Image segmentation is the process of partitioning the im- age into regions of interest in order to provide a meaningful represen- tation of information. Nowadays, segmentation has become a necessity in many practical medical imaging methods as locating tumors and dis- eases. Hidden Markov Random Field model is one of several techniques used in image segmentation. It provides an elegant way to model the segmentation process. This modeling leads to the minimization of an ob- jective function. Conjugate Gradient algorithm (CG) is one of the best known optimization techniques. This paper proposes the use of the non- linear Conjugate Gradient algorithm (CG) for image segmentation, in combination with the Hidden Markov Random Field modelization. Since derivatives are not available for this expression, finite differences are used in the CG algorithm to approximate the first derivative. The approach is evaluated using a number of publicly available images, where ground truth is known. The Dice Coefficient is used as an objective criterion to measure the quality of segmentation. The results show that the proposed CG approach compares favorably with other variants of Hidden Markov Random Field segmentation algorithms.

1. IFIP International Conference on Computational Intelligence and Its Applications (IFIP CIIA 2018)
Conjugate Gradient method for Brain
Magnetic Resonance Images Segmentation
EL-Hachemi Guerrout Samy Ait-Aoudia Dominique Michelucci Ramdane Mahiou
1

2. 1. Introduction
2. Segmentation
3. Hidden Markov Random Field
4. HMRF-CG
5. Experimental Results
6. Conclusion & Perspective
2

3. Introduction

4. Problematic & Solution
We face the huge amount of data produced
by imaging devices
Manual analysis and interpretation is
a tedious task
Automatic segmentation is the solution
3

5. Segmentation

6. What is the segmentation ?
Image segmentation is:
The process of partitioning the image into regions of interest in order to
provide a meaningful representation of information
4

7. A segmentation methods
• Thresholding based methods
• Clustering based methods
• Edge detection based methods
• Region-growing based methods
• Watersheds based methods
• Model based methods
• Hidden Markov Random Field based methods
We have chosen HMRF as a model to perform segmentation
5

8. Hidden Markov Random Field

9. Why Hidden Markov Random Field ?
• Provides an elegant way to model the segmentation problem
• Provides an algorithm robust to noise
• Provides a high quality of segmentation
The good analysis and interpretation means
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10. Hidden Markov Random Field
The image to segment y = {ys}s∈S
into K classes is a realization of Y
• Y = {Ys}s∈S is a family of random
variables
• ys ∈ [0 . . . 255]
The segmented image into K classes
x = {xs}s∈S is realization of X
• X = {Xs}s∈S is a family of random
variables
• xs ∈ {1, . . . , K}
An example of segmentation into
K = 4 classes
x∗
= argx∈Ω max {P[X = x | Y = y]}
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11. Hidden Markov Random Field
• This elegant model leads to the optimization of an energy function
Ψ(x, y) = s∈S ln(σxs
) +
(ys −µxs )2
2σ2
xs
+ β
T c2={s,t} (1 − 2δ(xs, xt))
• 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
• The main idea is to focus on the means adjustment instead of
treating pixels adjustment
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12. Hidden Markov Random Field
• Now, we seek for u∗



µ∗
= argµ∈[0...255]K min {Ψ(µ)}
Ψ(µ) =
K
j=1 f (µj )
f (µj ) =
s∈Sj
[ln(σj ) +
(ys −µj )2
2σ2
j
] + β
T
c2={s,t}
(1 − 2δ(xs, xt))
• To apply optimization techniques, we redefine the function Ψ(µ) for
µ ∈ RK
instead µ ∈ [0 . . . 255]K
.
9

13. Hidden Markov Random Field
Ψ(µ) =
K
j=1
F(µj ) where µj ∈ R
F(µj ) =



f (0) − uj ∗ 103
if µj < 0
f (µj ) if µj ∈ [0 . . . 255]
f (255) + (uj − 255) ∗ 103
if µj > 255
10

14. HMRF-CG

15. Conjugate Gradient algorithm
• Let µ0
be the initial point and d0
= −Ψ (µ0
) be the first direction
search.
• Calculate the step size αk
that minimizes ϕk (α). It is found by
ensuring that the gradient is orthogonal to the search direction dk
.
ϕk (α) = Ψ(µk
+ αdk
)
• At the iteration k + 1, calculate µk+1
as follows:
µk+1
= µk
+ αk
dk
• Calculate the residual or the steepest direction:
rk+1
= −Ψ (µk+1
)
• Calculate the search direction dk+1
as follows:
dk+1
= rk+1
+ βk+1
dk
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16. Conjugate Gradient algorithm
In conjugate gradient method there are many variants to compute βk+1
,
for example:
• The Fletcher-Reeves conjugate gradient method:
βk+1
=
rk+1 T
rk+1
(rk )
T
rk
• The Polak-Ribi`ere conjugate gradient method:
βk+1
= max
rk+1 T
rk+1
− rk
(rk )
T
rk
, 0
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17. HMRF-CG
• To use conjugate gradient algorithm, we need the first derivative
Ψ (µ)) = (∆1, . . . , ∆i , . . . , ∆K ).
• In our tests, we have used a centered difference approximation to
compute the first derivative as follows:
∆i =
Ψ(µ1, . . . , µi + ε, . . . , µn) − Ψ(µ1, . . . , µi − ε, . . . , µn)
2ε
• The good approximation of the first derivative relies on the choice of
the value of the parameter ε. Through the tests conducted, we have
selected 0.01 as the best value.
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18. Experimental Results

19. 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
(perfect segmentation)
2. DC equals 0 in the worst case
(every pixel is misclassified)
Figure 1: The Dice Coefficient
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20. Well Known databases
IBSR
The Internet Brain Segmentation Repository
(IBSR) provides manually-guided expert segmentation
results along with magnetic resonance brain image data
BrainWeb
images are simulated MRI volumes for normal brain
In this database, an image can be selected by setting:
noise
modality
slice thickness
intensity non-uniformity
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21. Well Known methods
MRF-ACO-Gossiping
K-means
LGMM
HMRF-EM
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22. Results -DC - IBSR
Methods
Dice Coefficient
GM WM CSF Mean
K-means 0.500 0.607 0.06 0.390
MRF-ACO-Gossiping 0.778 0.827 0.262 0.623
HMRF-CG 0.859 0.855 0.381 0.698
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23. Results - DC - BrainWeb
Tissue Method
Dice Coefficient
(0%,0%) (3%,20%) (5%,20%)
GM
HMRF-CG 0.970 0.945 0.921
LGMM 0.697 0.905 0.912
FSL FAST 0.727 0.737 0.735
WM
HMRF-CG 0.990 0.971 0.954
LGMM 0.667 0.940 0.951
FSL FAST 0.877 0.862 0.860
CSF
HMRF-CG 0.961 0.942 0.926
LGMM 0.751 0.897 0.893
FSL FAST 0.635 0.647 0.643
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24. Example of segmentation using HMRF-CG - IBSR
IBSR 1-24/18
IBSR 1-24/34
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25. Example of segmentation using HMRF-CG - BrainWeb
(0%,0%)
(3%,20%)
(5%,20%)
20

26. Conclusion & Perspective

27. Conclusion & Perspective
• HMRF-CG method shows a good results and it is very promising
• Nevertheless, the opinion of specialists must be considered in the
evaluation
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28. Thank you for your attention
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29. Questions?
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