Bipartite Graph Filter Banks: Polyphase Analysis and Generalization

The work by Narang and Ortega [“Perfect reconstruction two-channel wavelet filter banks for graph structured data,” <italic>IEEE Trans. Signal Process.</italic>, vol. 60, no. 6, pp. 2786–2799, Jun. 2012], [“Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs,” <italic>IEEE Trans. Signal Process.</italic>, vol. 61, no. 19, pp. 4673–4685, Oct. 2013] laid the foundations for the two-channel critically sampled perfect reconstruction filter bank for signals defined on undirected graphs. This basic filter bank is applicable only to bipartite graphs but using the notion of separable filtering, the basic filter bank can be applied to any arbitrary undirected graphs. In this paper, several new theoretical results are presented. In particular, the proposed polyphase analysis yields filtering structures in the downsampled domain that are equivalent to those before downsampling and, thus, can be exploited for efficient implementation. These theoretical results also provide new insights that can be exploited in the design of these systems. These insights allow us to generalize these filter banks to directed graphs and to using a variety of graph base matrices, while also providing a link to the <inline-formula><tex-math notation="LaTeX">$\text{DSP}_G$</tex-math></inline-formula> framework of Sandryhaila and Moura [“Discrete signal processing on graphs,” <italic>IEEE Trans. Signal Process.</italic>, vol. 61, no. 7, pp. 1644–1636, Apr. 2013], [“Discrete signal processing on graphs: Frequency analysis,” <italic>IEEE Trans. Signal Process.</italic>, vol. 62, no. 12, pp. 3042–3054, Jun. 2014]. Experiments show evidence that better nonlinear approximation and denoising results may be obtained by a better selection of these base matrices.