Perfect Reconstruction Two-Channel Filter Banks on Arbitrary Graphs Based on an Optimization Model

TL;DR

This paper introduces a novel method for constructing perfect reconstruction two-channel filterbanks on arbitrary graphs using an optimization approach, enabling effective multi-resolution analysis on complex graph data.

Contribution

It presents a new spectral folding property and an optimization-based framework for designing perfect reconstruction filterbanks on arbitrary graphs, extending beyond bipartite structures.

Findings

Effective filterbanks demonstrated on real-world data

Optimization algorithm achieves near-global optimal solutions

Multi-resolution analysis validates the approach

Abstract

In this paper, we propose the construction of critically sampled perfect reconstruction two-channel filterbanks on arbitrary undirected graphs.Inspired by the design of graphQMF proposed in the literature, we propose a general ``spectral folding property'' similar to that of bipartite graphs and provide sufficient conditions for constructing perfect reconstruction filterbanks based on a general graph Fourier basis, which is not the eigenvectors of the Laplacian matrix. To obtain the desired graph Fourier basis, we need to solve a series of quadratic equality constrained quadratic optimization problems (QECQPs) which are known to be non-convex and difficult to solve. We develop an algorithm to obtain the global optimal solution within a pre-specified tolerance. Multi-resolution analysis on real-world data and synthetic data are performed to validate the effectiveness of the proposed filterbanks.