Relaxing Topological Barriers in Geometry Processing

David R. Palmer

Abstract: Geometric optimization problems are full of topological barriers that hinder optimization, leading to nonconvexity, initialization-dependence, and local minima. This thesis explores convex relaxation as a powerful guide and tool for reframing such problems. We bring the tools of semidefinite relaxation to bear on challenging optimization problems in field-based meshing and unlock polynomial geometry kernels for physical simulation. We bring together frame fields with spectral representation of geometry. We use current relaxation to devise a new neural shape representation for surfaces with boundary as well as a convex relaxation of field optimization problems featuring singularities. Unifying these disparate problems is a focus on how the right choice of representation for geometry can simplify optimization algorithms.