Localized Manifold Harmonics for Spectral Shape Analysis

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 Lapl...

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Bibliographic Details
Published in:Computer graphics forum Vol. 37; no. 6; pp. 20 - 34
Main Authors: Melzi, S., Rodolà, E., Castellani, U., Bronstein, M. M.
Format: Journal Article
Language:English
Published: Oxford Blackwell Publishing Ltd 01.09.2018
Subjects:
3D shape matching
Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modelling—Shape Analysis, 3D Shape Matching, Geometric Modelling
computational geometry
Computer graphics
Eigenvectors
Fourier analysis
Manifolds (mathematics)
methods and applications
modelling
signal processing
ISSN:0167-7055, 1467-8659
Online Access:Get full text
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Summary: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. 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.
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ISSN:0167-7055
1467-8659
DOI:10.1111/cgf.13309