Scalable $M$-Channel Critically Sampled Filter Banks for Graph Signals

Abstract

We investigate a scalable M -channel critically sampled filter bank for graph signals, where each of the M filters is supported on a different subband of the graph Laplacian spectrum. For analysis, the graph signal is filtered on each subband and downsampled on a corresponding set of vertices. However, the classical synthesis filters are replaced with interpolation operators. For small graphs, we use a full eigendecomposition of the graph Laplacian to partition the graph vertices such that the m th set comprises a uniqueness set for signals supported on the m th subband. The resulting transform is critically sampled, the dictionary atoms are orthogonal to those supported on different bands, and graph signals are perfectly reconstructable from their analysis coefficients. We also investigate fast versions of the proposed transform that scale efficiently for large, sparse graphs. Issues that arise in this context include designing the filter bank to be more amenable to polynomial approximation, estimating the number of samples required for each band, performing non-uniform random sampling for the filtered signals on each band, and using efficient reconstruction methods. We empirically explore the joint vertex-frequency localization of the dictionary atoms, the sparsity of the analysis coefficients for different classes of signals, the reconstruction error resulting from the numerical approximations, and the ability of the proposed transform to compress piecewise-smooth graph signals. The proposed filter bank also yields a fast, approximate graph Fourier transform with a coarse resolution in the spectral domain.

A literature survey of matrix methods for data science

“…It is clear that polynomial filters j are easy to apply either directly, by multiplying with the matrices, or via an (approximate) eigendecomposition of the Laplacian. Many filters and efficient methods for their computations have been suggested and we refer to [6,138,161,180,183,253,293,312] for some of them.…”

A Modified Spline Graph Filter Bank

“…To compare the complexity of our proposed SGFBSS with other existing solutions in [13], [15], [20], [22], we focus on filtering and sampling. Both the spectral approaches of SGFBSS and the method in [20] have the same complexity for filtering in the analysis section and sampling.…”

“…They applied local GFT to each subgraph and obtained a GFB with PR property (SubGFB). An M -channel critically sampled GFB (CSFB) on arbitrary graphs was introduced in [13], where the synthesis filters in each subband were replaced with interpolation operators. Authors in [14] proposed a critical sampling method for two-channel filter bank on an arbitrary graph where the PR condition was only satisfied for bipartite graphs.…”

On the Use of Vertex-Frequency Analysis for Anomaly Detection in Graph Signals

“…1 depicts the approximation of cubic splines, Meyer kernels, Hann kernels, and ideal filters by a 50 th -order Chebyshev polynomial. Ideal filters are approximated by the Jackson-Chebyshev polynomial [16], which is able to cancel the ripples at the cost of expanding the transition band. This approximation is useful to enhance the stop-band attenuation.…”