Signal processing on graphs: Transforms and tomograms

TL;DR

This paper extends graph signal processing by developing generalized transforms, including wavelet-like transforms and tomograms, for signals on graphs, providing robust, probabilistic tools for analyzing complex network data.

Contribution

It introduces a framework for constructing wavelet-like transforms and tomograms on graphs, generalizing classical signal processing tools to complex network structures.

Findings

Developed graph wavelet-like transforms based on spectral mappings.

Extended tomogram transforms to signals on arbitrary graphs.

Demonstrated robustness of these transforms in noisy environments.

Abstract

Using projections on the (generalized) eigenvectors associated to matrices that characterize the topological structure, several authors have constructed generalizations of the Fourier transform on graphs. By exploring mappings of the spectrum of these matrices we show how to construct more general transforms, in particular wavelet-like transforms on graphs. For time-series, tomograms, a generalization of the Radon transforms to arbitrary pairs of non-commuting operators, are positive bilinear transforms with a rigorous probabilistic interpretation which provide a full characterization of the signals and are robust in the presence of noise. Here the notion of tomogram transform is also extended to signals on arbitrary graphs