Efficient and Robust Seamless Parametrization

Seamless surface parametrizations support many important geometry processing tasks, including quad meshing, texture mapping, and patch-based surface representations. Recent intrinsic approaches based on Penner coordinates provide a robust mechanism for constructing feasible, seamless parametrizations with prescribed topological structure, determined by a guiding field on a surface and detected feature lines, overcoming many robustness limitations of previous methods. However, these methods rely on highly accurate nonlinear solves, which present a significant obstacle to scalability on large, densely sampled meshes. We address this challenge by observing that Penner-coordinate methods are primarily needed to establish an initial feasible solution whose structure depends on overall guiding field topology and surface topology rather than fine geometric detail, and that high-precision optimization is unnecessary for achieving high-quality results at full resolution, as long as the coarse initial solution can be transferred to the fine levels, while maintaining constraints. We propose a three-scale pipeline that combines an efficient hierarchical approach with robust Penner-coordinate methods. Our method computes guiding fields and holonomy on a medium-resolution mesh, solves the Penner-coordinate feasibility problem efficiently on a coarse mesh that permits connectivity changes, and transfers the resulting structure to the original fine mesh, where only a final, scalable optimization and quantization are performed. Experiments on large benchmark and real-world models demonstrate substantial speedups with no change in robustness in the processing of meshes with millions of triangles.