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

94 Chapter 4 Statistical shape model of Hippocampus
with (Wb) s ∈ R 3 indicating the displacement of the s-th model point given the
vector b.
Noting that A is row-stochastic, i.e. ∑
t A ∗ st = 1, it can be transformed into a least
square estimation
where
T ∗ ∑ 1
= arg min
T s kσ2 T (¯xs ) − p s Yc2 Yc = E(A|T )Y and p s ∑
Yc =
t
A ∗ stp t Y
(4.65)
(4.66)
are the correspondence of model points ¯ X in the scene Y weighted by the con-
ditional expectation of A. The optimization of T in the M-step can be divided
further into:
M.1 The least square estimation of deformation b
b ∗ ∑ 1
= arg min
b s kσ2 TA(¯p s X + (Wb) s ) − p s Yc2 , (4.67)
which has the least square solution b ∗ = W T (T −1
A (Yc) − ¯ X), where T −1
A is
the inverse of the affine component in the current estimation;
M.2 The least square estimation of the pose TA
T ∗ A = arg min
TA
∑
TA(¯p
s
s X + (Wb ∗ ) s ) − p s Yc2 . (4.68)
The algorithm for deforming the SSM to fit the scene Y is listed in Algorithm 10.
4.2.2 Estimation with symmetrical consistency and shape priors
The energy function (4.63) to be minimized in M.1 is asymmetric since it is the
mean square distance from each point in T ( ¯ X) to Yc. Symmetrically, a mean
square data term
1
kY σ2 ∑
B
s,t
∗
ts T (¯p s X) − p t
Y 2
(4.69)