Hidden Markov Random Field model and BFGS algorithm for Brain Image Segmentation

Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective

1th Mediterranean Conference on Pattern Recognition and Artificial Intelligence

EL-Hachemi Samy Dominique RamdaneGuerrout Ait-Aoudia Michelucci Mahiou

Hidden Markov Random Field model and BFGSalgorithm for Brain Image Segmentation

LMCS Laboratory, ESI, Algeria & LE2I Laboratory, UB, France

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1 Introduction

2 Hidden Markov Random Field

3 BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm

4 Experimental Results

5 Conclusion & Perspective

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Problematic & Solution

1 Nowadays, We face a huge number of medical images

2 Manual analysis and interpretation became a tedious task

3 Automatic image analysis and interpretation is a necessity

4 To simplify the representation of an image into items meaningfuland easier to analyze, we need a segmentation method

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A segmentation methods

Segmentation methods can be classified in four main categories :

1 Threshold-based methods

2 Region-based methods

3 Model-based methods4 Classification methods

1 HMRF - Hidden Markov Random Field2 etc

We have chosen HMRF as a model to perform segmentation

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Hidden Markov Random Field

1 HMRF provides an elegant way to model the segmentationproblem

2 HMRF is a generalization of Hidden Markov Model

3 Each pixel is seen as a realization of Markov random variable

4 Each image is seen as a realization of set or family of Markovrandom variables

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The image to segment y = yss∈S

into K classes is a realization of Y

1 Y = Yss∈S is a family ofrandom variables

2 ys ∈ [0 . . .255]

The segmented image into K classesx = xss∈S is realization of X

1 X = Xss∈S is a family ofrandom variables

2 xs ∈ 1, . . . ,K

An example of segmentation intoK = 4 classes

The goal of HMRF is looking for x∗

x∗ = argx∈Ω max P[X = x | Y = y ]

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1 This elegant model leads to the optimization of an energy function

Ψ(x ,y) = ∑s∈S

[ln(σxs ) +

(ys−µxs )2

2σ2xs

]+ β

T ∑c2=s,t (1−2δ(xs,xt))

2 Our way to look for the minimization of Ψ(x ,y) is to look for theminimization Ψ(µ), µ = (µ1, . . . ,µK ) where µi are means of grayvalues of class i

3 The main idea is to focus on the means adjustment instead oftreating pixels adjustment

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1 Now, we seek for u∗

µ∗ = argµ∈[0...255]K minΨ(µ)

Ψ(µ) = ∑Kj=1 f (µj)

f (µj) = ∑s∈Sj

[ln(σj) +(ys−µj )

2

2σ2j

] + β

T ∑c2=s,t

(1−2δ(xs,xt))

2 To apply optimization techniques, we redefine the function Ψ(µ)for µ ∈ RK instead µ ∈ [0 . . .255]K . For that, we distinguish twoforms Ψ1(µ) and Ψ2(µ).

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Form 1

Ψ1(µ) =

Ψ(µ) if µ ∈ [0 . . .255]K

+∞ otherwise

Ψ1 treats all points outside[0 . . .255] in the same way

Form 2

Ψ2(µ) = ∑Kj=1 F(µj) where µj ∈ R

F(µj) =

f (0)−uj if µj < 0

f (µj) if µj ∈ [0 . . .255]

f (255) + (uj −255) if µj > 2559 / 19


BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm

1 BFGS is one of the most powerful methods to solveunconstrained optimization problem

2 BFGS is the most popular quasi-Newton method

3 BFGS is based on the gradient descent to reach the localminimum

4 Main idea of descent gradient is :1 We start from the initial point µ0

2 At the iteration k + 1, the point µk+1 is calculated from the point µk

according to the following formula : µk+1 = µk + αk dk

- αk is the step size at the iteration k- dk the search direction at the iteration k

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BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm

Summary of BFGS algorithm

1 Initialization : Set k := 0,Choose µ0 close to the solutionSet H0 := I, Set α0 = 1Choose the required accuracy ε ∈ R,ε>0

2 At the iteration k :Compute Hessian matrix approximation Hk

Compute the inverse of Hessian matrixCompute the search direction dk

Compute the step size αk

Compute the point µk+1

3 The stopping criterion : If ‖Ψ′(µk )‖<ε then µ := µk

4 k := k + 1

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DC - The Dice Coefficient

The Dice coefficient measures how muchthe segmentation result is close to theground truth

DC =2|A∩B||A∪B|

1 DC equals 1 in the best case(perfect segmentation)

2 DC equals 0 in the worst case(every pixel is misclassified)

FIGURE – The Dice Coefficient

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BFGS in practice

1 We used the Gnu Scientific Library implementation of the BFGS

2 To apply BFGS, we need at least the first derivative

3 In our case, computing the first derivative is not obvious

4 We have used the centric form to compute an approximation ofthe first derivative

Centric form of the first derivativeΨ′(µ)) = ( ∂Ψ

∂µ1, . . . , ∂Ψ

∂µn)

∂Ψ∂µi

= Ψ(µ1,...,µi +ε,...,µn)−Ψ(µ1,...,µi−ε,...,µn)2ε

5 Good approximation of the first derivative relies on the choice ofthe value of the parameter ε

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BFGS in practice - Results

1 Through the numerous tests conducted we have selectedε = 0.01 as the best value for a good approximation of the firstderivative

2 We have tested two functions Ψ1 and Ψ2, Ψ1 treats all pointsoutside [0 . . .255] in the same way

3 In practice, Ψ1 and Ψ2 give nearly the same results

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HMRF-BFGS VS Classical MRF, MRF-ACO-Gossiping &MRF-ACO

MethodsDice coefficient

GM WM CSF MeanClassical-MRF 0.763 0.723 0.780 0.756MRF-ACO 0.770 0.729 0.785 0.762MRF-ACO-Gossiping 0.770 0.729 0.786 0.762HMRF-BFGS 0.974 0.991 0.960 0.975

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Results with (N-Noise,I-Intensity non-uniformity)

(N,I)The initial Dice coefficient

Time(s)point GM WM CSF Mean

(0 % , 0 %) µ0,1 0.974 0.991 0.960 0.975 27.544(2 % , 20 %) µ0,2 0.942 0.969 0.939 0.950 15.630(5 % , 20 %) µ0,3 0.919 0.952 0.920 0.930 84.967

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Example of segmentation using HMRF-BFGS

(0%,0%) (3%,20%) (5%,20%)

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Conclusion & Perspective

1 We have presented a combination method HMRF-BFGS2 Through the tests conducted,

1 We have figured out good parameter settings2 HMRF-BFGS method shows a good results

3 We conclude that HMRF-BFGS method it is very promising

4 Nevertheless, the opinion of specialists must be considered in theevaluation

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Thank youfor your attention

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Hidden Markov Random Field model and BFGS algorithm for Brain Image Segmentation