Model-Based Segmentation, Reconstruction and Analysis of the Vertebral Body from Spinal CT - Spinal Imaging and Image Analysis - page 408

C d p is a normally propagated wave from C OV , the probability of an iso-surface to be
VB decays exponentially as the discrete index d p increases. Therefore; we can use a
Poisson distribution to model the distance histogram, which is estimated as follows.
The value of the histogram at a distance d p can be calculated as

X

M

X

K

X

h dp ¼ dð p 2O ij Þ; ð 31 Þ

i¼1 j¼1 p 2 C dp

where d A is true, and zero

otherwise, M is the number of training data sets, K is the number of CT slices of
each data set, and

ðÞ is an indicator function equals 1 when the condition A

O ij is the VB region in the training set i and in the slice j. The
domain of the distance d p is the variability region. The histogram should be mul-
tiplied by the VB prior value, which can be estimated as follows:

X X X

M K

1
MK jj

p O ¼ dð 2O ij Þ: ð Þ

p 32

i¼1 j¼1 p 2V

Same computations can be done to estimate the marginal density of VB ' s

background

Distance Probabilistic Model

Assuming the conditional distribution P ð d j f Þ is an independent random field of

distances, then

Y

P ð df j Þ ¼ ð Pd p f p Þ: ð 33 Þ

p 2V

We model the distance marginal density of each class P ð d p j f p Þ as a Poisson

ned by K f p positive and K f p negative discrete Gaussians compo-
nents. So the distance marginal density of each class can be written as follows:

distribution re

K fp

K fp

X X

Þ ¼ #ð d p k f p Þþ

w f p ; r uð d p h f p ; r w f p ; l uð d p h f p ; l P ð d p f p Þ Þ; ð 34 Þ

r¼1 l¼1

where #ð d p jk f p Þ is a Poisson density with rate k , uð:jhÞ is a Gaussian density with

parameter h ðl; r 2 with mean l and variance r 2 . w f p ; r means the rth positive

weight in class f p and w f p ; l means the lth negative weight in class f p . This weights
have a restriction P

Þ

r¼1 w f p ; r P

K f p K f p

l¼1 w f p ; l ¼

1.