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

74 Chapter 4 Statistical shape model of Hippocampus
correspondence by reparameterization, we first consider the rotation degree of
freedom of the reparameterization (§4.1.2), and then solve the reparameterizations
{˜γi} by an optimization approach minimizing the Minimum Description Length
(MDL) of the shape model (§4.1.3).
4.1.1 Parameterization of the shape surfaces
A natural choice of coordinate system on S 2 is to use spherical polar coordinates
(θ, φ) ∈ [0, π] × [−π, π) for the parameterization. A numerical solution to the
spherical parameterization of genus 0 surfaces is developed by Brechbühler et al.
(1995) which is used to map the hippocampal surfaces to the unit sphere. Since
the topology of the surface is used in the parameterization algorithm, the mesh
representation of the input surface X is used to preserve both geometrical and
topological information. For a given mesh representation for the input X, the
parameterization algorithm computes the latitude θ(v) and the longitude φ(v) for
each vertex v by solving the Laplacian equations
∇ 2 θ = 0, (4.6)
∇ 2 φ = 0 (4.7)
diffusing the coordinates (θ, φ) over the mesh X. The diffusion boundary condition
is set by initializing a north pole and a south pole on the mesh to be parameter-
ized. The assignment of latitude and longitude is then optimized by minimizing
the distortion in the parameterization. Thus the parameterization f(θ, φ) of the
surface X is defined such that
gives the spatial position p v for all the vertex v.
f(θ(v), φ(v)) = p v ∈ X (4.8)