Computing locally injective mappings by advanced MIPS

Abstract

Computing locally injective mappings with low distortion in an efficient way is a fundamental task in computer graphics. By revisiting the well-known MIPS (Most-Isometric ParameterizationS) method, we introduce an advanced MIPS method that inherits the local injectivity of MIPS, achieves as low as possible distortions compared to the state-of-the-art locally injective mapping techniques, and performs one to two orders of magnitude faster in computing a mesh-based mapping. The success of our method relies on two key components. The first one is an enhanced MIPS energy function that penalizes the maximal distortion significantly and distributes the distortion evenly over the domain for both mesh-based and meshless mappings. The second is a use of the inexact block coordinate descent method in mesh-based mapping in a way that efficiently minimizes the distortion with the capability not to be trapped early by the local minimum. We demonstrate the capability and superiority of our method in various applications including mesh parameterization, mesh-based and meshless deformation, and mesh improvement.

Voting for Distortion Points in Geometric Processing

“…E MIPS (J t ) reaches its minimum when J t represents a similarity transformation. The AMIPS energy [45] is defined as follows:…”

“…Since the main purpose of finding distortion points is to decrease the distortion in parameterizations, a simple quantitative evaluation method is to conduct parameterizations and then compute the resulting distortion distribution. Specifically, we adopt the isometric distortion metric defined in [45] to evaluate the quality of the parameterizations. Then we report the distortion distribution by offering the maximum (worst case), average, and standard deviations for all triangles, denoted as δ max , δ avg , and δ std , respectively.…”

Isometry‐Aware Preconditioning for Mesh Parameterization

“…Hormann and Greiner [HG00] utilize a single‐vertex‐at‐a‐time update to guarantee injective parameterizations, though such an optimization is slow in practice. Fu et al [FLG15] employ a highly‐parallel version of localized gradient descent to obtain reasonable optimization times. Smith and Schaefer [SS15] use LBFGS coupled with an explicit bound on the line search to guarantee a bijective parameterization.…”

Fast mapping and morphing for genus-zero meshes with cross spherical parameterization

“…Aigerman [8] presented a parameterization method for surface property mapping and achieved about 3.5K triangles/minute. Fu et al [33] parallelized the AMIP method using OpenMP in Cþ þ and had about 150k triangles/minute, and they had 30k triangles/minute in the serialized implementation. Nadeem et al [25] proposed a divide-and-conquer method to maintain a good balance between angle and area distortion, and they achieved about 2.5K triangles/minute for balanced mapping.…”