This thesis applies Bayesian analysis and Markov random field (MRF) models with MCMC-Gibbs sampling to segment brain MR images into three classes: gray matter, white matter, and cerebrospinal fluid. Synthetic data is used to evaluate the MRF method under different noise conditions. Real brain MR images are also segmented. The MRF model characterizes the spatial dependencies between image pixels. MCMC sampling is used to estimate the posterior distribution and perform segmentation. The results demonstrate the MRF approach can accurately segment images, with better performance on lower noise images.

1. Bayesian Image Analysis applied to Medical Magnetic Resonance Brain
Images
Master of Science Thesis
BY: Selamawit Serka
School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia
Advisor: Prof. Henning Omre
Department of Mathematical Sciences, NTNU
Co-Advisor: Endale Berhane, MSc.
School of Mathematical and Statistical Sciences, Hawassa University
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2. Abstract
In this work I present a method for the image analysis of Magnetic Resonance (MR) Imaging of brain.
The goal of this work is to summarize the statistical theory of Markov random field (MRF) and MCMC
simulation, to evaluate MRF methods for image segmentation on synthetic data and to apply MRF
models for brain image segmentation. I propose Bayesian analysis and MRF model with MCMC-Gibbs
sampler to segment the brain class in to three parts which is Gray matter, White matter and black. The
main target of this approach is to improve the performance of segmentation by emphasizing the level of
noise. The comparisons between the three noisy images show that the image with white noise can give
more accurate segmentation results. Furthermore in all three cases the image with low noise level can
give more accurate segmentation results.
1 Markov Random Field
The image analysis will be made in a Bayesian setting. Bayesian setting plays a great and important role in
image analysis.
1.1 The posterior model
Prob {L = |o} =
f(o| )Prob {L = }
f(o)
(1)
where, Prob {L = |o} is posterior model, f(o| ) is likelihood model, Prob {L = } is prior model and f(o)
is normalizing constant.
Likelihood model
f(o| ) =
x∈LB
1
√
2πσo|
exp
−1
2
(ox − µ x )2
σ2
o|
Where µ x are the response level for each brain class x ∈ Ω . The model parameters are µ x x = 1, 2, 3 and
σ2
o|
Prior model
Gibbs form:
Prob {L = } = const × exp {−h( ; θ)} (2)
with
h( ; θ) =
c∈C
νc( ; θ) (3)
where νc( ) is a function of u with u ∈ c only, and θ is a set of model parameters.
Markov form:
Prob {Lx = x|L−x = −x} = Prob {Lx = x|Lr = r; r ∈ ∂(x)} ; ∀x ∈ LB (4)
Definition 1. Neighborhood System
Define a subset of grid nodes ∂(x) ⊂ LB around each x ∈ LB. The ∂(x) is termed the neighborhood of x,
and the neighborhood system is: ∂ : {∂(x); x ∈ LB}.
Requirements for a neighborhood are:
2

3. 1. x /∈ ∂(x)- not a neighborhood of oneself
2. x ∈ ∂(t) ⇔ t ∈ ∂(x)- symmetric neighbors.
Definition 2. Clique System
Define a subset of grid nodes c ⊂ LB such that
u ∈ ∂(υ)
υ ∈ ∂(u).
(5)
for all u, υ ∈ c and u = υ. This subset c is termed as clique associated with neighborhood system ∂, and the
largest subset c is termed largest clique. The set of all largest cliques over LB is denoted clique system and
denoted C.
Theorem 1. Hammersley-Clifford
Hammersely-Clifford theorem states that there is a one-to-one correspondence between Markov RF with
respect to neighborhood system ∂ and Gibbs RF with respect to clique system C for correspondence ∂ and C.
2 Results
2.1 Synthetic test
The synthetic test is inspired by the true data and it is based on the synthetic image on a (40 × 40). Three
types of observations are simulated. The three types of observations are white noise, blurring and correlated
noise.
3

4. (a) True synthetic image (b) Observed image (c) Histogram of observed data
(d) Realiz 1 Posterior model (e) Realiz 52 Posterior model (f) Realiz 103 Posterior model
(g) Prediction naive treshold (h) Prediction MMAP post
mod
(i) Prob map post mod classW (j) Prob map post mod classG (k) Prob map post mod classB
Figure 1: Images of observed white noise with variance σ2
w = 15, σ2
g = 9 and σ2
b = 6 and β = 8
4

5. (a) True synthetic image (b) Observed image (c) Histogram of observed data
(d) Realiz 1 Posterior model (e) Realiz 52 Posterior model (f) Realiz 103 Posterior model
(g) Prediction naive treshold (h) Prediction MMAP post
mod
(i) Prob map post mod classW (j) Prob map post mod classG (k) Prob map post mod classB
Figure 2: Images of observed blurred noise with variance σ2
= 15 and β = 8
5

6. (a) True synthetic image (b) Observed image (c) Histogram of observed data
(d) Realiz 1 Posterior model (e) Realiz 52 Posterior model (f) Realiz 103 Posterior model
(g) Prediction naive treshold (h) Prediction MMAP post
mod
(i) Prob map post mod classW (j) Prob map post mod classG (k) Prob map post mod classB
Figure 3: Images of observed correlated noise with variance σ2
= 15 and β = 8
6

7. (a) Observed image (b) Histogram of observed data
(c) Realiz 1 Posterior model (d) Realiz 45 Posterior model (e) Realiz 90 Posterior model
(f) Prediction naive treshold (g) Prediction MMAP post
mod
(h) Prob map post mod classW (i) Prob map post mod classG (j) Prob map post mod classB
Figure 4: Images of cut data from the original with estimated variance σ2
w = 25, σ2
g = 20 and σ2
b = 15 and
β = 8
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8. 3 Acknowledgment
I am deeply grateful to my advisor, Professor Henning Omre of the University of NTNU, Professor Arnoldo
Frigessi University of Oslo,Endale Berhane, Doctor Ayele Taye Of Hawassa University and Guro Dørum for
their valuable comments and moral.
References
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