Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space

TL;DR

This paper develops graph signal processing tools based on the Edge-Laplacian for analyzing and denoising flow signals on graph edges, addressing limitations of traditional Laplacian methods.

Contribution

It introduces Edge-Laplacian-based filters for flow-conservation and combines them with classical methods, advancing edge signal processing in graphs.

Findings

Edge-Laplacian filters enforce flow-conservation.

Combined filtering improves denoising of traffic flow data.

Tools demonstrated on synthetic traffic data in London.

Abstract

This paper focuses on devising graph signal processing tools for the treatment of data defined on the edges of a graph. We first show that conventional tools from graph signal processing may not be suitable for the analysis of such signals. More specifically, we discuss how the underlying notion of a `smooth signal' inherited from (the typically considered variants of) the graph Laplacian are not suitable when dealing with edge signals that encode a notion of flow. To overcome this limitation we introduce a class of filters based on the Edge-Laplacian, a special case of the Hodge-Laplacian for simplicial complexes of order one. We demonstrate how this Edge-Laplacian leads to low-pass filters that enforce (approximate) flow-conservation in the processed signals. Moreover, we show how these new filters can be combined with more classical Laplacian-based processing methods on the line-graph. Finally, we illustrate the developed tools by denoising synthetic traffic flows on the London street network.