Oversampled Graph Laplacian Matrix for Graph Filter Banks

Abstract

We describe a method of oversampling signals defined on a weighted graph by using an oversampled graph Laplacian matrix. The conventional method of using critically sampled graph filter banks has to decompose the original graph into bipartite subgraphs, and a transform has to be performed on each subgraph because of the spectral folding phenomenon caused by downsampling of graph signals. Therefore, the conventional method cannot always utilize all edges of the original graph in a single stage transformation. Our method is based on oversampling of the underlying graph itself, and it can append nodes and edges to the graph somewhat arbitrarily. We use this approach to make one oversampled bipartite graph that includes all edges of the original non-bipartite graph. We apply the oversampled graph with the critically sampled graph filter bank or the oversampled one for decomposing graph signals and show the performances on some experiments.

Extending Classical Multirate Signal Processing Theory to Graphs—Part I: Fundamentals

“…Implementation in (b) exists without any restriction on the adjacency matrix. However, A should satisfy (15) in order to utilize the first noble identity for the implementation in (c).…”

<formula formulatype="inline"><tex Notation="TeX">$M$</tex> </formula>-Channel Oversampled Graph Filter Banks

“…However, an inappropriate choice of and/or will cause a performance loss compared to the original GLM. One possible recommendation, studied in our recent papers [21], [22], is that is to be (a part of) duplicated copy of and is determined according to the edges in the original graph.…”

“…The original bipartite graph is oversampled by adding the nodes just above and and connecting the appended nodes with corresponding original nodes. The effective oversampling methods for arbitrary (non-bipartite) graphs are studied in [21], [22]. Its strategy is used for the experiments in Section VI of the oversampled graph filter banks proposed in this paper.…”

“…To the best of our knowledge, it is the first attempt at using an oversampled graph Laplacian matrix for graph signal processing. Moreover, our recent work [21]- [23] describes graph expansion techniques in more detail.…”

Half Sampling on Bipartite Graphs

“…We refer the reader to [1,[3][4][5][6][7][8][14][15][16][17][18][19][20]25] for just a sampling of such results. There are also some related works on filter banks and wavelets on graphs, for example [12,13,23,24]. In general there is a space X where functions are defined and a sampling set E ⊆ X where the values of a function are given.…”