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

Chapter 4 Statistical shape model of Hippocampus 85
can be calculated on the image Si by interpolation.
4.1.3.3 Fluid regularization
Since the groupwise optimization of LMDL is generally ill-posed, regularization of
the solution is required. Modeling the deformation as viscous fluid, fluid registra-
tion method (Christensen et al., 1996) has been used by Davies et al. (2008b) in
solving the deformation u. The external force is calculated from the variation of
the object function
Fi(xıj) = − δLMDL
. (4.37)
δui(xıj)
which drives the velocity field v described by the Navier-Lamé equation for the
steady state defined by the Navier-Lamé operator L
L[v] ≡ −µ∇ 2 v − (µ + λ)∇(∇ · v) = F. (4.38)
where µ and λ are the Lamé coefficients, which are set to be equal in our appli-
cation. The Navier-Lamé equation (4.38) is solved using an LU decomposition
by Davies et al. (2008b). The solvability of the boundary value problems for the
system has been discussed by Dahlberg et al. (1988). A fast solution to the vis-
cous fluid PDE using discrete Fourier transform has been proposed by Bro-Nielsen
and Gramkow (1996) to solve the equation under the sliding boundary condition,
i.e. the normal components of v on the boundary ∂D vanishes, while the tangential
components are undetermined.
Since the boundaries of the shape image are cut from the original parameterization
(sphere or octahedron), we expect the boundary to be fixed in the reparameteriza-
tion. To solve the reparameterization on the shape image, the Navier-Lamé system
(4.38) is to be considered in conjunction with zero Dirichlet boundary condition,
also know as the ‘no-slip’ condition
v|∂D = 0. (4.39)