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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
1. Introduction
2. Segmentation
3. Hidden Markov Random Field
4. HMRF-CG
5. Experimental Results
6. Conclusion & Perspective
2
Introduction
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
Segmentation
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
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
Hidden Markov Random Field
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
6
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]}
7
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
8
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
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
HMRF-CG
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
11
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
12
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.
13
Experimental Results
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
14
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
15
Well Known methods
MRF-ACO-Gossiping
K-means
LGMM
HMRF-EM
16
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
17
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
18
Example of segmentation using HMRF-CG - IBSR
IBSR 1-24/18
IBSR 1-24/34
19
Example of segmentation using HMRF-CG - BrainWeb
(0%,0%)
(3%,20%)
(5%,20%)
20
Conclusion & Perspective
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
21
Thank you for your attention
21
Questions?
21

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Conjugate Gradient method for Brain Magnetic Resonance Images Segmentation

  • 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
  • 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
  • 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
  • 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 6
  • 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]} 7
  • 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 8
  • 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
  • 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 11
  • 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 12
  • 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. 13
  • 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 14
  • 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 15
  • 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 17
  • 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 18
  • 24. Example of segmentation using HMRF-CG - IBSR IBSR 1-24/18 IBSR 1-24/34 19
  • 25. Example of segmentation using HMRF-CG - BrainWeb (0%,0%) (3%,20%) (5%,20%) 20
  • 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 21
  • 28. Thank you for your attention 21