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76 Chapter 4 Statistical shape model of Hippocampus
the Laplacian matrix for the mesh X, and the vector b is set by the boundary
conditions, and θ = (θ(v1), θ(v2), · · · , θ(vk)) T is the vector of the values for all the
vertex θ(v). The algorithm setting up the matrix A, and the vector b is listed in
Algorithm 5.
Algorithm 5 Linear system for the initial diffusion of θ, adapted from Brechbühler
et al. (1995)
1: {Setting up the matrix A}
2: for all vertex vi, i = 1, · · · , k do
3: Aii ← number of direct neighbors of vi
4: for all vj which is direct neighbor of vi do
5: Aij ← −1
6: end for
7: end for
8: {Setting up the vector b}
9: for all i = 1, · · · , k do
10: bi ← 0
11: if vi is direct neighbor of vsouth then
12: bi ← π
13: end if
14: end for
To compute the diffusion of the longitude, a date line connecting the north and
south pole is chosen as the path with steepest latitude ascent. Analogous to the
International Date Line on the globe, there is a 2π-discontinuity in φ across the
date line, which is used as the boundary condition for φ. Since the longitude is
undefined for the poles, the neighbors of both poles are disconnected to the pole
in the computation of the longitudes. The Laplacian matrix A for solving the
φ = (φ(v1), φ(v2), · · · , φ(vk)) T is thus modified. The algorithm modifying A and
setting b is listed in Algorithm 6.
4.1.1.2 Distortion minimization
The initial assignment of parameters (θ(v), φ(v)) is optimized to reduce the dis-
tortion. The optimization is carried out under the constraint of area preserving,
i.e. the area of each facet on X must map to a region of proportional area in
parameter space S 2 . The optimization process aims to minimize the distortion