Model-Based Segmentation, Reconstruction and Analysis of the Vertebral Body from Spinal CT - Spinal Imaging and Image Analysis - page 376

2.1 Overview

In this chapter, three pieces of information (intensity, spatial interaction, and shape)
are modelled to obtain the optimum segmentation. The data is assumed to have two
classes: background and object regions which are represented as

“ ” “ ” “ ”

b or 0 and o

or “ 1 ” , respectively. So, let L ¼ f 0 ; 1 g denotes the set of labels. In this work, the

given VB s volume, the shape model and the desired map (labeled volume) are

described by a joint Markov-Gibbs random field (MGRF) model. We can define the
gray level volume I by the mapping

'

P! G and its desired map f by the mapping

P!L is the set of voxels and G is the set of gray levels. Shape

information is represented by the set of distances of variability region

, where P

is voxels d
(more details explained later). Since I and d are independent, a conditional distri-
bution model of input volume, its desired map, and the shape constraints can be
written by the as follows:

'

ð j ; Þ ð j Þ ð j Þ ð Þ; ð Þ

P f I d P I f P d f P f 1

where P ð I j f Þ and P ð f Þ represents appearance models, and the conditional distri-

bution P is the shape model. Given I and d, the map f can be obtained using

Bayesian maximum-a posteriori estimate as follows:

ð d j f Þ

f ¼

ð ; ; Þ; ð Þ

arg max
f 2F

L I d f 2

is the set of all possible f 0 s, and L I where F ð ; d ; f Þ is the log-likelihood function,

which can be written as follows:

L ð I ; d ; f Þ/ logPðfÞ: ð d j f Þþ logPðfÞ: ð I j f Þþ logPðfÞ: ð f Þ: ð 3 Þ

The parameters of the shape model P ð d j f Þ and the volume appearance models

should be identi ne this log-likelihood function.

Intensity and interaction models may not be enough to obtain optimum seg-
mentation. To segment the VB, a new shape based methods which integrate the
models of the intensity, spatial interaction, and shape prior information is proposed.
The proposed method presents several advantages which can be written as: (i) the
probabilistic shape model is automatically registered to the testing image, hence
manual interaction is eliminated, (ii) the registration benefits from the segmented
region to be used in the shape representation, and (iii) the probabilistic shape model
re

ed, to completely de

nes the initial segmentation result using the registered variability volume.
The segmentation part has following steps: (1) initial segmentation using only
intensity and spatial interaction information (this step is needed to obtain the feature
correspondence between the image domain and shape model), (2) shape model
registration, and (3) the

final segmentation using three models.
Next section, we describe the spinal cord extraction stage as a preprocessing step.