Ryan Viertel | Searcharxiv

arXiv2019

Coarse Quad Layouts Through Robust Simplification of Cross Field Separatrix Partitions

Streamline-based quad meshing algorithms use smooth cross fields to partition surfaces into quadrilateral regions by tracing cross field separatrices. In practice, re-entrant corners and misalignment of singularities lead to small regions and limit cycles, negating some of the benefits a quad layout can provide in quad meshing. We introduce three novel methods to improve on a pipeline for coarse quad partitioning. First, we formulate an efficient method to compute high-quality cross fields on curved surfaces by extending the diffusion generated method from Viertel and Osting (SISC, 2019). Next, we introduce a method for accurately computing the trajectory of streamlines through singular triangles that prevents tangential crossings. Finally, we introduce a robust method to produce coarse quad layouts by simplifying the partitions obtained via naive separatrix tracing. Our methods are tested on a database of 100 objects and the results are analyzed. The algorithm performs well both in terms of efficiency and visual results on the database when compared to state-of-the-art methods.

cs.CG↗

arXiv2017

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

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.

cs.CG↗