Here we describe the particular model that has been used. We start by giving assumptions about the slice regions and the tissue types, thereafter specify the prior model chosen for the contours and finally the model for the observed image.
Assumptions about the slice regions and the tissue types
Firstly, we assume an axial ventricular level region-model divided into four regions (Fig. 1). These regions are outside brain, i.e. arachnoid / dura mater (r=1), subarachnoid space (r=2), brain parenchyma (r=3), and the lateral ventricles (r=4). The boundaries between region 1 and 2, region 2 and 3, and region 3 and 4 are assumed to be simply closed non-intersecting curves.
Figure 1: A priori region model for
the brain.
The normal head mask R
contains four disjoint, connected regions 
, r=1,2,3,4, and
seven different tissue types.
Each region is characterized by its constituent tissue types, denoted
.
Region 1: outside brain (i.e. arachnoid / dura mater),
air and bone, muscle, fat, connective tissue
;
Region 2: arachnoid space, 
CSF;
Region 3: brain parenchyma, 
white matter,
gray matter;
Region 4: lateral ventricles, 
CSF.
Secondly, we assume the ventricular level slice image consists
of seven main tissue classes,
air and bone (c=1), muscle (c=2), fat (c=3),
connective tissue (c=4), CSF (c=5), and
white (c=6) and gray matter (c=7) making up
the brain parenchyma. Normal MRI anatomy constrains
region 1 to contain at most air and bone,
muscle, fat, and connective tissue; region 2 to
contain CSF only; region 3 to contain brain parenchyma; and
region 4 to contain CSF only. We let R
denote the region of interest (i.e. the head mask for a given slice
examination) partitioned by the three closed contours x into four disjoint
regions 
. The set of possible tissue classes within
region r is denoted , where the members of each are
listed above.
Prior model
The prior model is specified through an energy
function [44, 54],
where Z is a normalizing constant such that the prior is a proper probability distribution.
We have chosen 
to be related to the ``fractal'' geometric property of the closed contour by
where 
is a smoothing parameter.
By construction, attains lowest energy values for smooth and
circle-shaped contours x.
The choice of this energy function compared to the usual choices of stretching and bending energies was made after experimentation with both choices. The favorable result we obtained by using (2) is however highly related to the specific choice of representation of the boundary that we have used (see the Appendix) and a different conclusion could be obtained by other boundary representations.
For the particular problem considered, there are actually three contours
that are to be recognized.
We therefore generalize model (2)
to
Data model
Regarding the data, each class 
is assumed to have a
multivariate Gaussian
distribution with expectation vector
and covariance matrix 
. For the four-region model, the
conditional distribution function will be
where 
and are defined above, and
is the prior probability of class c inside region r
using non-informative
a priori probabilities 
,
.
Also in this case, it is possible to transform the
probability distribution f to an energy function, denoted 
,
defined as the negative log-likelihood
Given the total energy function, 
, we then seek to
find the minimum energy curves x', corresponding
to the maximum a posteriori (MAP) estimate of the curves.
In [50] and [49] an iterative
algorithm based on simulated annealing was constructed for solving this
optimization problem.
