Theses Doctoral

# Smoothness Energies in Geometry Processing

Stein, Oded

This thesis presents an analysis of several smoothness energies (also called smoothing energies) in geometry processing, and introduces new methods as well as a mathematical proof of correctness and convergence for a well-established method. Geometry processing deals with the acquisition, modification, and output (be it on a screen, in virtual reality, or via fabrication and manufacturing) of complex geometric objects and data. It is closely related to computer graphics, but is also used by many other fields that employ applied mathematics in the context of geometry.

The popular Laplacian energy is a smoothness energy that quantifies smoothness and that is closely related to the biharmonic equation (which gives it desirable properties). Minimizers of the Laplacian energy solve the biharmonic equation. This thesis provides a proof of correctness and convergence for a very popular discretization method for the biharmonic equation with zero Dirichlet and Neumann boundary conditions, the piecewise linear Lagrangian mixed finite element method. The same approach also discretizes the Laplacian energy. Such a proof has existed for flat surfaces for a long time, but there exists no such proof for the curved surfaces that are needed to represent the complicated geometries used in geometry processing. This proof will improve the usefulness of this discretization for the Laplacian energy.

In this thesis, the novel Hessian energy for curved surfaces is introduced, which also quantifies the smoothness of a functions, and whose minimizers solve the biharmonic equation. This Hessian energy has natural boundary conditions that allow the construction of functions that are not significantly biased by the geometry and presence of boundaries in the domain (unlike the Laplacian energy with zero Neumann boundary conditions), while still conforming to constraints informed by the application. This is useful in any situation where the boundary of the domain is not an integral part of the problem itself, but just an artifact of data representation---be it, because of artifacts created by an imprecise scan of the surface, because information is missing outside of a certain region, or because the application simply demands a result that should not depend on the geometry of the boundary. Novel discretizations of this energy are also introduced and analyzed.

This thesis also presents the new developability energy, which quantifies a different kind of smoothness than the Laplacian and Hessian energies: how easy is it to unfold a surface so that it lies flat on the plane without any distortion (surfaces for which this is possible are called developable surfaces). Developable surfaces are interesting, as they can be easily constructed from cheap material such as paper and plywood, or manufactured with methods such as 5-axis CNC milling. A novel definition of developability for discrete triangle meshes, as well as a variety of discrete developability energies are also introduced and applied to problems such as approximation of a surface by a piecewise developable surface, and the design and fabrication of piecewise developable surfaces. This will enable users to more easily take advantages of these cheap and quick fabrication methods.

The novel methods, algorithms and the mathematical proof introduced in this thesis will be useful in many applications and fields, including numerical analysis of elliptic partial differential equations, geometry processing of triangle meshes, character animation, data denoising, data smoothing, scattered data interpolation, fabrication from simple materials, computer-controlled fabrication, and more.