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

86 Chapter 4 Statistical shape model of Hippocampus
A fast solver to the boundary value problem of Navier-Lamé system under Dirichlet
boundary condition based on discrete sine transform (Martucci, 1994) has devel-
oped by Cahill et al. (2007). The adjoint of the operator in matrix form
⎛
L † = −µ∇ 2 ⎜
− (µ + λ) ⎝
∂ 2
∂y 2
− ∂2
∂x∂y
is multiplied to both side of the equation (4.38)
where the matrix determinant of L has the form
The sinusoidal waves on D
− ∂2
∂x∂y
∂ 2
∂x 2
⎞
⎟
⎠ (4.40)
L † L[v] = det(L)[v] = L † [F] (4.41)
det(L) = µ(2µ + λ)∇ 2 ∇ 2 . (4.42)
ωa,b(x) = sin aπx sin bπy, x = (x, y) ∈ D, and a, b ∈ N, (4.43)
are the eigen-functions of det(L) under the ‘no-slip’ condition (4.39), which moti-
vates the fast solution of the boundary value problem via discrete sine transform
listed in Algorithm 8.
Algorithm 8 Solving the velocity field with fluid regularization on shape image.
Adapted from Cahill et al. (2007).
1: Compute L † [F](xı,j), ∀ı, j = 1, 2, · · · , 2N + 1 by direct convolution of F with
linear differential filter
2: Compute the discrete sine transform F(a, b) of L † [F]
3: Compute the coefficients
F(a, b)
v(a, b) ←
4µ(2µ + λ) (
cos ( )
aπ + cos 2N
( )
bπ − 2 2N
) 2
4: Compute the velocity v(xı,j) from the inverse discrete sine transform of v(a, b).