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

Chapter 4 Statistical shape model of Hippocampus 89
Algorithm 9 Groupwise optimization with fluid regularization. Adapted and
modified from Davies et al. (2008b).
1: Initialize with ui ← 0, vi ← 0, i = 1, · · · , n.
2: while u not converging do
3: for all shapes images Si, i = 1, · · · , n do
4: Compute Fi given ui (eq. 4.37)
5: Compute vi given Fi (eq. 4.41, Cahill et al., 2007)
6: Update ui: ← ui + τ(vi − (vi · ∇)ui) (eq. 4.45)
7: end for
8: if the minimum Jacobian of any ui is below threshold then
9: for all shapes images Si, i = 1, · · · , n do
10: Si(x) ← Si(x − ui) (eq. 4.46)
11: ui ← 0, vi ← 0
12: end for
13: end if
14: if reorientation required then
15: for all shapes images Si, i = 1, · · · , n do
16: Si(x) ← Si(x − ui) (eq. 4.46)
17: reorient the octahedron unfolding gi
18: update Si
19: ui ← 0, vi ← 0
20: end for
21: end if
22: end while
where
(x s i , y s i , z s i ) T = fi ◦ γi(θs, φs) = Si ◦ ψi(ıs, js), s = 1, 2, · · · , k (4.48)
are the sampled points. For sampling on subdivided octahedron k = 4N 2 +2. The
mean
gives the data matrix
¯X = 1
n
n∑
Xi, (4.49)
i=1
X = (X1 − ¯ X, X2 − ¯ X, · · · , Xn − ¯ X). (4.50)
A PCA is performed such that each shape can be approximated by a subspace
spanned by the eigenvectors of the larger covariance matrix
X = ¯ X + Wb, (4.51)