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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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
1 Introduction
2 HMRF
3 Direct Search MethodsNelder-Mead methodTorczon method
4 Experimental Results
5 Conclusion
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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
Hidden Markov Random Field
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
Hidden Markov Random Field
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
Torczon method
1- Evaluate
1 Compute Ψi := Ψ(Vi)
2 Determine the index l from Ψl := mini
(Ψi)
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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
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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Introduction HMRF Direct Search Methods Experimental Results Conclusion
Thank youfor your attention
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