Scalable Locally Injective Mappings

Abstract

InitializationIteration 20 Iteration 40Figure 1: A locally injective, energy minimizing parameterization of a mesh with over 25 million triangles computed with our algorithm in 80 minutes. The algorithm starts from a highly distorted locally injective initialization and in only 40 iterations, each requiring to solve a sparse linear system, it converges to a highly isometric map that is guaranteed to be free of inverted elements. AbstractWe present a scalable approach for the optimization of flippreventing energies in the general context of simplicial mappings, and specifically for mesh parameterization. Our iterative minimization is based on the observation that many distortion energies can be optimized indirectly by minimizing a simpler proxy energy and compensating for the difference with a reweighting scheme. Our algorithm is simple to implement and scales to datasets with millions of faces. We demonstrate our approach for the computation of maps that minimize a conformal or isometric distortion energy, both in two and three dimensions. In addition to mesh parameterization, we show that our algorithm can be applied to mesh deformation and mesh quality improvement.

Isometry‐Aware Preconditioning for Mesh Parameterization

“…This paper contributes several new ideas to parameterization and related optimization problems in geometry. We stress that our method matches or exceeds the performance of existing methods [KGL16, RPPSH17] while enjoying additional theoretical understanding and a remarkably simple matrix construction. Specific contributions include the following:…”

“…In this paper, we take inspiration from recent preconditioners that accelerate first‐order (gradient‐based) optimization for mesh parameterization [SS15, KGL16, RPPSH17]. We formulate a new preconditioner specifically designed for parameterization problems, using the language of vector field design.…”

“… A positive (semi‐)definite, rotation‐invariant preconditioner for gradient fields that penalizes non‐isometric deformations and is suited to minimization problems involving distortion energies Simple, easy‐to‐code, closed‐form expressions for our preconditioner both in the planar and volumetric cases in terms of basic mesh elements, which can be assembled in parallel without matrix multiplication [SBCBG11a] or per‐element eigenvector/SVD computation [RPPSH17] Interpretation of our preconditioner as the gradient with respect to a natural AKVF‐based metric on the “space of parameterizations” of a mesh with fixed topology Application of our method to mesh parameterization problems, with improved performance over state‐of‐the‐art Application to volumetric problems over tetrahedral meshes …”

Orientation preserving mesh optimisation and preconditioning

“…This means that the preconditioner will fail to give good results if the stability term of the functional becomes dominant.On the other hand, the numerical experience in these situations is that the mesh is already nearly deteriorated if the stability term becomes dominant, which usually only happens if, for example, the boundary deformation is about to produce self‐intersections at the boundary. This indicates problems in whatever is governing the boundary deformation; thus, the arising ill‐conditioning of the mesh optimisation problem is usually the least of all worries. The quadratic proxy distortion measures in the work of Rabinovich et al are quite similar to a preconditioner.…”

“…For velocity‐based methods, successful approaches for generating orientation preserving mesh deformations have been for the purpose of using an attraction/repulsion model for vertices, for geometric conservation law‐based, or for directly specifying the determinant of the deformation gradient, but without being able to control the cell shapes . Examples for location‐based methods are using the (parabolic) Monge‐Ampère equation or the biharmonic equation or modelling the mesh as a hyperelastic material; see also the work of Rabinovich et al for surface mesh parametrisation using similar deformation energies and that of Persson and Peraire for a hyperelasticity‐based approach to higher‐order mesh generation. An in‐depth literature review can be found in the work of Budd et al Generally speaking, most mesh optimisation methods do not enforce the orientation preserving property directly (see the work of Shontz and Vavasis for a combination of a linear method and a postprocessing step) or even identify it as the major culprit, as they are more concerned with optimising the cell size distribution in a very regular domain.…”