A Joint Bayesian Framework for MR Brain Scan Tissue and Structure Segmentation Based on Distributed Markovian Agents - Computational Intelligence in Healthcare: Advanced Methodologies - page 83

background class. As parameters θ , we consider the parameters describing the
intensity distributions for the K = 3 tissue classes. They are denoted by θ =
{ θ i ,i .Wewriteforall k =1 ...K , θ k = θ i ,i

∈ V, k =1 ...K } { ∈ V } and for all

i V , θ i = t ( θ i ,k =1 ...K )( t means transpose). Note that we describe here

the most general setting in which the intensity distributions can depend on voxel
i and vary with its location. Standard approaches usually consider that intensity
distributions are Gaussian distributions for which the parameters depend only
on the tissue class. Although the Bayesian approach makes the general case
possible, in practice we consider θ i 's equal for all voxels i in some prescribed
regions. More specifically, our local approach consists in dividing the volume V
into a partition of subvolumes and consider the θ i

∈

constant over each subvolume

(see Section 4.2).
To explicitly take into account the fact that tissue and structure classes are
related, a generative approach would be to define a complete probabilistic model,
namely p ( y , t , s ,θ ). To define such a joint probability is equivalent to define the
two probability distributions p ( y )and p ( t , s ,θ

y ). However, in this work, we
rather adopt a discriminative approach in which a conditional model p ( t , s ,θ

|

y )
is constructed from the observations and labels but the marginal p ( y )isnot
modeled explicitly. In a segmentation context, the full generative model is not
particularly relevant to the task of inferring the class labels. This appears clearly
in equations (2) and (3) where the relevant distributions are conditional. In
addition, it has been observed that conditional approaches tend to be more
robust than generative models [13], [14]. Therefore, we focus on p ( t , s ,θ

|

y )asthe
quantity of interest. It is fully specified when the two conditional distributions
p ( t , s

|

y , t , s ) are defined. The following subsections 4.1 and 4.2
specify respectively these two distributions.

| y ,θ )and p ( θ |

4.1 A Conditional Model for Tissues and Structures

The distribution p ( t , s | y ,θ ) can be in turn specified by defining p ( t | s , y ,θ )and
p ( s | t , y ,θ ). The advantage of the later conditional models is that they can cap-
ture in an explicit way the effect of tissue segmentation on structure segmentation
and vice versa. Note that on a computational point of view there is no need at
this stage to describe explicitly the joint model that can be quite complex. In
what follows, notation t xx denotes the scalar product between two vectors x and
x . Notation U ij ( t i ,t j ; η T )and U ij ( s i ,s j ; η S ) denotes pairwise potential functions
with interaction parameters η T and η S . Simple examples for U ij ( t i ,t j ; η T )and
U ij ( s i ,s j ; η S ) are provided by adopting a Potts model which corresponds to

U ij ( t i ,t j ; η T )= η T

t t i t j

and U ij ( s i ,s j ; η S )= η S

t s i s j .

Such a model captures, within each label set t and s , interactions between neigh-
boring voxels. It implies spatial interaction within each label set.