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

Chapter 4 Statistical shape model of Hippocampus 75
In order to calculate the position f(θ, φ) for an arbitrary pair parameters (θ, φ),
we first locate the parameters on the unit sphere, where
r(θ, φ) = (sin θ cos φ, sin θ sin φ, cos θ) T , (4.9)
is contained inside the triangle with vertices
r1 = (sin θ(vi1) cos φ(vi1), sin θ(vi1) sin φ(vi1), cos θ(vi1)) T ,
r2 = (sin θ(vi2) cos φ(vi2), sin θ(vi2) sin φ(vi2), cos θ(vi2)) T ,
r3 = (sin θ(vi3) cos φ(vi3), sin θ(vi3) sin φ(vi3), cos θ(vi3)) T .
The barycentric coordinate
for r can be calculated, such that
(4.10)
(w1, w2, w3), ∀i = 1, 2, 3 : 0 ≤ wi ≤ 1 (4.11)
w1r1 + w2r2 + w3r3 = r. (4.12)
The corresponding position for (θ, φ) can be obtained by a linear interpolation
4.1.1.1 Initial parameterization by diffusion
f(θ, φ) = w1p vi 1 + w2p vi 2 + w3p vi 3
(4.13)
Given the mesh representation for the input X of k vertices, two of the vertices
vnorth, vsouth are selected as the north pole and the south pole
θ(vnorth) = 0, (4.14)
θ(vsouth) = π (4.15)
which is used as Dirichlet condition of the Lapacian equation ∇ 2 θ = 0. The
PDE is approximated on the mesh X by a linear system Aθ = b, where A is