Seamless Parametrization with Arbitrarily Prescribed Cones

TL;DR

This paper proves the existence of seamless surface parametrizations with prescribed cones and introduces a constructive algorithm combining convex optimization and linear systems to compute such mappings.

Contribution

It establishes the universal existence of seamless parametrizations with arbitrary prescribed cones and provides a practical algorithm for their computation.

Findings

A constructive proof of the existence of seamless parametrizations with prescribed cones.

An algorithm combining convex conformal maps and linear modifications for computing these parametrizations.

The method enables initialization for distortion optimization in geometry processing applications.

Abstract

Seamless global parametrization of surfaces is a key operation in geometry processing, e.g. for high-quality quad mesh generation. A common approach is to prescribe the parametric domain structure, in particular the locations of parametrization singularities (cones), and solve a non-convex optimization problem minimizing a distortion measure, with local injectivity imposed through either constraints or barrier terms. In both cases, an initial valid parametrization is essential to serve as feasible starting point for obtaining an optimized solution. While convexified versions of the constraints eliminate this initialization requirement, they narrow the range of solutions, causing some problem instances that actually do have a solution to become infeasible. We demonstrate that for arbitrary given sets of topologically admissible parametric cones with prescribed curvature, a global seamless parametrization always exists (with the exception of one well-known case). Importantly, our proof is constructive and directly leads to a general algorithm for computing such parametrizations. Most distinctively, this algorithm is bootstrapped with a convex optimization problem (solving for a conformal map), in tandem with a simple linear equation system (determining a seamless modification of this map). This initial map can then serve as valid starting point and be optimized with respect to application specific distortion measures using existing injectivity preserving methods.