Joonas Vanninen Antonio Palomino Alarcos. One of the objectives of biomedical image analysis The characteristics of the regions are examined later.

Joonas VanninenAntonio Palomino Alarcos

One of the objectives of biomedical image analysis

The characteristics of the regions are examined later in detail

Segmentation is the process of dividing an image into different parts by Discontinuity Similarity

Many of the methods can be used more generally in detection of features

If the gray levels of the objects of interest are known, the image can be thresholded to include only them

Doesn’t generally produce uniform regions

Can be useful in noise removal and the analysis of particles

Isolated points can be detected with the following convolution mask

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•Can be thresholded

•Straight lines can be detected with masks

An edge: A large change in the gray level The change is in a particular direction,

depending upon the orientation of the edge Can be measured for example with derivatives

or gradients:

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Derivatives and gradients can be approximated from differences with convolution masks

Prewitt operators:

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Sobel operators have larger weights for the pixels in the same row / column:

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•Roberts operators use 2 x 2 neighborhoods to compute cross-differences

With two masks we can get a vector value for the gradient:

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A second-order difference operator

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•Omnidirectional: sensitive to edges in all directions but can’t detect the direction of the edge

•Sensitive to noise since there is no averaging

•Positive and negative values for each edge

•Zero crossings in between, can be used to find the local maximas of the first-order gradients

The noise in an image can be reduced by first convolving it with a Gaussian:

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erGeyxG

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•The order of the operators can be changed:

The result is called the Laplacian of Gaussian operator, LoG

Often referred as the Mexican hat function

Can be approximated by the difference between two gaussians, DoG operator

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eyx

yxG

Uses zero-crossings of the image convolved with the LoG-operator to represent edges

Problems: If the edges are not well separated, zero

crossings may also represent local minimas (false zero crossings)

The edge localization may be poor

Different structures are visible with different scales, parameter σ of the Gaussian

Ideally an edge would be seen with as many scales as possible

Stability map measures the persistence of boundaries over a range of filter scales

Ideal detector for step-type edges corrupted with additive white noise by three criterias:

Detection: no false or missing edges Localization: detected edges are spatially

near the real ones A single output for a single edge

The image is convolved with a Gaussian The gradient is estimated for each pixel

• The direction of the gradient is the normal of the edge

• The amplitude of the gradient is the strength of the edge

Non-maximal suppression: the values of gradients that are not local maximas are set to zero

The gradients are hysteresis thresholded: a pixel is considered to be an edge pixel if• It has a gradient value larger than the higher

threshold, • It has a gradient value larger than the lower

threshold and it is spatially connected to another edge pixel

The zero-crossings of the second derivative in the direction of the normal can also be used• This can be used for sub-pixel accuracy

Highpass filters in the Fourier-domain can be used to find edges

High-frequency noise → use a bandpass filter

LoG –filter: a high-frequency emphasising Laplacian and a Gaussian lowpass filter

Use of frequency domain may be computationally advantageous if the LoG is specified with a large array (large σ)

Edges are usually not linked The similarity of edge pixels can be

measured by: The strenght of the gradient The direction of the gradient

Most similar pixels should be used to link edges to each other

The original image

Edge detection with Prewitt mask, T=10

Edge detection with Prewitt mask, T=15

Edge detection with Prewitt mask, T=30

Edge detection with Sobel mask, T=15

Edge detection with Roberts mask, T=20

Edge detection with Marr-Hildreth -method (LoG), T=1.2

Edge detection with Canny method, Tl=0.05, Th=0.2

Dividing the image into regions that could correspond to ROIs is an important prerequisite apply the image analysis techniques

Computer analysis of images usually starts with segmentation Reduces pixel data to region-based information

about the objects present in the image

Thresholding techniques Assumption: all pixels whose values lie within

a certain range belong to the same class Threshold may be determined based upon

the histogram of the image

Boundary-based methods Assumption: pixel values change rapidly at

the boundaries between regions Intensity discontinuities lying at the

boundaries between objects and backgrounds must be detected

Region-based methods Assumption: neighboring pixels within a

region have similar values May be divided into two groups

Region Splitting and Merging Region Growing

Hybrid techniques Combine boundary and region criteria

Noise modify the gray levels to distributions represented by Gaussian PDFs

Probability of erroneus classification is

Differentiating whith respect to T, equating the result to zero and taking some simplifications (σ1=σ2=σ)

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This method partitions R (entire space of the given image) into n subregions such that: is a connected region for i=1,2,…,n

Results are highly dependent upon the procedure used to select the seed pixels and the inclusion criteria used

RRini 1iR

jijiRR ji ,,TRUERF i )(

jijiFALSERRF ji ,,)(

Initially, we will divide the given image arbitrarily into a set of disjoint quadrants

If F(Ri)=FALSE for any quadrant, subdivide that quadrant into subquadrants

Iterate the procedure until no further changes are made

Splitting procedure could result in adjacent regions that are similar, a merging step would be required, as follows: F(RiURk)=TRUE

Iterate until no further merging is possible

A neighboring pixel f(m,n) is appended to the region if:

T Ξ ‘Additive Tolerance Level’

Problem: The size and shape of the region depend on the seed pixel selected

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Running-mean algorithm The new pixel is compared with the mean gray

level (running mean) of the region being grown

”Current center pixel” method After a pixel C is appended to the region, its 4

(or 8) connected neighbours would be checked for inclusion in the region as follows:

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nmfN

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TCN i

A relative difference, based upon a ”multiplicative tolerance level” (τ) could be employed:

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nmf

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or

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),(2

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•μRcΞ original seed pixel value current center pixel value running-mean gray level

Last methods presents difficulties in the selection of the range of the tolerance value Possible solution: make use of some of the

characteristics of the HVS

New parameter ”Just-noticeable difference” (JND) is used:

JND=L.CT

L Ξ background luminance CT Ξ threshold contrast

Determination of the JND as a function of background gray level is needed to apply this method

It is possible to determine this relationship based upon psychophysical esperiments

1. It starts with a 4-connected neighbor-pixel grouping. The condition is defined as:

2. Removal of small regions is performed

3. Merging of connected regions is performed if any of two neighboring regions meet the JND condition

4. The procedure is iterated until no nieghboring region satisfies the JND condition

)(,)(min 2121 pJNDpJNDpp

Region grown using an additive tolerance T=10

Region grown using running mean tolerance T=10

Region grown using multiplicative mean tolerance T=0.1

Hough domain: straight lines are characterized by the pair of parameters (m,c)

m is the slope c is the position

Disadvantage: m and c have unbounded ranges

Parametric representation

θ limited to [0,π] (or to [0,2π]) ρ limited by the size of the image Limits of (ρ,θ) affected by the choice of the origin

cmxy

sincos yx

If normal parameters of the line are (ρ0,θ0)

Derived properties: Point in (x,y) space corresponds to a sinusoidal curve in the

(ρ,θ) space Point in the (ρ,θ) space correspond to a straigh line in the

(x,y) space Points in the same straigh line in the (x,y) space

corresponds to curves through a common point in the (ρ,θ) space

Points on the same curve in the parameter space corrspond to lines through a common point in the (x,y) space

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1. Discretize the (ρ,θ) space into accumulator cells by quantizing ρ and θ

2. Accumulator cells are increased by one (new curve ’ρ=x(n)cosθ+y(n)cosθ’ found) for each pixel with a value of 1

3. Cordinates of points of intersection of the curves in the parameter space provide the parameters of the line

Image: Wikipedia

Any circle in the (x,y) space is represented by a single point in the 3D (a,b,c) parameter space

Points along the perimeter of the circle describe a circular cone in the (a,b,c) space

Algorithm for detection of straight lines may be extended for the detection of circles

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The methods here are quite general, their applications are not limited to just segmentation

The purpose of any image processing method is to obtain a result that can be presented to humans or used in a further analysis The result should be consistent with a human

observer’s assessment A priori information about the shapes and

features in an image is important in segmentation