A Time-Vertex Signal Processing Framework: Scalable Processing and Meaningful Representations for Time-Series on Graphs

An emerging way to deal with high-dimensional noneuclidean data is to assume that the underlying structure can be captured by a graph. Recently, ideas have begun to emerge related to the analysis of time-varying graph signals. This paper aims to elevate the notion of joint harmonic analysis to a ful...

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Published in:	IEEE transactions on signal processing Vol. 66; no. 3; pp. 817 - 829
Main Authors:	Grassi, Francesco, Loukas, Andreas, Perraudin, Nathanael, Ricaud, Benjamin
Format:	Journal Article
Language:	English
Published:	IEEE 01.02.2018
Subjects:	Fourier transforms graph signal processing Graphical models Harmonic analysis Laplace equations partial differential equations Signal processing algorithms Time-vertex signal processing
ISSN:	1053-587X, 1941-0476
Online Access:	Get full text
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Abstract	An emerging way to deal with high-dimensional noneuclidean data is to assume that the underlying structure can be captured by a graph. Recently, ideas have begun to emerge related to the analysis of time-varying graph signals. This paper aims to elevate the notion of joint harmonic analysis to a full-fledged framework denoted as time-vertex signal processing, that links together the time-domain signal processing techniques with the new tools of graph signal processing. This entails three main contributions: a) We provide a formal motivation for harmonic time-vertex analysis as an analysis tool for the state evolution of simple partial differential equations on graphs; b) we improve the accuracy of joint filtering operators by up-to two orders of magnitude; c) using our joint filters, we construct time-vertex dictionaries analyzing the different scales and the local time-frequency content of a signal. The utility of our tools is illustrated in numerous applications and datasets, such as dynamic mesh denoising and classification, still-video inpainting, and source localization in seismic events. Our results suggest that joint analysis of time-vertex signals can bring benefits to regression and learning.
AbstractList	An emerging way to deal with high-dimensional noneuclidean data is to assume that the underlying structure can be captured by a graph. Recently, ideas have begun to emerge related to the analysis of time-varying graph signals. This paper aims to elevate the notion of joint harmonic analysis to a full-fledged framework denoted as time-vertex signal processing, that links together the time-domain signal processing techniques with the new tools of graph signal processing. This entails three main contributions: a) We provide a formal motivation for harmonic time-vertex analysis as an analysis tool for the state evolution of simple partial differential equations on graphs; b) we improve the accuracy of joint filtering operators by up-to two orders of magnitude; c) using our joint filters, we construct time-vertex dictionaries analyzing the different scales and the local time-frequency content of a signal. The utility of our tools is illustrated in numerous applications and datasets, such as dynamic mesh denoising and classification, still-video inpainting, and source localization in seismic events. Our results suggest that joint analysis of time-vertex signals can bring benefits to regression and learning.
Author	Grassi, Francesco Ricaud, Benjamin Loukas, Andreas Perraudin, Nathanael
Author_xml	– sequence: 1 givenname: Francesco orcidid: 0000-0002-2360-0354 surname: Grassi fullname: Grassi, Francesco email: francesco.grassi@polito.it organization: Politecnico di Torino, Torino, Italy – sequence: 2 givenname: Andreas surname: Loukas fullname: Loukas, Andreas email: andreas.loukas@epfl.ch organization: Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland – sequence: 3 givenname: Nathanael orcidid: 0000-0001-8285-1308 surname: Perraudin fullname: Perraudin, Nathanael email: nathanael.perraudin@sdsc.ethz.ch organization: Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland – sequence: 4 givenname: Benjamin surname: Ricaud fullname: Ricaud, Benjamin email: benjamin.ricaud@epfl.ch organization: Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
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