Compressed Manifold Modes for Mesh Processing

Abstract

This paper introduces compressed eigenfunctions of the Laplace‐Beltrami operator on 3D manifold surfaces. They constitute a novel functional basis, called the compressed manifold basis, where each function has local support. We derive an algorithm, based on the alternating direction method of multipliers (ADMM), to compute this basis on a given triangulated mesh. We show that compressed manifold modes identify key shape features, yielding an intuitive understanding of the basis for a human observer, where a shape can be processed as a collection of parts. We evaluate compressed manifold modes for potential applications in shape matching and mesh abstraction. Our results show that this basis has distinct advantages over existing alternatives, indicating high potential for a wide range of use‐cases in mesh processing.

Survey on sparsity in geometric modeling and processing

“…Mesh editting is a quite popular method in geometry modeling. Local deformation is strongly needed to preserve last editting, and thus sparsity regularization is a very suitable tool to achieve this purpose [67,68,71,72].…”

“…Low Rank [69] intuitive, relatively robust Tensor Rank [70] capturing global symmetry, relatively robust Other Applications CMM [71] local controllability LBC [72] local controllability Skeleton Extraction [73] robust to noise and outlier 3D Printing [74] reduce the material largely LRSCPK [75] sharp feature preserving Point Cloud Compression [76] high compression ratio, robust to noise sharp feature preserving, robust to noise and outlier, and Reconstruction [16] unifying geometry and connectivity (2). Instead, Zhang et al [15] adopt the sparsity of face normals differences and propose a two-phase method including f ace normal filtering and vertex updating.…”

“…But as we have known that many times locality may induce better result, like deformation [72] in Section 3.5.1. To produce an intrinsic shape basis with local spatial support and take advantage of MHB, Neumann et al [71] propose the compressed manif old basis (CMB) obtained with a sparsity inducing 1 norm of the the individual basis functions {φ k } as following:…”

“…As such, it can be used for shape matching which involves robust feature detection. [71]. The proposed compressed manifold modes (CMMs) have local support and are confined to specific local features like protrusions and ridges.…”

Adaptive multi‐scale analysis for point‐based surface editing

“…However, this does not permit to directly edit a precise feature at a given scale as such a task requires to determine the respective set of eigenfunctions. Despite a very recent effort in that direction [NVT*14], this still constitute a challenging problem.…”