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

Chapter 4 Statistical shape model of Hippocampus 77
Algorithm 6 Linear system for the initial diffusion of φ, adapted from Brechbühler
et al. (1995)
1: {Modification of the matrix A}
2: for all direct neighbors vi of the north or the south pole do
3: Aii ← Aii − 1
4: end for
5: A11 ← A11 + 2 (arbitrary)
6: {Setting up the vector b}
7: for all i = 1, · · · , k do
8: bi ← 0
9: end for
10: previous ← inorth
11: i ← 1
12: maximum ← 0.0
13: while i = isouth do
14: for all vj which is direct neighbor of vi do
15: if θ(vj) > maximum then
16: maximum ← θ(vj)
17: next ← j
18: end if
19: if j = previous then
20: p previous ← p vj
21: end if
22: end for
23: for all vj which is direct neighbor of vi, clockwise between p previous and p next
do
24: bj ← bj + 2π
25: bi ← bi − 2π
26: end for
27: previous ← i
28: i ← next
29: end while
of the mapping. Each square facet on X should map to a spherical quadrilateral
close to a spherical square on S 2 , by maximizing the objective function ∑ 4 i=1 cos ei,
where ei, i = 1, 2, 3, 4 are the 4 arcs of the mapped spherical quardrilateral (see
Figure 4.2). The cosine of the arc can be computed as the dot product of two
vectors from the centre of the sphere (origin) to the two vertices. The constrained
optimization algorithm is solved by a Newton-Lagrange algorithm. An alternative
implementation is proposed by Weistrand (2005) which computes the distortion
on triangles instead of on quadrilaterals.