Seamless Parametrization with Arbitrary Cones for Arbitrary Genus

Abstract

Fig. 1. Overview of our method's stages: a) Cutgraph on a surface, consisting of handle loops, connectors, and one additional path. b) Conformal parametrization which maps the cutgraph's branches to axis-aligned straight segments in the parametric domain. This map is only rotationally seamless. c) This map modified to be fully seamless by map padding; notice that this map, though locally highly distorted, is actually seamless across the red cutgraph. d) The final map optimized for low isometric distortion, starting from the valid map in (c). Zoom-ins that more clearly expose the effect of padding are shown in Figure 2.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.

Combinatorial Construction of Seamless Parameter Domains

“…Note that in the process of metapolygon construction, we have no explicit control over the final side lengths. A very similar obstacle was described in [CSZZ19], where conformal map domains have analogous scale incompatibilities. In that work a padding technique is described to modify a parametrization using stretch and shift maps, equalizing the lengths of identified domain side.…”

“…This strategy, unfortunately, cannot strictly preserve the prescribed singularity configuration in certain complicated cases. The recent method in [CSZZ19], which also handles the general case, does strictly respect the prescribed singularities. It employs a convex optimization problem—however, a nonlinear one (discrete conformal mapping), bringing about potential numerical challenges in practice.…”

“…Recently, an algorithm relying on convex optimization, based on discrete conformal mapping, was proposed in [CSZZ19]. This algorithm ensures that prescribed cones are respected in a fully general setting.…”

Deep geometric texture synthesis

“…Naively, mapping a topological disc with boundary onto a manifold without boundaries, necessarily yields noticeable seams, where the boundaries are mapped and form discontinuities. Various works dealing with special textures with symmetries have developed continuous seamless mappings between closed surfaces (i.e., no boundaries) which have compatible genus Knöppel et al 2015;Campen et al 2018].…”

Practical Fabrication of Discrete Chebyshev Nets

“…Curl correction is proposed to optimize for integrability [RLL*06]. Since frame fields are in general regarded as the references for the gradients of parameterization coordinate functions, methods for computing global seamless parameterizations also seek integrable fields [FL16, CSZZ19, MZ13, MPZ14, BCW17]. However, these frame fields may deviate from the unit‐length target.…”