Docteur de l'université Automatic Segmentation and Shape Analysis ...

92 Chapter 4 Statistical shape model of Hippocampus
consistency and regularization on the estimator. Since PCA provides a linear pa-
rameterization of the variation and non-rigid deformation in the shape space, the
formulation of symmetric data terms for consistent estimation has a closed form
least square solution. The deformation of the shape model under the guidance of
shape components also reduces significantly the dimension of the linear system.
The shape parameters are associated with a priori distribution due to the statis-
tical nature of SSM, which facilitates the extension of ML estimator to maximum
a posteriori (MAP) by adding a Tikhonov regularization term.
4.2.1 Gaussian mixture model and EM algorithm
The point set registration problem is to transform the point set {¯p s X, s = 1, · · · , k}
of the model ¯ X (the mean of the SSM in this case) to a target scene Y consisting
of kY landmarks {p t Y , t = 1, · · · , kY }. The Gaussian mixture model for point set
registration models the transformed points T ( ¯ X) as samples from a mixture of
Gaussian distribution with mean at target points {p t Y }
p(T ( ¯ X), A|Y ) = ∏
s,t
(
πst · p (
T (¯p s ) ))
Ast t
pY , (4.55)
where p( · |p t Y ) is the density of the Gaussian distribution in 3D, and A = (Ast) is
the binary matching matrix of which each entry Ast = 1 if point p t Y corresponds
to ¯p s on the model with a priori probability
P (Ast = 1) = πst, ∑
t
πst = 1, s = 1, · · · , k. (4.56)
By applying Bayes’ rule, the distribution density of the matching matrix A given
the transformation T
p(A|T ( ¯ X), Y ) = ∏
(
πst · p(T (¯p
s,t
s X)|pk Y )
∑
l πsl · p(T (¯p s X)|pl ) Ast
Y )
(4.57)