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

80 Chapter 4 Statistical shape model of Hippocampus
Figure 4.3: Multi-resolution subsampling by icosahedron subdivision. Left to
right: icosahedron, subdivisions by factor 2, 4 and 6. Image credit: Styner et al.
(2006).
Algorithm 7 Multi-resolution reparameterization by rotation.
1: for i = 1, · · · , n do
2: Γi ← Id
3: end for
4: for each trial do
5: select an arbitrary parameterization fj as the template
6: {(θs, φs)} ← coordinates of icosahedron vertices
7: for each level of resolution do
8: for all fi = fj do
9:
10:
initialize the optimizer with current Γi
∑ks=1 update Γi ← arg min fi(Γ(θs, φs)) − fj(Γj(θs, φs))
Γ∈SO(3)
2
11: end for
12: end for
13: end for
4.1.3 Groupwise optimization on shape images
Given the rotationally reparameterized surfaces {fi ◦ Γi, i = 1, · · · , n}, the re-
maining component of the homeomorphisms {˜γi} are to be computed by the
optimization of the MDL of the collection {fi ◦ Γi ◦ ˜γi, i = 1, · · · , n}. Instead
of reparameterizing on S 2 , which involves intersection and interpolation on the
sphere, Davies et al. (2008b) re-map the parameterization to an image in R 2 to fa-
cilitate the manipulation of the shape representation. The reparameterization on
the image representation of shapes is turned to a problem similar to the non-rigid
registration. Groupwise optimization is used to reparameterize each individual
shape, and fluid regularization is applied.