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

Chapter 4 Statistical shape model of Hippocampus 81
.
Figure 4.4: Cutting the octahedron and unfold to a plane. The edges cut
are color coded. Left: octahedron, right: unfolded octahedron. Adapted from
Praun and Hoppe (2003).
4.1.3.1 Image representation and manipulation of shapes
In previous steps, the shape surfaces are parameterized by spherical coordinates on
S 2 . By sampling on S 2 by a subdivided octahedron, shape surface Xi is sampled
as
.
fi ◦ Γi(θs, φs), s = 1, · · · , k (4.20)
where (θs, φs), s = 1, · · · , k are sampled parameters on S 2 by the octahedron. The
octahedron can be mapped to a 2D grid bijectively by cutting the edges of the
octahedron, and unfolding it to the plane (see Figure 4.4, Praun and Hoppe, 2003).
Due to cutting of the surface, some nodes in the octahedron are duplicated when
unfolded on the plane. Subdivided octahedron of k = 4N 2 + 2 nodes is unfolded
to a (2N + 1) × (2N + 1) grid. Each node (ı, j) ∈ N 2 on the grid is associated with
a node s(ı, j) on the subdivided octahedron. The mapping
g : D = [0, 1] × [0, 1] ⊂ R 2 ↦→ S 2
(4.21)
from a image domain D to the parameter space S 2 , which can be interpolated
from the values of the grid nodes
g(xı,j) = (θs(ı,j), φs(ı,j)) (4.22)