Hidden Markov Random Field model and BFGS algorithm for Brain Image Segmentation
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Transcript of Hidden Markov Random Field model and BFGS algorithm for Brain Image Segmentation
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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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
Hidden Markov Random Field
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
Hidden Markov Random Field
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
Hidden Markov Random Field
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
Hidden Markov Random Field
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
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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
Example of segmentation using HMRF-BFGS
(0%,0%) (3%,20%) (5%,20%)
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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
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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Introduction Hidden Markov Random Field BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm Experimental Results Conclusion & Perspective
Thank youfor your attention
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