Perfect Reconstruction Two-Channel Filter Banks on Arbitrary Graphs

TL;DR

This paper generalizes perfect reconstruction two-channel filter banks from bipartite to arbitrary graphs, providing theoretical foundations, design algorithms, and analyzing their properties.

Contribution

It extends the theory of two-channel filter banks to non-bipartite graphs, including new design algorithms and analysis of their properties.

Findings

Established the frame of filter banks on arbitrary graphs.

Derived conditions for perfect reconstruction.

Designed algorithms for orthogonal and biorthogonal filter banks.

Abstract

This paper extends the existing theory of perfect reconstruction two-channel filter banks from bipartite graphs to non-bipartite graphs. By generalizing the concept of downsampling/upsampling we establish the frame of two-channel filter bank on arbitrary connected, undirected and weighted graphs. Then the equations for perfect reconstruction of the filter banks are presented and solved under proper conditions. Algorithms for designing orthogonal and biorthogonal banks are given and two typical orthogonal two-channel filter banks are calculated. The locality and approximation properties of such filter banks are discussed theoretically and experimentally.