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Chapter 4 Statistical shape model of Hippocampus 87
4.1.3.4 Optimization process
Once the velocity v is solved, the update of u follows the Eulerian derivative
˙u(x, τ) = v(x, τ) − (v · ∇)u(x, τ), (4.44)
which gives the increment of u in each time step τ
τ(vi − (vi · ∇)ui). (4.45)
In the process of optimization, re-gridding is performed to avoid folding (Chris-
tensen et al., 1996). The Jacobian in folded deformation is negative, thus the
condition of performing re-gridding is determined by checking the Jacobian of the
deformation. If the deformation of any shape image leads to the local Jacobian
below a positive threshold, the regrid is performed by reparameterizing all the
shape images based on the current u
S ′ i(x) = Si(x − ui(x)) (4.46)
and the displacement u is reset. In order to avoid the bias due to the choice of the
orientation of the octahedron in cutting and unfolding, the octahedron param-
eterization is periodically reoriented and unfolded to the plane. The 6 possible
orientations (∼ 6 vertices of the octahedron) and the unfolding are shown in Fig-
ure 4.7. The algorithm of groupwise optimization for shape images is listed in
Algorithm 9.
4.1.4 Validation and evaluation of shape models
For reparameterized shape surfaces {fi ◦ γi, i = 1, · · · , n} or image representation
{Si ◦ ψi, i = 1, · · · , n}, each shape surface is sampled and represented in SSM by
a vector concatenating the coordinates of sampled points
Xi = (
x 1 i , y 1 i , z 1 i , x 2 i , y 2 i , z 2 i , · · · x k i , y k i , z k ) T
i
∈ R 3k , (4.47)