Globally Injective Geometry Optimization with Non‐Injective Steps

Abstract

We present a method to minimize distortion and compute globally injective mappings from non‐injective initialization Many approaches for distortion minimization subject to injectivity constraints require an injective initialization and feasible intermediate states. However, it is often the case that injective initializers are not readily available, and many distortion energies of interest have barrier terms that stall global progress. The alternating direction method of multipliers (ADMM) has recently gained traction in graphics due to its efficiency and generality. In this work we explore how to endow ADMM with global injectivity while retaining the ability to traverse non‐injective iterates. We develop an iterated coupled‐solver approach that evolves two solution states in tandem. Our primary solver rapidly drives down energy to a nearly injective state using a dynamic set of efficiently enforceable inversion and overlap constraints. Then, a secondary solver corrects the state, herding the solution closer to feasibility. The resulting method not only compares well to previous work, but can also resolve overlap with free boundaries.

VOLMAP: a Large Scale Benchmark for Volume Mappings to Simple Base Domains

“…However, explicitly imposing injectivity in this formulation yields cubic constraints, making the numerical problem impossible to optimize with any existing solver. A variety of relaxed formulations that promote injectivity without strictly imposing it have been proposed over the years [AL13, KABL14, KABL15, FL16, SFL19, SLS22, ASGS22, NZZ20,OKN21,POK23,DAZ*20,GKK*21], often showcasing remarkable results even on extremely challenging tests. In Section 8, we tested the authors' implementation of [DAZ*20] on a portion of our dataset: in 34% of the cases it fails to produce an injective map.…”

Metric Optimization in Penner Coordinates

“…For example, [Du et al 2020] introduces a novel total lifted content energy which has the property that its global minimum is an injective embedding, if one exists. [Overby et al 2021] proposes another method in this category. It demonstrates an impressive practical success rate for meshes with fixed connectivity and fixed boundary, but the existence of a solution or that a local minimum is found, is not guaranteed.…”

Practical construction of globally injective parameterizations with positional constraints

“…However, the non-linear intersection-free constraint is not straightforward to handle, as the intersecting boundary edge pairs usually change during optimization. We note that computing intersectionfree and flip-free parameterizations without positional constraints is a popular research topic [10][11][12][13], while optimizing the positional constraints as a soft energy may lead to self-locking situations [14,15].…”