Splines and Wavelets on Circulant Graphs

TL;DR

This paper introduces new wavelet filterbanks for circulant graphs that leverage the graph Laplacian's properties to analyze polynomial signals, with extensions to multi-dimensional and arbitrary graphs.

Contribution

It presents novel higher-order graph spline wavelet filterbanks based on the e-graph Laplacian, enabling polynomial signal analysis on circulant graphs and generalizations via graph products.

Findings

Design of vertex-localized, critically-sampled wavelet filterbanks

Ability to reproduce and annihilate polynomial signals

Extension to multi-dimensional graph structures

Abstract

We present novel families of wavelets and associated filterbanks for the analysis and representation of functions defined on circulant graphs. In this work, we leverage the inherent vanishing moment property of the circulant graph Laplacian operator, and by extension, the e-graph Laplacian, which is established as a parameterization of the former with respect to the degree per node, for the design of vertex-localized and critically-sampled higher-order graph (e-)spline wavelet filterbanks, which can reproduce and annihilate classes of (exponential) polynomial signals on circulant graphs. In addition, we discuss similarities and analogies of the detected properties and resulting constructions with splines and spline wavelets in the Euclidean domain. Ultimately, we consider generalizations to arbitrary graphs in the form of graph approximations, with focus on graph product decompositions. In particular, we proceed to show how the use of graph products facilitates a multi-dimensional extension of the proposed constructions and properties.