Sum-of-Squares Geometry Processing

TL;DR

This paper introduces a sum-of-squares optimization framework to address complex geometry processing problems on higher-order surfaces, enabling solutions like collision detection and closest point queries with minimal adjustments.

Contribution

It presents a unified sum-of-squares based approach that simplifies solving diverse geometry processing tasks on complex higher-order surfaces.

Findings

Successfully applied to collision detection on curved patches

Enabled closest point queries on higher-order geometries

Reduced problem complexity to a single degree parameter

Abstract

Geometry processing presents a variety of difficult numerical problems, each seeming to require its own tailored solution. This breadth is largely due to the expansive list of geometric primitives, e.g., splines, triangles, and hexahedra, joined with an ever-expanding variety of objectives one might want to achieve with them. With the recent increase in attention toward higher-order surfaces, we can expect a variety of challenges porting existing solutions that work on triangle meshes to work on these more complex geometry types. In this paper, we present a framework for solving many core geometry processing problems on higher-order surfaces. We achieve this goal through sum-of-squares optimization, which transforms nonlinear polynomial optimization problems into sequences of convex problems whose complexity is captured by a single degree parameter. This allows us to solve a suite of problems on higher-order surfaces, such as continuous collision detection and closest point queries on curved patches, with only minor changes between formulations and geometries.