Model-Based Segmentation, Reconstruction and Analysis of the Vertebral Body from Spinal CT - Spinal Imaging and Image Analysis - page 409

Fig. 32 The distance
probabilistic models of VB
and its background in the
variability region

The maximum likelihood estimator is used to estimate the Poisson distribution
parameter. Where the modi ed EM algorithm [ 38 ] is used to estimate the param-

eters of Gaussians components. Figure 32 illustrates the estimated densities of VB
and its background.

2.5.2 The Gray Level Probabilistic Model

Also, assuming the conditional distribution of the original volume given the map is
an independent random

field of gray levels with different gray value distributions.

Y

P(If j Þ ¼ P ð I p f p Þ: ð 35 Þ

p 2P

, we use our
LCG model [ 35 , 36 ] with C p ; f p positive and C n ; f p negative components. Thus; the
gray level marginal density of each class can be written as follows:

To approximate the gray level marginal density of each class P ð I p j f p Þ

C p ; fp C n ; fp

X X

ð Þ ¼

w p ; r ; f p uð I p h p ; r ; f p

Þ

w n ; s ; f p uð I p h n ; s ; f p

Þ ð Þ

P I p f p 36

r¼1 s¼1

is a Gaussian density with parameter h (mean l and variance r 2 ),
w p ; r ; f p means the rth positive weight in class f p , w n ; s ; f p means the sth negative weight
in class f p . Also the weights should satisfy P C p ; fp

where, uð I p jhÞ

r¼1 w p ; r ; f p P C n ; fp

s¼1 w n ; s ; f p ¼

1.

ed EM algorithm [ 38 ], which deals with the positive and neg-
ative components, is used to estimate the parameters of the LCG model. Figure 33

Also, the modi