Perfect Matching Quad Layouts for Manifold Meshes

Abstract

This paper introduces a new approach to automatically generate pure quadrilateral patch layouts on manifold meshes. The algorithm is based on a careful construction of a singularity graph of a given input frame field or a given periodic global parameterization. A pure quadrilateral patch layout is then derived as a constrained minimum weight perfect matching of that graph. The resulting layout is optimal relative to a balance between coarseness and geometric feature alignment. We formulate the problem of finding pure quadrilateral patch layouts as a global optimization problem related to a well‐known concept in graph theory. The main advantage of the new method is its simplicity and its computation speed. Patch layouts generated by the present algorithm are high quality and are very competitive compared to current state of the art.

Tracing Field‐Coherent Quad Layouts

“…The approaches proposed in [RRP15,ZZY16] are most related to our proposal because they share the idea of finding the best subset of separatrices that produces a quad layout by resolving a global 0-1 programming problem. However our method has several noticeable advantages: Less dependence on input data: The approach of [RRP15] needs as input a globally smooth parametrization without foldovers.…”

“…Simpler tracing: Designing a robust algorithm to trace paths in parametric space (as in [RRP15]) is a challenging task, especially in the presence of a complex parametric domain. Better and more controllable result: The quad layouts of [RRP15,ZZY16] can be arbitrarily misaligned with respect to the original cross field. As opposite, we trace field-aligned separatrices directly on the triangle mesh by using a new graph-based strategy.…”

“…Methods generating a globally smooth parametrization usually introduce discontinuities (jumps) and overlapping regions in parametric space, increasing the complexity of the tracing procedure. More flexible: Both methods in [RRP15,ZZY16] require to trace the complete set of candidate separatrices to guarantee a feasible solution. Better and more controllable result: The quad layouts of [RRP15,ZZY16] can be arbitrarily misaligned with respect to the original cross field.…”

“…With respect to methods based on global optimization of separatrices [RRP15] we do not need an initial globally smooth parametrization as input, and our formulation allows us to work also with an incomplete graph in input. We compute a super-graph of potential separatrices H, which is reduced to the optimal graph of separatrices by resolving a binary LP problem.…”

FlexMolds

“…Cignoni et al [2014] extended this concept and placed the elements driven by a feature-aligned input cross-field [Ray et al 2009;Bommes et al 2009], a mathematical entity that catches the overall structure and curvature of the object. Inspired by this concept, we compute potential candidates for cuts along smooth polylines that emerge from a field-aligned patch layout decomposition of the initial mesh [Tarini et al 2011;Campen et al 2012;Myles et al 2014;Razafindrazaka et al 2015].…”