Spectral folding and two-channel filter-banks on arbitrary graphs

TL;DR

This paper extends bipartite filter-bank theories to arbitrary graphs using a new graph Fourier transform definition, enabling perfect reconstruction filter-banks for complex graph signals like large 3D point clouds.

Contribution

It introduces a novel spectral folding approach and a new GFT definition that generalizes filter-bank design to all graphs, not just bipartite ones.

Findings

Effective signal representation on large 3D point clouds

Constructed orthogonal and bi-orthogonal perfect reconstruction filter-banks

Enhanced computational efficiency for graph signal processing

Abstract

In the past decade, several multi-resolution representation theories for graph signals have been proposed. Bipartite filter-banks stand out as the most natural extension of time domain filter-banks, in part because perfect reconstruction, orthogonality and bi-orthogonality conditions in the graph spectral domain resemble those for traditional filter-banks. Therefore, many of the well known orthogonal and bi-orthogonal designs can be easily adapted for graph signals. A major limitation is that this framework can only be applied to the normalized Laplacian of bipartite graphs. In this paper we extend this theory to arbitrary graphs and positive semi-definite variation operators. Our approach is based on a different definition of the graph Fourier transform (GFT), where orthogonality is defined with the respect to the Q inner product. We construct GFTs satisfying a spectral folding property, which allows us to easily construct orthogonal and bi-orthogonal perfect reconstruction filter-banks. We illustrate signal representation and computational efficiency of our filter-banks on 3D point clouds with hundreds of thousands of points.