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

Chapter 4 Statistical shape model of Hippocampus 83
(a) reparameterization ˜γ
(b) reparameterization ψ = g −1 ◦ ˜γ ◦ g
Figure 4.6: Reparameterization on the spherical parameterization and on the
shape image. Left: the reparameterization ˜γ on the sphere; right: the reparameterization
ψ = g −1 ◦ ˜γ ◦ g on the shape image, which is equivalent to a
deformation on a 2D image. Image credit: Davies et al. (2008a).
with the deformation
4.1.3.2 MDL cost function and its gradient
ψ(x) = x − u(x). (4.27)
The original MDL cost function for optimizing correspondences was proposed by
Davies et al. (2002, 2003), which was simplified by Thodberg (2003). With each
shape surface in the collection {Xi, i = 1, · · · , n} sampled by octahedron and
mapped to 2D images, the mean ¯ S and the covariance matrix Σ for the shape
sample can be computed
Σij =
1
(2N + 1) 2
2N+1
∑
ı=1
2N+1
∑
j=1
¯S(x) = 1
n
The MDL cost function is defined as
LMDL = ∑
n∑
Si(x), (4.28)
i=1
(
Si(xı,j) − ¯ S(xı,j) )
· (
Sj(xı,j) − ¯ S(xı,j) )
. (4.29)
λm≥λc
(
1 + log λm
)
+ ∑
λc
λm