Abstract
The work by Narang and Ortega [“Perfect reconstruction two-channel wavelet filter banks for graph structured data,” IEEE Trans. Signal Process. , vol. 60, no. 6, pp. 2786–2799, Jun. 2012], [“Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs,” IEEE Trans. Signal Process. , 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 $\text{DSP}_G$ framework of Sandryhaila and Moura [“Discrete signal processing on graphs,” IEEE Trans. Signal Process. , vol. 61, no. 7, pp. 1644–1636, Apr. 2013], [“Discrete signal processing on graphs: Frequency analysis,” IEEE Trans. Signal Process. , 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.
Highlights
There has been great interest amongst signal processing researchers to develop techniques for processing signal defined on graphs, i.e. graph signal processing (GSP)
One view of GSP is that it is a merging of graph theory with concepts and techniques from regular domain signal processing
Spectral graph theory [7] provides a natural extension of the notion of frequency and frequency domain to the spectral domain for undirected graph signals [5]
Summary
There has been great interest amongst signal processing researchers to develop techniques for processing signal defined on graphs, i.e. graph signal processing (GSP). The main contributions are: i) a new polyphase framework for the analysis of the GFB; ii) based on this new framework, a derivation of filtering structures in the downsampled domain (filtering after downsampling) that are equivalent to filtering operations in the original graph (filtering before downsampling); leading to significantly reduced implementation complexity; iii) a study of the connections between the characteristics of the downsampled graph, and its corresponding filters and signals, and those of the original graph, filters and signals, which sheds some light on the relationship between original and downsampled domain, and could be exploited to design improved multirate systems; and iv) a generalization of the approaches in [1], [2] to directed graphs, and other graph matrices, as well as a link to the DSPG framework of Sandryhaila and Moura [3], [4].
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