Multiscale Discrete Framelet Transform for Graph-Structured Signals

Abstract

Graph-structured signal enables rich description of data defined in the domain with irregular structure, which has seen its rapid growth in many applications including social, energy, transportation, sensor, neuronal networks, and many others. This paper aims at generalizing discrete framelet transform defined for regular grids in Euclidean space to finite undirected weighted graphs. By leveraging the intuition from classic framelet transform for signals on regular grids, we proposed an approach for constructing multiscale undecimal framelet transform for signals defined on finite graphs with a perfect reconstruction property. The proposed method is based on the definition of basic blocks involved in framelet transform, including graph shift operator, convolution, and band-limited down/up-sampling. These blocks enable a painless construction of a class of multilevel undecimal framelet transforms in vertex domain by directly calling wavelet filter banks of existing wavelet biframes and tight frames. The proposed discrete framelet transform on graphs keeps most desired properties of its counterpart on regular grids, and one can see its usage in applications.