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.

Geometric optimization via composite majorization

“…Furthermore, by exploiting the fixed Hessian approximation (i.e., independent of x) they are able to devise particularly efficient iterations. Rabinovich et al [2017] are inspired by the fast initial progress of global-local iterations and advocate the use of a reweighted Laplacian for the minimization of nonlinear energies on meshes. Their algorithm is shown to be very efficient in minimizing the energy from arbitrary feasible initializations, but it significantly slows down near an optimum, and so additional Newton iterations might be required if an accurate solution is pursued.…”

“…We used our analytic Hessian formula in equation (9) to efficiently compute the indefinite Hessian (by simply dropping the clamping). We observed that computation of Hessians using our analytic formulation works an order of magnitude faster than the automatic differentiation typically used for this task Rabinovich et al 2017]. Scalable Locally Injective Mappings -We implemented the approach of [Rabinovich et al 2017], wherein a modified Laplacian takes the place of the Hessian (SLIM).…”

“…We tested our approach on surface parameterization computed by minimizing the symmetric Dirichlet energy, equation (24). We followed the standard protocol of using Tutte's embedding for computing a bijective initial parameterization [Kovalsky et al 2016;Rabinovich et al 2017;Smith and Schaefer 2015]. In turn, we note that the resulting parameterizations are guaranteed to be locally injective, as we minimize an inversion-resisting energy and accordingly restrict our line-search.…”

“…The figure shows a snapshot of the state each method achieved at the time marked on the graph. Our Composite Majorization (CM-ours) converges faster than Projected Newton [Teran et al 2005], SLIM [Rabinovich et al 2017] and AQP [Kovalsky et al 2016].…”

“…These algorithms take advantage of the particular structure of the energies used in geometry processing and introduce, albeit sometimes implicitly, a convex osculating quadric used to determine their iterations. For instance, [Kovalsky et al 2016;Liu et al 2008;Sorkine and Alexa 2007] essentially replace the Hessian with the mesh Laplacian, further use low-rank quasi-Newton updates to better approximate the Hessian, and [Rabinovich et al 2017] reweigh the Laplacian to improve the effectiveness of their iterations. These are all first order methods, meaning they do not directly use second order derivatives of the energy, and therefore generally fail to achieve high convergence rate, particularly as they approach convergence.…”