Compact Support Biorthogonal Wavelet Filterbanks for Arbitrary Undirected Graphs

TL;DR

This paper introduces a biorthogonal wavelet filterbank design for arbitrary undirected graphs that achieves perfect reconstruction and compact support, improving upon previous orthogonal designs by allowing localized wavelets.

Contribution

It proposes a biorthogonal wavelet filterbank construction for arbitrary graphs that relaxes orthogonality constraints, enabling compact support and perfect reconstruction.

Findings

Filterbanks are effective for graph signal processing.

Proposed design achieves compact support and perfect reconstruction.

Preliminary results show practical utility for graph signals.

Abstract

In our recent work, we proposed the design of perfect reconstruction orthogonal wavelet filterbanks, called graph- QMF, for arbitrary undirected weighted graphs. In that formulation we first designed "one-dimensional" two-channel filterbanks on bipartite graphs, and then extended them to "multi-dimensional" separable two-channel filterbanks for arbitrary graphs via a bipartite subgraph decomposition. We specifically designed wavelet filters based on the spectral decomposition of the graph, and stated necessary and sufficient conditions for a two-channel graph filter-bank on bipartite graphs to provide aliasing-cancellation, perfect reconstruction and orthogonal set of basis (orthogonality). While, the exact graph-QMF designs satisfy all the above conditions, they are not exactly k-hop localized on the graph. In this paper, we relax the condition of orthogonality to design a biorthogonal pair of graph-wavelets that can have compact spatial spread and still satisfy the perfect reconstruction conditions. The design is analogous to the standard Cohen-Daubechies-Feauveau's (CDF) construction of factorizing a maximally-flat Daubechies half-band filter. Preliminary results demonstrate that the proposed filterbanks can be useful for both standard signal processing applications as well as for signals defined on arbitrary graphs. Note: Code examples from this paper are available at http://biron.usc.edu/wiki/index.php/Graph Filterbanks