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

Chapter 4 Statistical shape model of Hippocampus 93
gives the conditional expectation
E(Ast|T ) = πst · p(T (¯p s X)|pk Y )
∑
l πsl · p(T (¯p s X)|pl Y )
(4.58)
Viewing the correspondence matching matrix A as hidden variable, EM algorithm
can be used to minimize the free energy
F (A, T ) = −EA(log p(T (¯x), A|y)) + EA(log p(A)). (4.59)
In implementation, isotropic Gaussian noise with covariance σ 2 I is assumed
p(p ′ |p) =
1
(2π) 3 exp
2 σ3 (
− 1
2σ2 p − p′ 2
)
(4.60)
where p − p ′ is the Euclidean distance between the point p and p ′ . The prior
distribution is set to be uniform
πst = 1
kY
, ∀s, t. (4.61)
The EM algorithm estimates the latent match A and the transformation T alterna-
tively. In the Expectation-step (E-step), the algorithm estimates the expectation
of the match matrix A based on the current estimate of T
A ∗ st = E(Ast|T ) = e−T (¯ps X )−pt Y 2 /2σ2 ∑
l e−T (¯ps X )−pl Y 2 , (4.62)
/2σ2 In the Maximization-step (M-step), the transformation T is estimated by mini-
mizing the energy
T ∗ = arg min
T
1
kσ2 ∑
s,t
A ∗
st
T (¯p s X) − p t
Y
2
, (4.63)
where the transformation T = (TA, b) is composed of an affine transformation TA,
and the deformation of the SSM along the vector b (eq. 4.51)
T ( ¯ X) = TA( ¯ X + Wb) and T (¯p s X) = TA (¯p s X + (Wb) s ) , (4.64)