Compact Support Biorthogonal Wavelet Filterbanks for Arbitrary Undirected Graphs

This paper extends previous results on wavelet filterbanks for data defined on graphs from the case of orthogonal transforms to more general and flexible biorthogonal transforms. As in the recent work, the construction proceeds in two steps: first we design one-dimensional two-channel filterbanks on bipartite graphs, and then extend them to multi-dimensional separable two-channel filterbanks for arbitrary graphs via a bipartite sub-graph decomposition. We specifically design wavelet filters based on the spectral decomposition of the graph, and state sufficient conditions for the filterbanks to be perfect reconstruction and orthogonal. While our previous designs, referred to as graph-QMF filterbanks, are perfect reconstruction and orthogonal, they are not exactly k-hop localized, i.e., the computation at each node is not localized to a small k-hop neighborhood around the node. In this paper, we relax the condition of orthogonality to design a biorthogonal pair of graph-wavelets that are k-hop localized with compact spectral 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.