Inversion-free geometric mapping construction: A survey

Abstract

A geometric mapping establishes a correspondence between two domains. Since no real object has zero or negative volume, such a mapping is required to be inversion-free. Computing inversion-free mappings is a fundamental task in numerous computer graphics and geometric processing applications, such as deformation, texture mapping, mesh generation, and others. This task is usually formulated as a non-convex, nonlinear, constrained optimization problem. Various methods have been developed to solve this optimization problem. As well as being inversion-free, different applications have various further requirements. We expand the discussion in two directions to (i) problems imposing specific constraints and (ii) combinatorial problems. This report provides a systematic overview of inversion-free mapping construction, a detailed discussion of the construction methods, including their strengths and weaknesses, and a description of open problems in this research field.

Hex-Mesh Generation and Processing: a Survey

“…Alternatively to volumetric deformation, one could in principle use a surface method to define a polycube-surface map (e.g., with [Yang et al 2019]) and then solve for a compatible volumetric mapping between the two shapes. Again, however, despite the high level of practical robustness showcased by recent approaches [Du et al 2020;Garanzha et al 2021], the fully reliable automatic generation of constrained volumetric maps without flips remains an open problem [Fu et al 2021] Motivated by this difficulty, an interactive polycube construction pipeline that puts the user in the loop has been recently proposed [Li et al 2021]. Users are allowed extensive control over each stage, such as editing the polycube structure, positioning vertices, and exploring the trade-off among competing quality metrics, while also providing automatic alternatives.…”

Symmetric Volume Maps: Order-Invariant Volumetric Mesh Correspondence with Free Boundary

“…A parameterization is a deformation of a volume to a simpler domain, such as a topological ball Garanzha et al 2021;Paillé and Poulin 2012;Wang et al 2003;Yueh et al 2019] or a polycube [Aigerman and Lipman 2013;Li et al 2021;Paillé and Poulin 2012;Wang et al 2008b;Xia et al 2010]. The better-studied instance of parameterization in graphics maps two-dimensional surfaces (rather than volumes) into the plane; see [Floater and Hormann 2005;Fu et al 2021;Sheffer et al 2007] for discussion of this broad area of research.…”

Metric Optimization in Penner Coordinates

“…These methods are distinguished by the choice of variables, optimization objectives, type of constraints considered, guarantees these methods provide and initialization requirements. We briefly review most closely related work, and refer to recent surveys [Naitsat et al 2021] and [Fu et al 2021] for a more comprehensive review.…”