 L3
m m
ba = -max + y
Average line in coordinate space
3D Computer Visionand Video Computing Post Hough
n Image space localization is lost:
n Consequently, we still need to do some image space manipulations, e.g., something like an edge 'connected components' algorithm.
n Heikki Kälviäinen, Petri Hirvonen, L. Xu and Erkki Oja, “Probabilistic and nonprobabilistic Hough Transforms: Overview and comparisons”, Image and vision computing, Volume 13, Number 4, pp. 239-252, May 1995.
both sets contribute to the same Hough maxima.
3D Computer Visionand Video Computing Hough Fitting
n Sort the edges in one Hough clusterl rotate edge points according to ql sort them by (rotated) x coordinate
n Look for Gapsl have the user provide a “max gap” thresholdl if two edges (in the sorted list) are more than max gap
apart, break the line into segmentsl if there are enough edges in a given segment, fit a
straight line to the points
3D Computer Visionand Video Computing Generalizations
n Hough technique generalizes to any parameterized curve:
n Success of technique depends upon the quantization of the parameters:l too coarse: maxima 'pushed' togetherl too fine: peaks less defined
n Note that exponential growth in the dimensions of the accumulator array with the the number of curve parameters restricts its practical application to curves with few parameters
parameter vector (axes in Hough space)
f(x,a) = 0
3D Computer Visionand Video Computing Example: Finding a Circle
n Circles have three parametersl Center (a,b)l Radius r
n Circle f(x,y,r) = (x-a)2+(y-b)2-r2 = 0n Task:
n Given an edge point at (x,y) in the image, where could the center of the circle be?
Find the center of a circle with known radius r given an edge image with no gradient direction information (edge location only)
3D Computer Visionand Video Computing Finding a Circle
(i-a)2+(j-b)2-r2 = 0
Image
fixed (i,j)
Parameter space (a,b)
Parameter space (a,b)
Parameter space (a,b)
Circle Center(lots of votes!)
3D Computer Visionand Video Computing Finding Circles
n If we don’t know r, accumulator array is 3-dimensionaln If edge directions are known, computational
complexity if reducedl Suppose there is a known error limit on the edge
direction (say +/- 10o) - how does this affect the search?n Hough can be extended in many ways….see, for
example:l Ballard, D. H. Generalizing the Hough Transform to
Detect Arbitrary Shapes, Pattern Recognition 13:111-122, 1981.
l Illingworth, J. and J. Kittler, Survey of the Hough Transform, Computer Vision, Graphics, and Image Processing, 44(1):87-116, 1988