An Automated System for 3D Segmentation of CT Angiograms - Visual Computing: Scientific Visualization and Imaging Systems - page 100

5.3.1 Coronary Arteries Segmentation

We commence our analysis by assuming that voxels in contrast-enhanced CTA
images fall into three categories, i.e. the air in the lungs, soft tissues and blood-
filled regions. Then, we use a Gaussian Mixture Model (GMM) to fit the histogram
of the input CTA volume to estimate the probability density function for each
class, as shown in Fig. 5.4 a. The mean and variance for each class are estimated
using the Expectation-Maximization (EM) method. We use prior anatomical
knowledge that coronaries are located on the outer surface of the heart, and thus,
we neglect the class corresponds to the air to obtain a bi-modal histogram (see
Fig. 5.4 b). The first peak (T 1 ) in the fitted histogram corresponds to soft tissues in
the heart, which reflect the intensity distribution of the background pixels.
According to the assumption that voxels with intensity values less than T 1 as
belonging to the background, while voxels with intensity values greater than this
threshold are treated as potential objects of interest (i.e. blood-filled regions), we
assign each voxel in the volumetric data with a fuzzy label, which indicates the
probability of the voxel belonging to the object.
In this research, we formulate the labelling function as a normalised cumulative
density function of the histogram. We normalise the labelling function between -1
and 0 for voxels with intensity values between 0 and T 1 , and the output of the
labelling function bounded between 0 and 1 for the input voxels with intensity
values greater than T 1 . Thus, the function is defined as follows:

(

½ 1 ; 0 Þ ; if x belongs to the background

L ð x Þ ¼ ð 5 : 1 Þ

½0 ; 1 ; if x is a potential 'object'

Let X x denote a neighbourhood with a radius r centred at x on the active
contour C(x).The localised image, X x , can be partitioned into two sub-regions by
the active contour, i.e. the regions inside and outside the active contour, respec-
tively. Hence, we define the probability of a voxel being classified as belonging to
the region X i as follows:

!

exp ð l i I ð y ÞÞ 2
2r i

1
2p

P i ¼ P ð I ð y Þj y 2 X i \ X x Þ ¼ p ð 5 : 2 Þ

r i

where X i j i ¼ 1 ; f g denote the regions inside and outside the contour. I(y) is the
image intensity at y, l i and r i represent the mean and the variance derived from
region X i , respectively. Note that, we use x and y as two independent spatial
variables to represent a single point in the image domain. Let C(x) denotes a
contour, representing the boundary of the object to be segmented. For each point
along the contour, given its local image X x and the labelling function L(y), the
posterior probability of a voxel y being classified as belonging to the sub-region
X i \ X x can be defined as: