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

Chapter 4 Statistical shape model of Hippocampus 79
where dΩ = sin θdθdφ is the surface element on S 2 . The condition of isometry is
satisfied for rotation Γ
fi ◦ Γ − fj ◦ Γ 2 ∫
=
∫
=
∫
=
S 2
S 2
S 2
fi(Γ(θ, φ)) − fj(Γ(θ, φ)) 2 dΩ
fi(Γ(θ, φ)) − fj(Γ(θ, φ)) 2 J −1 (Γ(θ, φ))dΩ(Γ(θ, φ))
fi(Γ(θ, φ)) − fj(Γ(θ, φ)) 2 dΩ(Γ(θ, φ))
= fi − fj 2
(4.17)
since the Jacobian J for rotation is 1 on S 2 . Assuming the shape surfaces {Xi} are
aligned (e.g. by ICP algorithm) via similarity transforms to a common template,
the rotation Γi can be found by minimizing the pairwise distance
Γi = arg min fi ◦ Γ − f1
Γ∈SO(3)
2 , (4.18)
where f1 is an arbitrarily selected parameterization as the template. Multiple runs
with different choices of template may be performed in order to avoid bias towards
the choice of template.
In implementation, the L2 distance may be evaluated by sampling on S 2
fi − fj 2 4π
k
k∑
fi(θs, φs) − fj(θs, φs)
s=1
2 , (4.19)
where (θs, φs), s = 1, · · · k are uniformly sampled on S 2 by Platonic solids such
as octahedron, dodecahedron or icosahedron. By parameterizing the rotation Γ
using Rodrigues parameters or quaternions, the optimization problem in (4.18)
thus becomes a least-square problem with closed form solution, which can be
solved by Nelder-Mead simplex or other standard optimizers.
A multi-resolution scheme can be easily implemented by subdividing the initial
sampling (Figure 4.3). The multi-resolution algorithm for rotational reparameter-
ization is listed in Algorithm 7.