Hidden Markov Random Fields and Direct Search Methods for Medical Image Segmentation
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The document discusses advanced image segmentation techniques using Hidden Markov Random Fields (HMRF) and optimization methods like Nelder-Mead and Torczon to enhance medical image analysis. It presents experimental results indicating that HMRF methods significantly improve segmentation accuracy and processing time compared to traditional methods. The findings suggest promising performance in medical image segmentation tasks, warranting further consideration from specialists in the field.
![Introduction HMRF Direct Search Methods Experimental Results Conclusion
Hidden Markov Random Field
The image to segment into K classes
y = {ys}s∈S is seen as a realization of
a random field Y = {Ys}s∈S
1 Each ys is a realization of a
random variable Ys
2 Each ys ∈ [0...255]
The segmented image x = {xs}s∈S is
seen as a realization of a random field
X = {Xs}s∈S
1 Each xs is a realization of a
random variable Xs
2 Each xs ∈ {1,...,K}
An example of segmentation with K
= 4
6 / 26](https://image.slidesharecdn.com/icpram2016pre-160401214328/85/Hidden-Markov-Random-Fields-and-Direct-Search-Methods-for-Medical-Image-Segmentation-6-320.jpg)
![Introduction HMRF Direct Search Methods Experimental Results Conclusion
Hidden Markov Random Field
Ψ(µ) = ∑K
j=1 ∑
s∈Sj
[ln(σj )+
(ys−µj )2
2σ2
j
]+ 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}
8 / 26](https://image.slidesharecdn.com/icpram2016pre-160401214328/85/Hidden-Markov-Random-Fields-and-Direct-Search-Methods-for-Medical-Image-Segmentation-8-320.jpg)
Hidden Markov Random Fields and Direct Search Methods for Medical Image Segmentation
- 1. Introduction HMRF Direct Search Methods Experimental Results Conclusion 5th The International Conference on Pattern Recognition Applications and Methods EL-Hachemi Samy Dominique Ramdane Guerrout Ait-Aoudia Michelucci Mahiou Hidden Markov Random Fields and Direct Search Methods for Medical Image Segmentation LMCS Laboratory, ESI, Algeria & LE2I Laboratory, UB, France 1 / 26
- 2. Introduction HMRF Direct Search Methods Experimental Results Conclusion 1 Introduction 2 HMRF 3 Direct Search Methods Nelder-Mead method Torczon method 4 Experimental Results 5 Conclusion 2 / 26
- 3. Introduction HMRF Direct Search Methods Experimental Results Conclusion Problematic 1 Huge number of medical images 1 MRI - Magnetic Resonance Imaging 2 CT - Computed Tomography 3 Radiography 4 Digital mammography 5 . . .etc 2 The manual analysis and interpretation is a difficult task 3 / 26
- 4. Introduction HMRF Direct Search Methods Experimental Results Conclusion Solution The automatic extraction of meaningful information is one among the segmentation challenges The goal of image segmentation ? Simplify the representation of an image to items meaningful and easier to analyze 4 / 26
- 5. 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 5 / 26
- 6. Introduction HMRF Direct Search Methods Experimental Results Conclusion Hidden Markov Random Field The image to segment into K classes y = {ys}s∈S is seen as a realization of a random field Y = {Ys}s∈S 1 Each ys is a realization of a random variable Ys 2 Each ys ∈ [0...255] The segmented image x = {xs}s∈S is seen as a realization of a random field X = {Xs}s∈S 1 Each xs is a realization of a random variable Xs 2 Each xs ∈ {1,...,K} An example of segmentation with K = 4 6 / 26
- 7. Introduction HMRF Direct Search Methods Experimental Results Conclusion Hidden Markov Random Field 1 This elegant model leads to the optimization of an energy function Ψ(x,y) 2 Our way to look for the minimization of Ψ(x,y) is to look for the minimization Ψ(µ), µ = (µ1,...,µK ) where µi are means of gray values of class i 3 The main idea is instead to work directly on the pixels adjustment of x, to work on the means adjustment first. 7 / 26
- 8. Introduction HMRF Direct Search Methods Experimental Results Conclusion Hidden Markov Random Field Ψ(µ) = ∑K j=1 ∑ s∈Sj [ln(σj )+ (ys−µj )2 2σ2 j ]+ 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} 8 / 26
- 9. 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 gravity center of K best vertices by the operations : 1 Reflection 2 Expansion 3 Contraction 5 The vertex with the worse function value is replaced with the new vertex if its value is better. Otherwise the simplex is shrunk 9 / 26
- 10. 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 := max i=h (Ψi ) 3 Ψl := min i (Ψi ) 3 Compute ¯V := 1 K ∑ i=h Vi Example in R2 FIGURE : Gravity center 10 / 26
- 11. Introduction HMRF Direct Search Methods Experimental Results Conclusion Nelder-Mead method 2- Reflect 1 Compute the reflection vertex Vr from : Vr := 2¯V −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 11 / 26
- 12. 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 := 3¯V −2Vh 2 Evaluate Ψe := Ψ(Ve) 3 If Ψe < Ψr 1 Replace Vh by Ve 2 Terminate the iteration 4 Otherwise (if Ψe ≥ Ψr ) 1 Replace Vh by Vr 2 Terminate the iteration Example in R2 FIGURE : Expansion 12 / 26
- 13. 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 := 3 2 ¯V − 1 2 Vh 2 Evaluate Ψc := Ψ(Vc) 3 If Ψc < Ψr 1 Replace Vh by Vc 2 Terminate the iteration 4 Otherwise ( If Ψc ≥ Ψr ) 1 Go to the step 5 (shrink) Example in R2 FIGURE : Outside contraction 13 / 26
- 14. 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 := 1 2 (Vh + ¯V) 2 Evaluate Ψc := Ψ(Vc) 3 If Ψc < Ψh 1 Replace Vh by Vc 2 Terminate the iteration 4 Otherwise ( If Ψc ≥ Ψh ) 1 Go to the step 5 (shrink) Example in R2 FIGURE : Inside contraction 14 / 26
- 15. Introduction HMRF Direct Search Methods Experimental Results Conclusion Nelder-Mead method 5- Shrink Replace all vertices according to the following formula : Vi := 1 2 (Vi +Vl ) Example in R2 FIGURE : Shrink 15 / 26
- 16. Introduction HMRF Direct Search Methods Experimental Results Conclusion Torczon method Torczon method 1 Torczon method is an improvement of Nelder-Mead method 2 The differences are : 1 All the vertices are concerned by the operations (reflect, expand and contract) 2 All simplexes in Torczon method are homothetic to the first one 3 No degeneracy (i.e., flat simplex) can occur in Torczon method 16 / 26
- 17. Introduction HMRF Direct Search Methods Experimental Results Conclusion Torczon method 1- Evaluate 1 Compute Ψi := Ψ(Vi ) 2 Determine the index l from Ψl := min i (Ψi ) 17 / 26
- 18. Introduction HMRF Direct Search Methods Experimental Results Conclusion Torczon method 2- Reflect 1 Compute the reflected vertices Vr i := 2Vl −Vi 2 Evaluate Ψr i := Ψ(Vr i ) 3 If min i {Ψr i } < Ψl go to step 3 4 Otherwise, go to the step 4 Example in R2 FIGURE : Reflection 18 / 26
- 19. Introduction HMRF Direct Search Methods Experimental Results Conclusion Torczon method 3- Expand 1 Compute the expanded vertices Ve i = 3Vl −2Vi 2 Evaluate Ψe i := Ψ(Ve i ) 3 If min i {Ψe i } < min i {Ψr i } 1 Replace all vertices Vi by the expanded vertices Ve i 4 Otherwise 1 Replace all vertices Vi by the reflected vertices Vr i 5 Terminate the iteration. Example in R2 FIGURE : Expansion 19 / 26
- 20. Introduction HMRF Direct Search Methods Experimental Results Conclusion Torczon method 4- Contract 1 Compute the contracted vertices Vc i = 1 2 (Vi +Vl ) 2 Replace all vertices Vi by the contracted vertices Vc i Example in R2 FIGURE : Contraction. 20 / 26
- 21. Introduction HMRF Direct Search Methods Experimental Results Conclusion DC - The Dice Coefficient The Dice coefficient measures how much the segmentation result is close to the ground 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 21 / 26
- 22. Introduction HMRF Direct Search Methods Experimental Results Conclusion Results - DC TABLE : The Mean Kappa Index Values Methods Kappa Index GM WM CSF Mean Classical-MRF 0.763 0.723 0.780 0.756 MRF-ACO 0.770 0.729 0.785 0.762 MRF-ACO-Gossiping 0.770 0.729 0.786 0.762 HMRF-Nelder-Mead 0.952 0.975 0.939 0.955 HMRF-Torczon 0.975 0.985 0.956 0.973 22 / 26
- 23. Introduction HMRF Direct Search Methods Experimental Results Conclusion Results - Time TABLE : The Mean Segmentation Time Methods Time (s) Classical-MRF 3318 MRF-ACO 418 MRF-ACO-Gossiping 238 HMRF-Nelder-Mead 12.24 HMRF-Torczon 5.55 23 / 26
- 24. 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-Torczon methods on a slice of BrainWeb database 24 / 26
- 25. 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 very promising on : time and quality of the segmentation 4 Nevertheless, the opinion of specialists must be considered in the evaluation 25 / 26
- 26. Introduction HMRF Direct Search Methods Experimental Results Conclusion Thank you for your attention 26 / 26