Hidden Markov Random Fields and Direct Search Methods for Medical

Image Segmentation

El-Hachemi Guerrout

, Samy Ait-Aoudia

, Dominique Michelucci

and Ramdane Mahiou

Laboratoire LMCS, Ecole Nationale Sup

erieure en Informatique, Oued-Smar, Algiers, Algeria

Laboratoire LE2I, Universit

e de Bourgogne, Dijon, France

Keywords:

Medical Image Segmentation, Hidden Markov Random Field, Nelder-Mead, Torczon, Kappa Index.

Abstract:

The goal of image segmentation is to simplify the representation of an image to items meaningful and easier to

analyze. Medical image segmentation is one of the fundamental problems in image processing ﬁeld. It aims to

provide a crucial decision support to physicians. There is no one way to perform the segmentation. There are

several methods based on HMRF. Hidden Markov Random Fields (HMRF) constitute an elegant way to model

the problem of segmentation. This modelling leads to the minimization of an energy function. In this paper

we investigate direct search methods that are Nelder-Mead and Torczon methods to solve this optimization

problem. The quality of segmentation is evaluated on grounds truths images using the Kappa index called also

Dice Coefﬁcient (DC). The results show the supremacy of the methods used compared to others methods.

1 INTRODUCTION

MRI - Magnetic Resonance Imaging, CT - Com-

puted tomography, Radiography, digital mammogra-

phy, and other imaging modalities have indispensable

role in disease diagnosis. However, they produce a

huge number of images for which the manual analy-

sis and interpretation has become a difﬁcult task. The

automatic extraction of meaningful information is one

among the segmentation challenges. In the literature

several approaches have been proposed such as ac-

tive contour models (Kass et al., 1988), edge detec-

tion (Canny, 1986), thresholding (Sahoo et al., 1988),

region growing (Adams and Bischof, 1994), MRF -

Markov Random Fields (Manjunath and Chellappa,

1991), etc.

HMRF - Hidden Markov Random Field, a gener-

alization of Hidden Markov Model (Baum and Petrie,

1966) provides an elegant way to model the seg-

mentation problem. Since the seminal paper of Ge-

man and Geman (1984), Markov Random Fields

(MRF) models for image segmentation have been

investigated intensively (Manjunath and Chellappa,

1991; Panjwani and Healey, 1995; Held et al., 1997;

Hochbaum, 2001; Zhang et al., 2001; Deng and

Clausi, 2004; Kato and Pong, 2006; Youseﬁ et al.,

2012). The main idea underlying the segmentation

process using HMRF is: the image to segment (called

also the observed image) and the segmented image

(called also the hidden image) are seen like Markov

Random Field. The segmented image is computed

according to the MAP (Maximum A Posteriori) crite-

rion (Wyatt and Noble, 2003). MAP estimation leads

to the minimization of energy function (Szeliski et al.,

2008). This problem is computationally intractable.

In this paper we examine HMRF to model the

segmentation problem and the direct search meth-

ods: Nelder-Mead (Nelder and Mead, 1965) and Tor-

czon (Kolda et al., 2003), as the optimization tech-

niques. In order to evaluate the segmentation quality

we use the Kappa index criterion (Dice, 1945) also

called Dice Coefﬁcient (DC). The Kappa Index cri-

terion gives us how much the segmentation result is

close to the ground truth. The images have been ob-

tained from the Brainweb

database (Cocosco et al.,

1997), widely used by the neuroimaging community,

where the ground truth is known.

The achieved results are very satisfactory and

show a superiority of the our proposed methods:

HMRF-Nelder-Mead and HMRF-Torczon, compared

to other methods (Ouadfel and Batouche, 2003;

Youseﬁ et al., 2012) that are Classical MRF, MRF-

ACO (Ant Colony Optimization) and MRF-ACO-

Gossiping.

This paper is organized as follows. In section 2,

we provide some concepts of Markov Random Field

http://www.bic.mni.mcgill.ca/brainweb/

154

Guerrout, E-H., Ait-Aoudia, S., Michelucci, D. and Mahiou, R.

Hidden Markov Random Fields and Direct Search Methods for Medical Image Segmentation.

DOI: 10.5220/0005658501540161

In Proceedings of the 5th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2016), pages 154-161

ISBN: 978-989-758-173-1

model. Section 3 is devoted to Hidden Markov Field

model and direct search methods combination to per-

form the segmentation. We give in section 4 some

experimental results. Finally, the section 5 is devoted

to conclusions.

2 HIDDEN MARKOV RANDOM

FIELD MODEL (HMRF)

2.1 Markov Random Field

Let X = {X

,...,X

} be a family of random vari-

ables on the lattice S. Each random variable takes val-

ues in the discrete space Λ = {1,2,...,K}. The fam-

ily X is a random ﬁeld with conﬁguration set Ω = Λ

A random ﬁeld X is said to be an MRF (Markov

Random Field) on S, with respect to a neighborhood

system V (S), if and only if:







∀x ∈ Ω, P[X = x] > 0

∀s ∈ S, ∀x ∈ Ω,P[X

= x

,t 6= s] =

P[X

= x

,t ∈ V

(S)]

(1)

The Hammersley-Clifford theorem (Hammersley &

Clifford, 1971) establishes the equivalence between

Gibbs ﬁelds and Markov ones. The Gibbs distribu-

tion is characterized by the following relations:











P[X = x] = Z

−1

−U(x)

Z =

∑

ξ∈Ω

−U(ξ)

U(x) =

∑

c∈C

(x)

(2)

where T is a global control parameter, called tem-

perature, and Z is a normalizing constant, called

the partition function. Calculating Z is prohibitive.

Card(Ω) = 256

512×512

= 2

2097152

for a 512x512 gray

level image. U(x) is the energy function of the Gibbs

ﬁeld deﬁned as a sum of potentials over all the possi-

ble cliques C.

The local interactions between the neighboring

sites properties (gray levels for example) can be ex-

pressed as a clique potential.

2.2 Standard Markov Random Field

2.2.1 Ising Model

This model was proposed by Ernst Ising for ferro-

magnetism studies in statistical physics. The Ising

model involves discrete variables s (spins), placed

on a sampling grid. Each spin can take two values,

Λ = {−1, 1}. The spins interact in pairs. The ﬁrst or-

der clique potential is deﬁned by −Bx

and the second

order clique potential is deﬁned by:

{

s,t

}

) =



−β if x

= x

+β if x

6=x

(3)

{

s,t

}

) = −βx

(4)

The total energy is deﬁned by:

U(x) = −

∑

={s,t}

βx

−

∑

={s}

(5)

The coupling constant β, between neighboring sites,

regularize the model and B represents an extern mag-

netic ﬁeld.

2.2.2 Potts Model

The Potts model is a generalization of the Ising

model. Instead of Λ = {−1,1}, each spin is assigned

an integer value Λ = {1,2,...,K}. In the context of

image segmentation, the integer values are gray levels

or labels. The total energy is deﬁned by:

U(x) = β

∑

={s,t}

(1 − 2δ(x

)) (6)

where δ is the Kronecker’s delta:

δ(a,b) =



1 if a = b

0 if a6=b

(7)

When β > 0, the probable conﬁgurations correspond

to neighboring sites with same gray level or label.

This induces the constitution of large homogeneous

regions. The size of these regions is guided by the

value of β.

2.3 Hidden Markov Random Field

HMRF is a strong model for image segmentation. The

image to segment is seen as a realization y = {y

}

s∈S

of a Markov Random Field Y = {Y

}

s∈S

deﬁned on

the lattice S. Each realization y

of the random vari-

able Y

takes its values in the space of gray levels

obs

= {0 ...255}. The conﬁguration set of Y is noted

as Ω

obs

. The segmented image is seen as the real-

ization x = {x

}

s∈S

of another Markov Random Field

X = {X

}

s∈S

, deﬁned on the same lattice S. The real-

ization x

of the random variable X

takes its values in

the discrete space Λ = {1,2,...,K}. K is the number

of classes or homogeneous regions in the image. The

conﬁguration set of X is noted as Ω.

Figure 1 shows an example of the image to seg-

ment as a realization y of Y and its segmented image

with K = 4 seen as a realization x of X.

In the context of image segmentation, we face a

problem with incomplete data (Dempster et al., 1977).

To every site s ∈ S is associated different information:

Hidden Markov Random Fields and Direct Search Methods for Medical Image Segmentation

155

y: Observed Image x: Hidden Image

Figure 1: Observed and hidden image.

observed information expressed by the random vari-

able Y

; missed or hidden information, expressed by

the random variable X

. The Random Field X is called

Hidden Markov Random Field.

The segmentation process consists in ﬁnding a re-

alization x of X by observing the data of the realiza-

tion y, representing the image to segment.

We seek a labeling ˆx, which is an estimate of the

true labeling x

∗

, according to the MAP (Maximum

A Posteriori) criterion (maximizing the probability

P(X = x|Y = y)).

∗

= arg

x∈Ω

max

{

P[X = x|Y = y]

}

(8)

P[X = x|Y = y] =

P[Y = y|X = x]P [X = x]

P[Y = y]

(9)

The ﬁrst term of the numerator describes the probabil-

ity to observe the image y, knowing the x label. Based

on the assumption of conditional independence, we

get the following formula:

P[Y = y|X = x] =

∏

s∈S

P[Y

= y

= x

] (10)

The second term of the numerator describes the exis-

tence of the labeling x. The denominator is constant

and independent of x. We have then:

P[X = x|Y = y] = AP [Y = y|X = x]P [X = x] (11)

where A is a constant, from the equation 2 and 11 we

will have:

P[X = x|Y = y] = Ae

ln(P[Y =y|X=x])−

U(x)

(12)

P[X = x|Y = y] = Ae

−Ψ(x,y)

(13)

∗

= arg

x∈Ω

min

{

Ψ(x,y)

}

(14)

Maximizing the probability P(X = x |Y = y) is equiv-

alent to minimizing the function Ψ(x,y).

Ψ(x,y) = − ln(P [Y = y|X = x]) +

U(x)

(15)

We replace the equation 10 in the equation 15 and we

obtain:

Ψ(x,y) = − ln

∏

s∈S

P[Y

= y

= x

]

U(x)

(16)

Ψ(x,y) = −

∑

s∈S

ln(P[Y

= y

= x

]) +

U(x)

(17)

We assume that the variable [Y

= y

= x

] follows a

Gaussian distribution with parameters µ

(mean) and

(standard deviation). By giving the class label x

k, we have:

P[Y

= y

= k] =

2πσ

−(y

−µ

)

2σ

(18)

The Potts model is often used in image segmentation

to privilege large regions in the image. The energy is

then:

U (x ) = β

∑

={s,t}

(1 − 2δ (x

)) (19)

According to equations 17, 18 and 19 we get:

Ψ(x,y) =

∑

s∈S

∑

={s,t}

s,t

(20)

= [ln(σ

) +

− µ

)

2σ

]

s,t

= (1 − 2δ(x

))

Optimization techniques to get the estimation ˆx of the

labeling x

∗

are presented in next sections.

3 HMRF AND DIRECT SEARCH

METHODS

Applying the optimization techniques is not obvious.

Thus this section ﬁrst gives the big picture of our ap-

proach.

Let y = (y

,...,y

) be the image to seg-

ment into K classes. Instead of looking for the seg-

mented image x = (x

,...,x

) we look for

the vertex µ which contains the means of K classes

µ = (µ

,...,µ

). The segmented image x =

,...,x

) is computed by classifying y

nearest mean µ

i.e., if the nearest mean of y

is µ

then x

= j, in the case of tie between two means then

is classiﬁed to the smallest mean.

Let S

be the set all the sites s ∈ S in which the

nearest mean to y

is µ

= {s ∈ S | x

= j ⇔ the nearest mean of y

is µ

}

ICPRAM 2016 - International Conference on Pattern Recognition Applications and Methods

156

Let |S

| be the cardinal of S

. The standard devia-

tion σ

is calculated as follow:

| S

∑

s∈S

− µ

)

The image to segment y is known, so if we know

the means µ

, then we can compute: its standard devi-

ation σ = (σ

,...,σ

), the segmented image x

and the value of the function Ψ(x,y).

To segment the image, we look for the point

µ = (µ

, j = 1 . . . K) where the function Ψ is minimal.

This time, Ψ is seen as a function of µ:

Ψ(µ) =

∑

j=1

∑

s∈S

j,s

∑

={s,t}

s,t

(21)

j,s

= [ln(σ

) +

− µ

)

2σ

]

s,t

= (1 − 2δ(x

))

3.1 Direct Search Methods

In this section we explain what we need to apply the

direct search methods for solving our problem deﬁned

above.

The popular Nelder-Mead method or the simplex-

based direct search method was proposed by John

Nelder and Roger Mead (1965). The method starts

with (n + 1) vertices in R

that are viewed as the ver-

tices of a simplex. The process of minimizing is based

on the comparison of the function values at (n + 1)

vertices of the simplex, followed by replacing the ver-

tex which has the highest value of the function by

another vertex that can be obtained by operations of

reﬂection, expansion or contraction relatively to the

center of gravity of the n best vertices of the simplex.

When the substitution conditions are not satisﬁed, the

method will make the simplex shrink.

Torczon method (1989) (Kolda et al., 2003) comes

to ﬁx gaps of Nelder-Mead like degenerated ﬂat sim-

plices. In the Torczon method, operations are applied

to all vertices of the current simplex, thus by construc-

tion, all simplexes in Torczon method are homothetic

to the ﬁrst one, and no degeneracy (i.e., ., ﬂat simplex)

can occur.

The stopping criterion in direct search methods is

satisﬁed when the simplex vertices or their function

values are close.

In our case, we are looking for the best µ ∈

[0...255]

which minimizes the function Ψ. For that

we start with a non degenerate simplex with K +1 ver-

tices (coordinates are given in Table 1). When some

vertex is out the bounds (i.e., /∈ [0 . . . 255]

), the func-

tion Ψ is set to +∞ (in the practice a very big value).

3.2 HMRF and Nelder-Mead

Combination

We summarize hereafter the Nelder-Mead method

and give examples in two-dimensional space.

HMRF-Nelder-Mead Algorithm.

Repeat.

1. Evaluate

Compute Ψ

:= Ψ(V

)

Determine the indices h, s, l:

:= max

(Ψ

), Ψ

:= max

i6=h

(Ψ

), Ψ

:= min

(Ψ

)

Compute

V :=

∑

i6=h

Example in R

(See ﬁgure 2)

Figure 2: Center of gravity calculation.

2. Reﬂect

Compute the reﬂection vertex V

from

:= 2

V −V

Evaluate Ψ

:= Ψ(V

). If Ψ

≤ Ψ

< Ψ

, replace

by V

and terminate the iteration.

Example in R

(See ﬁgure 3)

Figure 3: Reﬂection.

3. Expand

If Ψ

< Ψ

, compute the expansion vertex V

from

:= 3

V − 2V

Evaluate Ψ

:= Ψ(V

If Ψ

< Ψ

replace V

by V

and terminate the it-

eration; otherwise (if Ψ

≥ Ψ

), replace V

by V

and terminate the iteration.

Example in R

(See ﬁgure 4)

Hidden Markov Random Fields and Direct Search Methods for Medical Image Segmentation

157

Figure 4: Expansion.

4. Contract

If Ψ

≥ Ψ

, compute a contraction between

V and

the best of V

and V

(a) Outside contraction

If Ψ

< Ψ

, compute an outside contraction V

from

V −

Evaluate Ψ

:= Ψ(V

If Ψ

< Ψ

replace V

by V

and terminate the

iteration; otherwise ( If Ψ

≥ Ψ

), go to the

step 5 (shrink).

Example in R

(See ﬁgure 5)

Figure 5: Outside contraction.

(b) Inside contraction

If Ψ

≤ Ψ

, compute an inside contraction V

from

V )

Evaluate Ψ

:= Ψ(V

If Ψ

< Ψ

replace V

by V

and terminate the

iteration; otherwise ( If Ψ

≥ Ψ

), go to the

step 5 (shrink).

Example in R

(See ﬁgure 6)

Figure 6: Inside contraction.

5. Shrink

Replace all vertices according to the following

formula V

)

Example in R

(See ﬁgure 7)

Figure 7: Shrink.

Until Satisfying a Stopping Criterion.

3.3 HMRF and Torczon Combination

We summarize hereafter the Torczon method and

give examples in two-dimensional space.

HMRF-Torczon Algorithm.

Repeat.

1. Evaluate

Compute Ψ

:= Ψ(V

)

Determine the indices l from Ψ

:= min

(Ψ

)

2. Reﬂect

Compute the reﬂected vertices, V

:= 2V

−V

Evaluate Ψ

:= Ψ(V

)

If min

{Ψ

} < Ψ

go to step 3; otherwise, go to the

step 4

Example in R

(See ﬁgure 8)

Figure 8: Reﬂection.

3. Expand

Compute the expanded vertices, V

= 3V

− 2V

Evaluate Ψ

:= Ψ(V

)

If min

{Ψ

} < min

{Ψ

}, replace all vertices V

the expanded vertices V

; otherwise replace all

vertices V

by the reﬂected vertices V

. In either

case, terminate the iteration.

Example in R

(See ﬁgure 9)

ICPRAM 2016 - International Conference on Pattern Recognition Applications and Methods

158

Table 1: Parameters used in our tests.

Method Parameters

Classical MRF T: Temperature=4

MRF-ACO T: Temperature=4, a: Pheromone info. Inﬂuence=1, b: Heuristic info. Inﬂuence=1,

q: Evaporation rate=0.1, w: Pheromone decay coefﬁcient=0.1

MRF-ACO-

Gossiping

T: Temperature=4, a: Pheromone info. Inﬂuence=1, b: Heuristic info. Inﬂuence=1,

q: Evaporation rate=0.1, w: Pheromone decay coefﬁcient=0.1, c

: Pheromone rein-

forcing coefﬁcient=10, c

: Pheromone reinforcing coefﬁcient=100

HMRF-Nelder-Mead

and HMRF-Torczon

T=4, β = 1, vertices of the initial simplex are











= (196,127,127,127)

= (127, 63,127,127)

= (127,127,127,127)

= (127,127,188,127)

= (127,127,127,195)

Figure 9: Expansion.

4. Contract

Compute the contracted vertices, V

)

Replace all vertices V

by the contracted vertices

Example in R

(See ﬁgure 10)

Figure 10: Contraction.

Until Satisfying a Stopping Criterion.

4 EXPERIMENTAL RESULTS

To assess the two investigated combination meth-

ods referred to as HMRF-Nelder-Mead and HMRF-

Torczon, we made a comparative study with three

segmentation algorithms operating on brain images

that are Classical MRF, MRF-ACO (Ant Colony Op-

timization) and MRF-ACO-Gossiping (Youseﬁ et al.,

2012). To perform a meaningful comparison, we

use the same medical images with ground truth from

Brainweb database. The comparison is based on the

Kappa Index (or Dice coefﬁcient). We give in table

1 parameters setting for each tested algorithm. The

Dice Coefﬁcient DC (Dice, 1945) or Kappa Index KI

given hereafter allows the visualization of the perfor-

mance of an algorithm. The Dice coefﬁcient between

two classes A and B equals 1 when they are identical

and 0 in the worst case, i.e. no match between A and

KI =

2|A ∩ B|

|A ∪ B|

2T P

2T P + FP + FN

(22)

where T P stands for true positive, FP for false posi-

tive and FN for false negative (See ﬁgure 11).

Figure 11: TP, FP and FN.

The tables 2 and 3 show respectively the mean

segmentation time and the mean Kappa Index values

of the three classes: GM (Grey Matter), WM (White

Matter), CSF (Cerebro Spinal Fluid) . We have used

the Brainweb database with the parameters: Modal-

ity= T1, Slice thickness = 1mm, Noise = 0% and In-

tensity non-uniformity = 0%. The slices chosen are

used in (Youseﬁ et al., 2012) which are: 85, 88, 90,

95, 97, 100, 104, 106, 110, 121 and 130.

Figure 12 shows the segmentation result of

HMRF-Nelder-Mead and HMRF-Torczon methods

on a sample of BrainWeb database.

Hidden Markov Random Fields and Direct Search Methods for Medical Image Segmentation

159

The slice number Original image Ground truth HMRF-Nelder-Mead HMRF-Torczon

Figure 12: Segmentation result of HMRF-Nelder-Mead and HMRF-Torczon methods on a sample of BrainWeb database.

Table 2: 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

5 CONCLUSION

In this paper, we have described methods that com-

bine Hidden Markov Random Fields and Direct

Search methods that are Nelder-Mead and Torc-

zon optimisation algorithms to perform segmentation.

Performance evaluation was carried out on sample

Table 3: 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

medical images from the Brainweb database. From

the tests we have conducted, the combination methods

outperform classical methods. The results are very

promising. Nevertheless, the opinion of specialists

must be considered in the evaluation when no ground

truth is available to have a more synthetic view of the

whole segmentation process.

ICPRAM 2016 - International Conference on Pattern Recognition Applications and Methods

160

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