Geometric Computing beyond the Laplacian

Abstract

The Laplace–Beltrami operator, or, Laplacian, is the central object of study in shape analysis, geometry processing, and scientific computing. Motivated by the ubiquity of Laplacian in geometric computing algorithms, this thesis initiates a program for systematically designing novel algorithms by replacing the Laplacian with better or even optimal alternative operators. Our approach is based on the observation that Laplacian-based algorithms can yield suboptimal results for geometric computing tasks, often due to, e.g., the inability to capture extrinsic geometry and/or the lack of a problem-specific metric under which the Laplacian is defined. We bridge the gap by proposing geometric computing algorithms built on operators or PDEs other than the ordinary Laplacian. Borrowing insights from optimal control and inverse PDE problems, we propose efficient numerical schemes to search for the metric or conformal structure whose associated (generalized) Laplacian is optimal for a given task, or explicitly design operators as suggested by modern spectral geometry. Concretely speaking, to represent an arbitrary diffeomorphism or injective map that is possibly non-conformal, we search for a generalized Laplacian or elliptic PDE that accounts for quasi-conformal deformation and satisfies a prescribed Cauchy boundary condition; for geometric data interpolation, we search for the generalized Laplacian whose behavior best approximates a higher-order variational problem; to design neural networks that directly operate on triangle meshes, we learn finite-element kernels that assemble the operators from data; and for extrinsic shape analysis, we consider an alternative operator, the Dirichlet-to-Neumann operator—the Schur complement of a higher dimensional Laplacian with the interior marginalized out. In addition, we develop discrete models of inverse elliptic problems, resembling core properties of the continuous counterparts. With extensive experimental evaluations, our formulations significantly improve over state-of-the- art algorithms for foundational considerations in geometry ranging from computing injective maps to interpolation on geometric domains.