Localized Manifold Harmonics for Spectral Shape Analysis

TL;DR

This paper introduces a new method for creating localized spectral bases on shapes, improving over traditional global Laplacian eigenfunctions for shape analysis tasks like approximation and correspondence.

Contribution

It proposes an efficient framework for constructing localized orthogonal bases by modifying the Laplacian, enhancing shape analysis capabilities.

Findings

Significant improvement over classical Laplacian eigenbases.

Effective for shape approximation and correspondence.

Theoretical and computational validation of the new framework.

Abstract

The use of Laplacian eigenfunctions is ubiquitous in a wide range of computer graphics and geometry processing applications. In particular, Laplacian eigenbases allow generalizing the classical Fourier analysis to manifolds. A key drawback of such bases is their inherently global nature, as the Laplacian eigenfunctions carry geometric and topological structure of the entire manifold. In this paper, we introduce a new framework for local spectral shape analysis. We show how to efficiently construct localized orthogonal bases by solving an optimization problem that in turn can be posed as the eigendecomposition of a new operator obtained by a modification of the standard Laplacian. We study the theoretical and computational aspects of the proposed framework and showcase our new construction on the classical problems of shape approximation and correspondence. We obtain significant improvement compared to classical Laplacian eigenbases as well as other alternatives for constructing localized bases.