Natural Graph Wavelet Packet Dictionaries

Abstract

We introduce a set of novel multiscale basis transforms for signals on graphs that utilize their “dual” domains by incorporating the “natural” distances between graph Laplacian eigenvectors, rather than simply using the eigenvalue ordering. These basis dictionaries can be seen as generalizations of the classical Shannon wavelet packet dictionary to arbitrary graphs, and do not rely on the frequency interpretation of Laplacian eigenvalues. We describe the algorithms (involving either vector rotations or orthogonalizations) to construct these basis dictionaries, use them to efficiently approximate graph signals through the best basis search, and demonstrate the strengths of these basis dictionaries for graph signals measured on sunflower graphs and street networks.

eGHWT: The Extended Generalized Haar-Walsh Transform

“…Another major contribution of our work is the software package we have developed. Based on the MTSG toolbox written in MATLAB by Je Irion [16], we have developed the MultiscaleGraphSignalTransforms.jl package [13] written entirely in the Julia programming language [1], which includes the new eGHWT implementation for 1D and 2D signals as well as the natural graph wavelet packet dictionaries that our group has recently developed [4]. We hope that interested readers will download the software themselves, and conduct their own experiments with it: https://github.com/UCD4IDS/MultiscaleGraphSignalTransforms.jl.…”

“…We note that our performance comparison is to emphasize the difference between the eGHWT and its close relatives. Hence we will not compare the performance of the eGHWT with those graph wavelets and wavelet packets of di erent nature; see, e.g., [4,18] for further information.…”

“…The relabeled GHWT basis vectors { , } generated by Algorithm 2; a nonnegative function ∶ ℝ → ℝ ≥0 to evaluate the cost of each expansion coe cient; an input graph signal ∈ ℝ . Output: The eGHWT best basis .process is similar to nding the best tilings for dyadic rectangles from those with half size in Thiele-Villemoes algorithm[45] as described in Sect 4…”

Two Channel Filter Banks on Arbitrary Graphs with Positive Semi Definite Variation Operators

“…Earlier filter bank designs for arbitrary graphs are difficult to invert (e.g., least squares reconstruction is needed) [63]- [65]. More recent multiresolution representations fail to be simultaneously perfect reconstruction and orthogonal [52], while others, require full eigendecomposition [53], [66]. Existing approaches that are valid for arbitrary graphs have several disadvantages.…”

“…On the one hand, approximation-based methods may reduce signal representation quality while also requiring additional computational resources (to select the best bipartite approximation) [22]- [24], [60]. On the other hand, methods that allow the original graph to be used can do so at the expense of other desirable features, such as low complexity [53], [66], perfect reconstruction or orthogonality [52], [62] (see Table I for a comparison of some of these approaches). Therefore, a theoretical formulation leading to critically sampled filter banks for arbitrary graphs, without the aforementioned disadvantages, can be an attractive alternative.…”