An Approach to Quad Meshing Based on Harmonic Cross-Valued Maps and the Ginzburg-Landau Theory

TL;DR

This paper introduces a mathematically grounded approach to quad meshing by linking cross field design to the Ginzburg-Landau problem, providing theoretical guarantees and an energy minimization technique for boundary-aligned meshes.

Contribution

It establishes a novel connection between cross field design for quad meshing and the Ginzburg-Landau theory, offering a new energy minimization method with provable guarantees.

Findings

Successfully generates boundary-aligned quad meshes

Provides bounds on mesh defect locations

Demonstrates effectiveness on various test domains

Abstract

A generalization of vector fields, referred to as N-direction fields or cross fields when N = 4, has been recently introduced and studied for geometry processing, with applications in quadrilateral (quad) meshing, texture mapping, and parameterization. We make the observation that cross field design for two-dimensional quad meshing is related to the well-known Ginzburg-Landau problem from mathematical physics. This yields a variety of theoretical tools for efficiently computing boundary-aligned quad meshes, with provable guarantees on the resulting mesh, such as the number of mesh defects and bounds on the defect locations. The procedure for generating the quad mesh is to (i) find a complex-valued "representation" field that minimizes the Ginzburg-Landau energy subject to a boundary constraint, (ii) convert the representation field into a boundary-aligned, smooth cross field, (iii) use separatrices of the cross field to partition the domain into four sided regions, and (iv) mesh each of these four-sided regions using standard techniques. Leveraging the Ginzburg-Landau theory, we prove that this procedure can be used to produce a cross field whose separatrices partition the domain into four sided regions. To minimize the Ginzburg-Landau energy for the representation field, we use an extension of the Merriman-Bence-Osher (MBO) threshold dynamics method, originally conceived as an algorithm to simulate mean curvature flow. Finally, we demonstrate the method on a variety of test domains.