Applied and computational harmonic analysis on graphs and networks

Jeff Irion; Naoki Saito

doi:10.1117/12.2186921

11 September 2015 Applied and computational harmonic analysis on graphs and networks

Jeff Irion, Naoki Saito

Author Affiliations +

Proceedings Volume 9597, Wavelets and Sparsity XVI; 95971F (2015) https://doi.org/10.1117/12.2186921
Event: SPIE Optical Engineering + Applications, 2015, San Diego, California, United States

Abstract

In recent years, the advent of new sensor technologies and social network infrastructure has provided huge opportunities and challenges for analyzing data recorded on such networks. In the case of data on regular lattices, computational harmonic analysis tools such as the Fourier and wavelet transforms have well-developed theories and proven track records of success. It is therefore quite important to extend such tools from the classical setting of regular lattices to the more general setting of graphs and networks. In this article, we first review basics of graph Laplacian matrices, whose eigenpairs are often interpreted as the frequencies and the Fourier basis vectors on a given graph. We point out, however, that such an interpretation is misleading unless the underlying graph is either an unweighted path or cycle. We then discuss our recent effort of constructing multiscale basis dictionaries on a graph, including the Hierarchical Graph Laplacian Eigenbasis Dictionary and the Generalized Haar-Walsh Wavelet Packet Dictionary, which are viewed as generalizations of the classical hierarchical block DCTs and the Haar-Walsh wavelet packets, respectively, to the graph setting. Finally, we demonstrate the usefulness of our dictionaries by using them to simultaneously segment and denoise 1-D noisy signals sampled on regular lattices, a problem where classical tools have difficulty.

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Jeff Irion and Naoki Saito "Applied and computational harmonic analysis on graphs and networks", Proc. SPIE 9597, Wavelets and Sparsity XVI, 95971F (11 September 2015); https://doi.org/10.1117/12.2186921

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KEYWORDS

Associative arrays

Wavelets

Transform theory

Data modeling

Denoising

Matrices

Radon

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