Perfect Reconstruction Two-Channel Wavelet Filter-Banks for Graph Structured Data

TL;DR

This paper introduces a novel framework for constructing perfect reconstruction two-channel wavelet filterbanks on arbitrary graphs, enabling advanced signal processing tasks on graph-structured data.

Contribution

It extends classical wavelet concepts to graph signals, introduces graph-QMF filters for bipartite graphs, and proposes a bipartite subgraph decomposition for arbitrary graphs.

Findings

Achieves perfect reconstruction with aliasing cancellation.

Provides conditions for orthogonality and perfect reconstruction.

Uses Chebyshev polynomial approximations for filter implementation.

Abstract

In this work we propose the construction of two-channel wavelet filterbanks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph. These graph based functions are referred to as graph-signals as we build a framework in which many concepts from the classical signal processing domain, such as Fourier decomposition, signal filtering and downsampling can be extended to graph domain. Especially, we observe a spectral folding phenomenon in bipartite graphs which occurs during downsampling of these graphs and produces aliasing in graph signals. This property of bipartite graphs, allows us to design critically sampled two-channel filterbanks, and we propose quadrature mirror filters (referred to as graph-QMF) for bipartite graph which cancel aliasing and lead to perfect reconstruction. For arbitrary graphs we present a bipartite subgraph decomposition which produces an edge-disjoint collection of bipartite subgraphs. Graph-QMFs are then constructed on each bipartite subgraph leading to "multi-dimensional" separable wavelet filterbanks on graphs. Our proposed filterbanks are critically sampled and we state necessary and sufficient conditions for orthogonality, aliasing cancellation and perfect reconstruction. The filterbanks are realized by Chebychev polynomial approximations.