author = {Olischl{\"{a}}ger, Nadine},
  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
	L2-gradient flow, the anisotropic Willmore flow, leads to a fourth-order
	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 = { 1},
  url = {}