```@phdthesis{Ol10,
title = {Processing Elastic Surfaces and Related Gradient Flows},
school = {University Bonn},
year = {2010},
type = {Dissertation},
abstract = {Surface processing tools and techniques have a long history in the
fields of computer graphics, computer aided geometric design and
engineering. In this thesis we consider variational methods and geometric
evolution problems for various surface processing applications including
surface fairing, surface restoration and surface matching. Geometric
evolution problems are often based on the gradient flow of geometric
energies. The Willmore functional, defined as the integral of the
squared mean curvature over the surface, is a geometric energy that
measures the deviation of a surface from a sphere. Therefore, it
is a suitable functional for surface restoration, where a destroyed
surface patch is replaced by a smooth patch defined as the minimizer
of the Willmore functional with boundary conditions for the position
and the normal at the patch boundary. However, using the Willmore
functional does not lead to satisfying results if an edge or a corner
of the surface is destroyed. The anisotropic Willmore energy is a
natural generalization of the Willmore energy which has crystal-shaped
surfaces like cubes or octahedra as minimizers. The corresponding
partial differential equation that can be written as a system of
two coupled second second order equations. Using linear Finite Elements,
we develop a semi-implicit scheme for the anisotropic Willmore flow
with boundary conditions. This approach suffer from significant restrictions
on the time step size. Effectively, one usually has to enforce time
steps smaller than the squared spatial grid size. Based on a natural
approach for the time discretization of gradient flows we present
a new scheme for the time and space discretization of the isotropic
and anisotropic Willmore flow. The approach is variational and takes
into account an approximation of the L2-distance between the surface
at the current time step and the unknown surface at the new time
step as well as a fully implicity approximation of the anisotropic
Willmore functional at the new time step. To evaluate the anisotropic
Willmore energy on the unknown surface of the next time step, we
first ask for the solution of an inner, secondary variational problem
describing a time step of anisotropic mean curvature motion. The
time discrete velocity deduced from the solution of the latter problem
is regarded as an approximation of the anisotropic mean curvature
vector and enters the approximation of the actual anisotropic Willmore
functional. The resulting two step time discretization of the Willmore
flow is applied to polygonal curves and triangular surfaces and is
independent of the co-dimension. Various numerical examples underline
the stability of the new scheme, which enables time steps of the
order of the spatial grid size. The Willmore functional of a surface
is referred to as the elastic surface energy. Another interesting
application of modeling elastic surfaces as minimizers of elastic
energies is surface matching, where a correspondence between two
surfaces is subject of investigation. There, we seek a mapping between
two surfaces respecting certain properties of the surfaces. The approach
is variational and based on well-established matching methods from
image processing in the parameter domains of the surfaces instead
of finding a correspondence between the two surfaces directly in
3D. Besides the appropriate modeling we analyze the derived model
theoretically. The resulting deformations are globally smooth, one-to-one
mappings. A physically proper morphing of characters in computer
graphic is capable with the resulting computational approach.},
pdf = {http://numod.ins.uni-bonn.de/research/papers/public/Ol10.pdf 1},
url = {http://hss.ulb.uni-bonn.de/2010/2201/2201.htm}
}
```