Wavelets on graphs via spectral graph theory

We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L . Gi...

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Bibliographic Details
Published in:Applied and computational harmonic analysis Vol. 30; no. 2; pp. 129 - 150
Main Authors: Hammond, David K., Vandergheynst, Pierre, Gribonval, Rémi
Format: Journal Article
Language:English
Published: Elsevier Inc 01.03.2011
Elsevier
Subjects:
Computer Science
Engineering Sciences
Graph theory
Overcomplete wavelet frames
Signal and Image Processing
Spectral graph theory
Wavelets
Wavelets
Graph theory
Overcomplete wavelet frames
Spectral graph theory
ISSN:1063-5203, 1096-603X
Online Access:Get full text
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Summary:We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L . Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator T g t = g ( t L ) . The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L . We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.
ISSN:1063-5203
1096-603X
DOI:10.1016/j.acha.2010.04.005