Nonsubsampled Graph Filter Banks: Theory and Distributed Algorithms

In this paper, we consider nonsubsampled graph filter banks (NSGFBs) to process data on a sparse graph. The analysis filter banks of NSGFBs have small bandwidth, pass/block the normalized constant signal, and have stability on ℓ². Given an analysis filter bank with small bandwidth, we introduce algebraic and optimization methods to construct well-localized synthesis filter banks such that the corresponding NSGFBs provide a perfect signal reconstruction in the noiseless setting. We also prove that the proposed NSGFBs can control the resonance effect in the presence of bounded noise and they can limit the influence of shot noise primarily to a small neighborhood near its location on the graph. We later introduce an iterative algorithm to implement the proposed NSGFBs in a distributed manner, and develop an NSGFB-based denoising technique which is demonstrated to have satisfactory performance on noise suppression.