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

Chapter 4 Statistical shape model of Hippocampus 73
U × V
γi ✲
γj
U × V
✲
.
.
.
U × V
fi ✲ Xi
✻
fj
❄
✲ Xj
correspondence
Figure 4.1: Parameterizations {fi} and reparameterizations {γi} of the shape
surfaces {Xi} and correspondences.
achieved by reparameterization of {fi}
on the parameter space, such that
γi : U × V ↦→ U × V, i = 1, · · · , n (4.2)
γi(uhi , vhi ) = γj(uhj , vhj ) = (uh, vh), (4.3)
and the result collection of parameterizations {fi◦γi} establish the correspondence
across the training data {Xi}.
Since the hippocampal segmentation are simply connected regions on MR images,
the closed surfaces extracted from the segmentations are homeomorphic to the S 2
sphere. The reparameterizations are thus homeomorphisms on S 2
in which a rotation component Γ can be separated
γi : S 2 ↦→ S 2 , i = 1, · · · , n, (4.4)
γi = Γi ◦ ˜γi
(4.5)
We introduce the calculation of the parameterization f in §4.1.1. To establish