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Fig. 5.4 The histogram of the CTA image. a The histogram and the fitted mixture model of the
CTA image. b The bi-modal histogram obtained within the heart region
P ð y 2 X i \ X x j I ð y Þ ; L ð y ÞÞ ¼ P ð I ð y Þ ; L ð y Þj y 2 X i \ X x Þ P ð y 2 X i \ X x Þ
P ð I ð y Þ ; L ð y ÞÞ
ð 5 : 3 Þ
where P(y [ X i \ X x i = 1,2) is the prior probability of the current voxel being
assigned to region X i among all the possible partitions within the local image X x .
This term can be ignored, if equal probabilities are assumed for all partitions of the
image. P(I(y), L(y)) denotes the joint probability density distribution of the grey
level value I(y) and the labelling function L(y), which is independent of the seg-
mentation of the image and can therefore be neglected. We assume that the voxel
labels and the grey level intensity distribution are independent. The posterior
probability for each voxel can thus be computed as:
P ð I ð y Þ ; L ð y Þj y 2 X i \ X x Þ ¼P ð I ð y Þj y 2 X i \ X x Þ P ð L ð y Þj y 2 X i \ X x Þ
ð 5 : 4 Þ
The prior probability of P(I(y)|y [ X i \ X x ) has been already defined in Eq. 5.2 .In
order to compute the posterior probabilities in Eq. 5.4 , the prior probability of the
labelling function should be known. In this research, we model the prior proba-
bility distribution of the labels as:
v
2 L ð x Þ R ð x Þ k r ð x ; y Þ
P ð L ð y Þj y 2 X i \ X x Þ/ exp
ð 5 : 5 Þ
where:
!
r exp ð x y Þ 2
k r ð x ; y Þ ¼ 1
p
2p
ð 5 : 6 Þ
2r 2
represents the weighting kernel, which is a decaying function of the distance
between x and y. v 2 is the overall weight that determines the influence of the labels
on the segmentation. R(x) is a normalised Boolean function indicating whether the
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