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

Abstract

We study the design of filter banks for signals defined on the nodes of graphs. We propose novel two channel filter banks, that can be applied to arbitrary graphs, given a positive semi definite variation operator, while using downsampling operators on arbitrary vertex partitions. The proposed filter banks also satisfy several desirable properties, including perfect reconstruction, and critical sampling, while having efficient implementations. Our results generalize previous approaches only valid for the normalized Laplacian of bipartite graphs. We consider graph Fourier transforms (GFTs) given by the generalized eigenvectors of the variation operator. This GFT basis is orthogonal in an alternative inner product space, which depends on the choices of downsampling sets and variation operators. We show that the spectral folding property of the normalized Laplacian of bipartite graphs, at the core of bipartite filter bank theory, can be generalized for the proposed GFT if the inner product matrix is chosen properly. We give a probabilistic interpretation to the proposed filter banks using Gaussian graphical models. We also study orthogonality properties of tree structured filter banks, and propose a vertex partition algorithms for downsampling. We show that the proposed filter banks can be implemented efficiently on 3D point clouds, with hundreds of thousands of points (nodes), while also improving the color signal representation quality over competing state of the art approaches.

Graph Filters for Signal Processing and Machine Learning on Graphs

“…The generalized critically-sampled filter banks of [145] take the filtering basis vectors {v i } i=1,2,...,N to be the solutions to the generalized eigenvalue problem…”

“…For example, in the classical logarithmic wavelet filter bank, at each level, another filter bank is applied to the downsampled output of the lowpass channel from the previous level [134]. Numerous works have investigated extensions to multi-level filter banks, lifting transforms, and pyramids for graph signals (e.g., [145], [146], [151], [152], [164], [167], [170]). In classical time series analysis or image processing, the structure of the underlying domain enables regular sampling (e.g., every other time sample) that preserves the notion of frequency entailed by filtering at each level of the multi-level filter bank.…”

“…This filter bank combines the perfect reconstruction biorthogonal filters of[144, Ex. 3] with the generalized filter bank approach of[145] for arbitrary graphs. The graph is partitioned into two approximately equal-sized complementary sets of vertices, V 1 and V 2 .…”

Multi-Channel Sampling on Graphs and Its Relationship to Graph Filter Banks

“…Sampling of graph signals is one of the central research topics in GSP [2]. Sampling theory for graph signals, hereafter we call it graph sampling theory, can be applied to various applications, including sensor placement [3], traffic monitoring [4], semi-supervised learning [5], [6], and graph filter bank designs [7]- [17].…”

“…One can notice that MCS is related to filter banks. In fact, sampling of full-band graph signals has been studied in a different line of research: Graph filter bank (GFB) designs [7]- [17]. GFBs are composed of multiple (typically low-and high-pass) graph filters and down-and up-sampling operators, which are also components in MCS.…”