Wavelets on graphs via spectral graph theory

• 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; 1096-603X
• DOI: 10.1016/j.acha.2010.04.005

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.