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

82 Chapter 4 Statistical shape model of Hippocampus
Figure 4.5: The construction of a shape image, from shape to sphere to embedded
octahedron to final shape image. The colour code denotes corresponding
regions. From left to right: physical surface (bunny), spherical parameterization,
octahedron embedding, shape image. Image credit: Praun and Hoppe
(2003); see also Davies et al. (2008b).
where each node (ı, j) is identified with a pixel xı,j ∈ [0, 1] × [0, 1]. A vector image
S can be defined on D
S = f ◦ Γ ◦ g : D ↦→ R 3
(4.23)
such that the homeomorphic reparameterization ˜γ on S 2 is equivalent to a defor-
mation ψ of the image domain
where
is a 2D deformation field.
S ◦ ψ = f ◦ Γ ◦ ˜γ ◦ g (4.24)
ψ = g −1 ◦ ˜γ ◦ g : D ↦→ D (4.25)
In the discussions of kinetics of continuous media and the modeling of deformation
in image registration, physical quantities such as the displacement and the velocity
field are more efficiently studied under Eulerian reference frame, in which the
frame of reference is fixed with respect to the laboratory system, in contrast to
the Lagrangian frame which follows the motion of the particles. Under Eulerian
reference frame, the deformation ψ is described by the displacement u(x, t) of the
particles passing through x at time t, originated from x − u(x, t) at t = 0. Given
the displacement field u, the shape image can be reparameterized
S ′ (x) = S(x − u(x)) (4.26)