Localized Manifold Harmonics for Spectral Shape Analysis

Self Cite

We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.

V false(x, y false) = \{\begin{matrix} left ν & left μ false(x, y false) \approx 0; \\ left 0 & left otherwise . \end{matrix}

where

ν \geq 1

.…”

Section: Spectral Map Processingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Functional Maps Representation On Product Manifolds

Rodolà

Lähner²,

Bronstein

et al. 2019

Self Cite

“…The Schrödinger operator augments the Laplacian with a potential, which can be specifically designed for the different applications. A related construction is introduced by Melzi et al [MRCB18]. Though we focus the presentation on the Laplace‐Beltrami operator, our our approach can be used for fast approximations of the spectra and eigen‐functions for these operators as well.…”

Section: Related Workmentioning

confidence: 99%

Fast Approximation of Laplace‐Beltrami Eigenproblems

Nasikun

Brandt

Hildebrandt

2018

The spectrum and eigenfunctions of the Laplace‐Beltrami operator are at the heart of effective schemes for a variety of problems in geometry processing. A burden attached to these spectral methods is that they need to numerically solve a large‐scale eigenvalue problem, which results in costly precomputation. In this paper, we address this problem by proposing a fast approximation algorithm for the lowest part of the spectrum of the Laplace‐Beltrami operator. Our experiments indicate that the resulting spectra well‐approximate reference spectra, which are computed with state‐of‐the‐art eigensolvers. Moreover, we demonstrate that for different applications that comparable results are produced with the approximate and the reference spectra and eigenfunctions. The benefits of the proposed algorithm are that the cost for computing the approximate spectra is just a fraction of the cost required for numerically solving the eigenvalue problems, the storage requirements are reduced and evaluation times are lower. Our approach can help to substantially reduce the computational burden attached to spectral methods for geometry processing.

“… Compute a small set of k 1 << n 1 and k 2 << n 2 basis functions on each shape. The most common choice consists in using the first k eigenfunctions of the Laplace‐Beltrami operator of each shape, although other bases derived from the Hamiltonian operator [CSBK18] and more localized basis functions [NVT*14, MRCB18] have also been used. Compute a set of descriptor functions on each shape, that are expected to be approximately preserved by the unknown map. Store their coefficients in the corresponding bases as columns of matrices A 1 , A 2 . Compute the optimal functional map C by solving the following optimization problem:

where the first term aims at the descriptor preservation: E desc ( C 12 ) = || C 12 A 1 – A 2 || 2 , whereas the second term regularizes the map by promoting the correctness of its overall structural properties.…”

Section: Introductionmentioning

confidence: 99%

Structured Regularization of Functional Map Computations

Ren

Panine

Wonka

et al. 2019

We consider the problem of non‐rigid shape matching using the functional map framework. Specifically, we analyze a commonly used approach for regularizing functional maps, which consists in penalizing the failure of the unknown map to commute with the Laplace‐Beltrami operators on the source and target shapes. We show that this approach has certain undesirable fundamental theoretical limitations, and can be undefined even for trivial maps in the smooth setting. Instead we propose a novel, theoretically well‐justified approach for regularizing functional maps, by using the notion of the resolvent of the Laplacian operator. In addition, we provide a natural one‐parameter family of regularizers, that can be easily tuned depending on the expected approximate isometry of the input shape pair. We show on a wide range of shape correspondence scenarios that our novel regularization leads to an improvement in the quality of the estimated functional, and ultimately pointwise correspondences before and after commonly‐used refinement techniques.