Windowed Fourier Analysis for Signal Processing on Graph Bundles

TL;DR

This paper introduces a method for representing signals on graph bundles using windowed Fourier analysis, enabling localized signal decomposition on complex graph structures with applications in stereochemistry.

Contribution

It presents a novel approach to analyze signals on graph bundles by leveraging their local product structure and partition of unity, extending Fourier analysis techniques to these complex graphs.

Findings

Effective signal representation on graph bundles demonstrated

Application to stereochemistry energy landscape analysis shown

Basis construction for component signals achieved

Abstract

We consider the task of representing signals supported on graph bundles, which are generalizations of product graphs that allow for "twists" in the product structure. Leveraging the localized product structure of a graph bundle, we demonstrate how a suitable partition of unity over the base graph can be used to lift the signal on the graph into a space where a product factorization can be readily applied. Motivated by the locality of this procedure, we demonstrate that bases for the signal spaces of the components of the graph bundle can be lifted in the same way, yielding a basis for the signal space of the total graph. We demonstrate this construction on synthetic graphs, as well as with an analysis of the energy landscape of conformational manifolds in stereochemistry.