Table Of Content

Graph Spectral Image Processing SCIENCES Image, Field Director – Laure Blanc-Feraud Compression, Coding and Protection of Images and Videos, Subject Head – Christine Guillemot Graph Spectral Image Processing Coordinated by Gene Cheung Enrico Magli First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2021 The rights of Gene Cheung and Enrico Magli to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2021932054 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78945-028-6 ERC code: PE7 Systems and Communication Engineering PE7_7 Signal processing Contents IntroductiontoGraphSpectralImageProcessing . . . . . . . . . . . . xi GeneCHEUNGandEnricoMAGLI Part1.FundamentalsofGraphSignalProcessing . . . . . . . . . . . 1 Chapter1.GraphSpectralFiltering . . . . . . . . . . . . . . . . . . . . . 3 YuichiTANAKA 1.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.Review: filteringoftime-domainsignals. . . . . . . . . . . . . . . . . . 4 1.3.Filteringofgraphsignals . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1.Vertexdomainfiltering . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2.Spectraldomainfiltering . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3.Relationshipbetweengraphspectralfilteringandclassical filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.Edge-preservingsmoothingofimagesasgraphspectralfilters . . . . . 11 1.4.1.Earlyworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.2.Edge-preservingsmoothing . . . . . . . . . . . . . . . . . . . . . . 12 1.5.Multiplegraphfilters: graphfilterbanks . . . . . . . . . . . . . . . . . . 15 1.5.1.Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.2.Perfectreconstructioncondition . . . . . . . . . . . . . . . . . . . 17 1.6.Fastcomputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6.1.Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6.2.Downsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6.3.PrecomputingGFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.6.4.Partialeigendecomposition . . . . . . . . . . . . . . . . . . . . . . 22 1.6.5.Polynomialapproximation . . . . . . . . . . . . . . . . . . . . . . 23 1.6.6.Krylovsubspacemethod. . . . . . . . . . . . . . . . . . . . . . . . 26 vi GraphSpectralImageProcessing 1.7.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.8.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Chapter2.GraphLearning . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 XiaowenDONG,DorinaTHANOU,MichaelRABBATandPascalFROSSARD 2.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.Literaturereview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2.1.Statisticalmodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2.2.Physicallymotivatedmodels . . . . . . . . . . . . . . . . . . . . . 35 2.3.Graphlearning: asignalrepresentationperspective. . . . . . . . . . . . 36 2.3.1.Modelsbasedonsignalsmoothness . . . . . . . . . . . . . . . . . 38 2.3.2.Modelsbasedonspectralfilteringofgraphsignals . . . . . . . . . 43 2.3.3.Modelsbasedoncausaldependenciesongraphs . . . . . . . . . . 48 2.3.4.Connectionswiththebroaderliterature . . . . . . . . . . . . . . . 50 2.4.Applicationsofgraphlearninginimageprocessing . . . . . . . . . . . 52 2.5.Concludingremarksandfuturedirections . . . . . . . . . . . . . . . . . 55 2.6.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Chapter3.GraphNeuralNetworks . . . . . . . . . . . . . . . . . . . . . . 63 GiuliaFRACASTOROandDiegoVALSESIA 3.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2.Spectralgraph-convolutionallayers . . . . . . . . . . . . . . . . . . . . 64 3.3.Spatialgraph-convolutionallayers . . . . . . . . . . . . . . . . . . . . . 66 3.4.Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.5.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Part2.ImagingApplicationsofGraphSignalProcessing . . . . . . . 73 Chapter4.GraphSpectralImageandVideoCompression . . . . . . 75 HilmiE.EGILMEZ,Yung-HsuanCHAOandAntonioORTEGA 4.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.1.1.Basicsofimageandvideocompression . . . . . . . . . . . . . . . 77 4.1.2.Literaturereview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.1.3.Outlineofthechapter . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.Graph-basedmodelsforimageandvideosignals . . . . . . . . . . . . . 79 4.2.1.Graph-basedmodelsforresidualsofpredictedsignals . . . . . . . 81 4.2.2.DCT/DSTsasGFTsandtheirrelationto1Dmodels . . . . . . . . 87 4.2.3.Interpretationofgraphweightsforpredictivetransformcoding. . 88 4.3.Graphspectralmethodsforcompression. . . . . . . . . . . . . . . . . . 89 4.3.1.GL-GFTdesign. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3.2.EA-GFTdesign. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.3.3.EmpiricalevaluationofGL-GFTandEA-GFT . . . . . . . . . . . 97 Contents vii 4.4.Conclusionandpotentialfuturework . . . . . . . . . . . . . . . . . . . 100 4.5.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Chapter5.GraphSpectral3DImageCompression . . . . . . . . . . . 105 ThomasMAUGEY,MiraRIZKALLAH,NavidMAHMOUDIANBIDGOLI,Aline ROUMYandChristineGUILLEMOT 5.1.Introductionto3Dimages . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.1.1.3Dimagedefinition . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.1.2.Pointcloudsandmeshes. . . . . . . . . . . . . . . . . . . . . . . . 106 5.1.3.Omnidirectionalimages . . . . . . . . . . . . . . . . . . . . . . . . 107 5.1.4.Lightfieldimages . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.1.5.Stereo/multi-viewimages . . . . . . . . . . . . . . . . . . . . . . . 110 5.2.Graph-based3Dimagecoding: overview . . . . . . . . . . . . . . . . . 110 5.3.Graphconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.3.1.Geometry-basedapproaches . . . . . . . . . . . . . . . . . . . . . 117 5.3.2.Jointgeometryandcolor-basedapproaches . . . . . . . . . . . . . 121 5.3.3.Separabletransforms. . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.4.Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.5.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Chapter6.GraphSpectralImageRestoration . . . . . . . . . . . . . . 133 JiahaoPANGandJinZENG 6.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.1.1.Asimpleimagedegradationmodel. . . . . . . . . . . . . . . . . . 133 6.1.2.Restorationwithsignalpriors . . . . . . . . . . . . . . . . . . . . . 135 6.1.3.Restorationviafiltering . . . . . . . . . . . . . . . . . . . . . . . . 137 6.1.4.GSPforimagerestoration . . . . . . . . . . . . . . . . . . . . . . . 140 6.2.Discrete-domainmethods . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.2.1.Non-localgraph-basedtransformfordepthimagedenoising . . . 141 6.2.2.DoublystochasticgraphLaplacian . . . . . . . . . . . . . . . . . . 142 6.2.3.Reweightedgraphtotalvariationprior . . . . . . . . . . . . . . . . 145 6.2.4.LefteigenvectorsofrandomwalkgraphLaplacian . . . . . . . . . 150 6.2.5.Graph-basedimagefiltering . . . . . . . . . . . . . . . . . . . . . . 155 6.3.Continuous-domainmethods . . . . . . . . . . . . . . . . . . . . . . . . 155 6.3.1.Continuous-domainanalysisofgraphLaplacianregularization . . 156 6.3.2.Low-dimensionalmanifoldmodelforimagerestoration . . . . . . 163 6.3.3.LDMMasgraphLaplacianregularization . . . . . . . . . . . . . . 165 6.4.Learning-basedmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.4.1.CNNwithGLR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.4.2.CNNwithgraphwaveletfilter . . . . . . . . . . . . . . . . . . . . 171 6.5.Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.6.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 viii GraphSpectralImageProcessing Chapter7.GraphSpectralPointCloudProcessing . . . . . . . . . . . 181 WeiHU,SihengCHENandDongTIAN 7.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.2.Graphandgraph-signalsinpointcloudprocessing . . . . . . . . . . . . 183 7.3.Graphspectralmethodologiesforpointcloudprocessing . . . . . . . . 185 7.3.1.Spectral-domaingraphfilteringforpointclouds . . . . . . . . . . 185 7.3.2.Nodal-domaingraphfilteringforpointclouds . . . . . . . . . . . 188 7.3.3.Learning-basedgraphspectralmethodsforpointclouds . . . . . . 189 7.4.Low-levelpointcloudprocessing . . . . . . . . . . . . . . . . . . . . . . 190 7.4.1.Pointclouddenoising . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.4.2.Pointcloudresampling . . . . . . . . . . . . . . . . . . . . . . . . 193 7.4.3.Datasetsandevaluationmetrics. . . . . . . . . . . . . . . . . . . . 198 7.5.High-levelpointcloudunderstanding . . . . . . . . . . . . . . . . . . . 199 7.5.1.Dataauto-encodingforpointclouds . . . . . . . . . . . . . . . . . 199 7.5.2.Transformationauto-encodingforpointclouds . . . . . . . . . . . 206 7.5.3.ApplicationsofGraphTERinpointclouds . . . . . . . . . . . . . 211 7.5.4.Datasetsandevaluationmetrics. . . . . . . . . . . . . . . . . . . . 211 7.6.Summaryandfurtherreading . . . . . . . . . . . . . . . . . . . . . . . . 213 7.7.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Chapter8.GraphSpectralImageSegmentation . . . . . . . . . . . . . 221 MichaelNG 8.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 8.2.Pixelmembershipfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 222 8.2.1.Two-classproblems . . . . . . . . . . . . . . . . . . . . . . . . . . 222 8.2.2.Multiple-classproblems . . . . . . . . . . . . . . . . . . . . . . . . 226 8.2.3.Multipleimages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.3.Matrixproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 8.4.Graphcuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 8.4.1.TheMumford–Shahmodel . . . . . . . . . . . . . . . . . . . . . . 234 8.4.2.Graphcutsminimization . . . . . . . . . . . . . . . . . . . . . . . 235 8.5.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8.6.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Chapter9.GraphSpectralImageClassification . . . . . . . . . . . . . 241 MinxiangYE,VladimirSTANKOVIC,LinaSTANKOVICandGeneCHEUNG 9.1.Formulationofgraph-basedclassificationproblems . . . . . . . . . . . 243 9.1.1.Graphspectralclassifierswithnoiselesslabels . . . . . . . . . . . 243 9.1.2.Graphspectralclassifierswithnoisylabels . . . . . . . . . . . . . 246 9.2.Towardpracticalgraphclassifierimplementation . . . . . . . . . . . . . 247 9.2.1.Graphconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 9.2.2.Experimentalsetupandanalysis . . . . . . . . . . . . . . . . . . . 249

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