Compact Support Biorthogonal Wavelet Filterbanks for Arbitrary Undirected Graphs

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.

Downsampling of Signals on Graphs Via Maximum Spanning Trees

“…A (biorthogonal) GWFB [5] transforms a signal living on a connected bipartite graph into wavelet coefficients of the same cardinality (critical sampling) that are localized in both vertexand frequency-domain. Like a classical discrete wavelet transform [18], a GWFB can be achieved by iterating (on the lowpass channels) a two-channel filter bank as shown in Fig.…”

“…For general connected graphs, it is proposed in [4], [5] to decompose the graph into a minimum number of bipartite subgraphs using Harary's algorithm. The two-channel filter bank is then applied separably to each subgraph at each level of the transform.…”

“…Unlike regular domains in classical signal processing, the irregular topology of the underlying graphs, on which the signals are indexed, poses many difficulties for even basic signal operations such as shifting, modulating, and downsampling [1]. The focus of this paper is on the design of efficient downsampling operators and graph multiresolution which are necessary components of any multiscale transforms such as the critical-sampling graph wavelet filter banks (GWFBs) [4], [5].…”

“…For general graphs, finding the best downsampling is equivalent to a max-cut problem which is NP-complete [8] and so intractable for large graphs. In 2012, Narang and Ortega [4] introduced the coloringbased downsampling as a component of the GWFBs, which are then subsequently developed in [5]. In this approach, the original graph is first decomposed into a sequence of bipartite subgraphs based on the graph coloring.…”

Detecting Localized Categorical Attributes on Graphs

“…It extends classical signal processing concepts such as signals, filters, Fourier transform, frequency response, low-and highpass filtering, from signals residing on regular lattices to data residing on general graphs; for example, a graph signal models the data value assigned to each node in a graph. Recent work involves sampling for graph signals [9], [10], [11], [12], recovery for graph signals [13], [14], [15], [16], representations for graph signals [17], [18] principles on graphs [19], [20], stationary graph signal processing [21], [22], graph dictionary construction [23], graph-based filter banks [24], [25], [26], [27], denoising on graphs [24], [28], community detection and clustering on graphs [29], [30], [31], distributed computing [32], [33] and graph-based transforms [34], [35], [36]. We here consider detecting localized categorical attributes on graphs.…”