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

Chapter 4 Statistical shape model of Hippocampus 95
Algorithm 10 EM-ICP estimation of the deformation and the pose of the SSM.
1: Estimate the pose TA by ICP
2: b ← 0
3: Initialization T 0 ← (T 0 A, b)
4: while T not converged do
5: {E-step}
6: A ∗ ← E(A|T ) (eq. 4.62)
7: {M-step}
8: for all s do
9: Yc ← A ∗ Y (eq. 4.66)
10: M.1 b ← W T (T −1
A (Yc) − ¯ X)
11: M.2 TA ← arg min
TA
12: end for
13: end while
TA( ¯ X + Wb) − Yc 2 (eq. 4.68)
from each point in Y to T ( ¯ X) can be added to the energy, where B = (Bts) is the
row-stochastic match matrix, which is updated in the E-step
B ∗ = E(B|T ) and B ∗ ts = E(Bts|T ) = e−T (¯ps X )−pt Y 2 /2σ2 ∑
l e−T (¯pl X )−pt Y 2 . (4.70)
/2σ2 The formulation of SSM allows the shape priors based on the distribution of b
parameters bm ∼ N (0, λm), where λm is the mth eigenvalue of the covariance
matrix. To include a priori information of parameters, a Tikhonov regularization
term is added to the energy function, which penalizes the coefficients of the lesser
components. Hence the cost function for symmetric consistent estimation with
regularization
E = 1
2σ2 1 ∑
A
k s,t
∗
st T (¯p s X) − p t
Y 2
+ 1
2σ2 α ∑
B
kY s,t
∗
ts T (¯p s X) − p t
Y 2
+ β ∑ b
2 m
2 m
,
λm
(4.71)
where α and β control the amount of symmetric consistency and regularization.
Fixing the affine part TA in T , and noting that all rows in A and B sum to 1, the