hidden markov random fields and direct search methods for medical image segmentation

Introduction HMRF Direct Search Methods Experimental Results Conclusion

5th The International Conference on Pattern Recognition Applications and Methods

EL-Hachemi Samy Dominique RamdaneGuerrout Ait-Aoudia Michelucci Mahiou

Hidden Markov Random Fields and Direct SearchMethods for Medical Image Segmentation

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

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1 Introduction

2 HMRF

3 Direct Search MethodsNelder-Mead methodTorczon method

4 Experimental Results

5 Conclusion

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Problematic

1 Huge number of medical images

1 MRI - Magnetic Resonance Imaging2 CT - Computed Tomography3 Radiography4 Digital mammography5 . . .etc

2 The manual analysis and interpretation is a difficult task

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Solution

The automatic extraction of meaningful information is one among thesegmentation challenges

The goal of image segmentation ?

Simplify the representation of an image to items meaningful and easierto analyze

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The segmentation, Howto ?

There are several techniques to perform the segmentation

1 Active contour

2 Edge detection

3 Thresholding

4 Region growing

5 HMRF - Hidden Markov Random Field

6 . . .etc

Our way to perform the segmentation relies on HMRF

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Hidden Markov Random Field

The image to segment into K classesy = {ys}s∈S is seen as a realization ofa random field Y = {Ys}s∈S

1 Each ys is a realization of arandom variable Ys

2 Each ys ∈ [0 . . .255]

The segmented image x = {xs}s∈S isseen as a realization of a random fieldX = {Xs}s∈S

1 Each xs is a realization of arandom variable Xs

2 Each xs ∈ {1, . . . ,K}

An example of segmentation with K= 4

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1 This elegant model leads to the optimization of an energyfunction Ψ(x ,y)

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 instead to work directly on the pixels adjustmentof x , to work on the means adjustment first.

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Ψ(µ) = ∑Kj=1 ∑

s∈Sj

[ln(σj) +(ys−µj)

2

2σ2j

] + β

T ∑c2={s,t} (1−2δ(xs,xt))

µ∗ = (µ∗1, . . . ,µ∗j , . . . ,µ

∗K ) = argµ∈[0...255]K min{Ψ(µ)}

µj = 1|Sj | ∑s∈Sj

ys

σj =√

1|Sj | ∑s∈Sj

(ys−µj)2

Sj = {s | xs = j}

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Nelder-Mead method

Nelder-Mead method

1 Proposed by John Nelder and Roger Mead (1965)

2 To minimize Ψ(µ) of K unknowns, we need K + 1 vertices ∈ RK

form non degenerate simplex (i.e., non flat)

3 Based on the comparison of Ψ(µ) values at K + 1 vertices

4 Each time a new vertex is generated relatively to the gravitycenter of K best vertices by the operations :

1 Reflection2 Expansion3 Contraction

5 The vertex with the worse function value is replaced with the newvertex if its value is better. Otherwise the simplex is shrunk

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Nelder-Mead method

1- Evaluate

1 Compute Ψi := Ψ(Vi)

2 Determine the indices h, s, l :1 Ψh := max

i(Ψi )

2 Ψs := maxi 6=h

(Ψi )

3 Ψl := mini

(Ψi )

3 Compute V̄ := 1K ∑

i 6=hVi

Example in R2

FIGURE : Gravity center

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Nelder-Mead method

2- Reflect

1 Compute the reflection vertex Vr

from : Vr := 2V̄ −Vh

2 Evaluate Ψr := Ψ(Vr )

3 If Ψl ≤Ψr < Ψs

1 Replace Vh by Vr

2 Terminate the iteration

Example in R2

FIGURE : Reflection

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Nelder-Mead method

3- Expand

1 If Ψr < Ψl

1 Compute the expansion vertex Ve

from : Ve := 3V̄ −2Vh

2 Evaluate Ψe := Ψ(Ve)3 If Ψe < Ψr

1 Replace Vh by Ve2 Terminate the iteration

4 Otherwise (if Ψe ≥Ψr )1 Replace Vh by Vr2 Terminate the iteration

Example in R2

FIGURE : Expansion

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Nelder-Mead method

4- Contract

1 If Ψr ≥Ψs and If Ψr < Ψh

1 Compute an outside contraction Vc

from : Vc := 32 V̄ − 1

2 Vh

2 Evaluate Ψc := Ψ(Vc)3 If Ψc < Ψr

1 Replace Vh by Vc2 Terminate the iteration

4 Otherwise ( If Ψc ≥Ψr )1 Go to the step 5 (shrink)

Example in R2

FIGURE : Outside contraction

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Nelder-Mead method

4- Contract

1 If Ψh ≤Ψr

1 Compute an inside contraction Vc

from : Vc := 12 (Vh + V̄ )

2 Evaluate Ψc := Ψ(Vc)3 If Ψc < Ψh

1 Replace Vh by Vc2 Terminate the iteration

4 Otherwise ( If Ψc ≥Ψh )1 Go to the step 5 (shrink)

Example in R2

FIGURE : Inside contraction

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Nelder-Mead method

5- Shrink

Replace all vertices according to thefollowing formula : Vi := 1

2 (Vi + Vl)

Example in R2

FIGURE : Shrink

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Torczon method

Torczon method

1 Torczon method is an improvement of Nelder-Mead method2 The differences are :

1 All the vertices are concerned by the operations (reflect, expandand contract)

2 All simplexes in Torczon method are homothetic to the first one3 No degeneracy (i.e., flat simplex) can occur in Torczon method

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Torczon method

1- Evaluate

1 Compute Ψi := Ψ(Vi)

2 Determine the index l from Ψl := mini

(Ψi)

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Torczon method

2- Reflect

1 Compute the reflected verticesV r

i := 2Vl −Vi

2 Evaluate Ψri := Ψ(V r

i )

3 If mini{Ψr

i }< Ψl go to step 3

4 Otherwise, go to the step 4

Example in R2

FIGURE : Reflection

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Torczon method

3- Expand

1 Compute the expanded verticesV e

i = 3Vl −2Vi

2 Evaluate Ψei := Ψ(V e

i )

3 If mini{Ψe

i }< mini{Ψr

i }1 Replace all vertices Vi by the

expanded vertices V ei

4 Otherwise1 Replace all vertices Vi by the

reflected vertices V ri

5 Terminate the iteration.

Example in R2

FIGURE : Expansion

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Torczon method

4- Contract

1 Compute the contracted verticesV c

i = 12 (Vi + Vl)

2 Replace all vertices Vi by thecontracted vertices V c

i

Example in R2

FIGURE : Contraction.

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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

2 DC equals 0 in the worst case FIGURE : The Dice Coefficient

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Results - DC

TABLE : The Mean Kappa Index Values

MethodsKappa Index

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-Nelder-Mead 0.952 0.975 0.939 0.955HMRF-Torczon 0.975 0.985 0.956 0.973

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Results - Time

TABLE : The Mean Segmentation Time

Methods Time (s)Classical-MRF 3318MRF-ACO 418MRF-ACO-Gossiping 238HMRF-Nelder-Mead 12.24HMRF-Torczon 5.55

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Results - Visual

Original image Ground truth HMRF-Nelder-Mead HMRF-Torczon

FIGURE : Segmentation result of HMRF-Nelder-Mead and HMRF-Torczonmethods on a slice of BrainWeb database

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Conclusion

1 We have described two methods :HMRF-Nelder-Mead and HMRF-Torczon

2 Performance evaluation relies on Brainweb database

3 From the tests we have conducted that the results are verypromising on : time and quality of the segmentation

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 fields and direct search methods for medical image segmentation

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