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

96 Chapter 4 Statistical shape model of Hippocampus
cost function can be rewritten as
E = TA
2kσ 2 ¯ X + Wb − T −1
A (Yc) 2
+ αTA
2kY σ 2 Xc − T −1
A (Y ) + BWb 2
+ β
2 bT Λ −1 b,
(4.72)
where TA is norm of the linear part of the affine transformation TA, which is the
scaling factor of TA,
Xc = B ∗ X, (4.73)
and Λ is the diagonal matrix of eigenvalues {λm} of SSM. As opposed to B ∈
R ky×k , B ∈ R 3kY ×3k is left multiplied to Wb ∈ R 3k , which is modified from B ∗
⎛
⎜ B
⎜
B = ⎜
⎝
∗ 11I3×3 B∗ 12I3×3 · · · B∗ B
1kI3×3
∗ 21I3×3 B∗ 22I3×3 · · · B∗ . .
. ..
2kI3×3
.
B∗ kY 1I3×3 B∗ kY 2I3×3 · · · B∗ ⎞
⎟
⎠
kY kI3×3
The minimum of the energy function is reached with the derivative
∂E
∂b
which is the solution to the linear system
(4.74)
= 0, (4.75)
(
1 α
I + W
k kY
T B T
BW σ
+ 2β TA Λ−1
)
b = 1
k WT (T −1
A (Yc) − ¯ X)
+ α
W
kY
T B T −1
(TA (Y ) − Xc).
(4.76)
To facilitate the computation of the term W T B T BW, sparsity on B can be im-
posed by restricting the search of correspondence in (4.70) within a neighborhood
of radius ρ.