Variational regularization for systems of inverse problems. Tikhonov regularization PDF
Preview Variational regularization for systems of inverse problems. Tikhonov regularization
Richard Huber Variational Regularization for Systems of Inverse Problems Tikhonov Regularization with Multiple Forward Operators BestMasters Mit „BestMasters“ zeichnet Springer die besten Masterarbeiten aus, die an renom mierten Hochschulen in Deutschland, Österreich und der Schweiz entstanden sind. Die mit Höchstnote ausgezeichneten Arbeiten wurden durch Gutachter zur Veröf fentlichung empfohlen und behandeln aktuelle Themen aus unterschiedlichen Fachgebieten der Naturwissenschaften, Psychologie, Technik und Wirtschaftswis senschaften. Die Reihe wendet sich an Praktiker und Wissenschaftler gleicherma ßen und soll insbesondere auch Nachwuchswissenschaftlern Orientierung geben. Springer awards “BestMasters” to the best master’s theses which have been com pleted at renowned Universities in Germany, Austria, and Switzerland. The studies received highest marks and were recommended for publication by supervisors. They address current issues from various fields of research in natural sciences, psychology, technology, and economics. The series addresses practitioners as well as scientists and, in particular, offers guidance for early stage researchers. More information about this series at http://www.springer.com/series/13198 Richard Huber Variational Regularization for Systems of Inverse Problems Tikhonov Regularization with Multiple Forward Operators Richard Huber Graz, Austria ISSN 26253577 ISSN 26253615 (electronic) BestMasters ISBN 9783658253899 ISBN 9783658253905 (eBook) https://doi.org/10.1007/9783658253905 Library of Congress Control Number: 2019931813 Springer Spektrum © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer Spektrum imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH part of Springer Nature The registered company address is: AbrahamLincolnStr. 46, 65189 Wiesbaden, Germany Acknowledgements Iwouldliketousethisopportunitytothankallthepeoplewhosupportedmeintheprocessof creatingthisthesis,andmorebroadlywithmyprofessionalcareerandthesuccessofmystudies. FirstImustthankProfessorKristianBredies,whoalwayssupportedmeandwasavailable for discussions, corrections or direction, and was always ready to challenge me to achieve the bestpossibleresults. Further,ImustexpressmygratitudetoMartinHoller,whowasalwaysavailableforquestions andproblemswhichoccurredduringtheworkprocess. Heplayedaparticularimportantrole inthecreationofthisthesis,andtheprojectonwhichitwasbased. IwouldalsoliketothanktheFELMI-ZFEforprovidingmewiththeprojectandtherequired data,andinparticularGeorgHaberfehlnermustbementioned,onwhoseexpertiseIcouldalways count. Moreover,therearecountlesscolleaguesandfriendswhomIhavetothankforsupportingme bothprofessionallyandmorally,whothoughtoomanytoallbementioned,nonethelessknow whotheyare. Finally,aheartfeltthanksmustbeexpressedtowardsmyparentsOttoandIlse,whoalways supportedmeinanywayimaginable. RichardHuber Contents I Introduction 1 I.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 I.2 MathematicalFoundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 I.2.1 Topologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 I.2.2 NormedVectorSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I.2.3 MeasureTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 I.2.4 ConvexAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 II GeneralTikhonovRegularisation 15 II.1 Single-DataTikhonovRegularisation . . . . . . . . . . . . . . . . . . . . . . . . . 15 II.1.1 ExistenceandStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 II.1.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 II.2 Multi-DataTikhonovRegularisation . . . . . . . . . . . . . . . . . . . . . . . . . 25 II.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 II.2.2 ParameterChoicesforVanishingNoise. . . . . . . . . . . . . . . . . . . . 27 II.2.3 Convergencerates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 IIISpecificDiscrepancies 39 III.1NormDiscrepancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 III.1.1 ClassicalNorms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 III.1.2 Subnorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 III.2Kullback-LeiblerDivergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 III.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 III.2.2 BasicProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 III.2.3 ContinuityResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 III.2.4 ApplicabilityasaDiscrepancy . . . . . . . . . . . . . . . . . . . . . . . . 57 IVRegularisationFunctionals 63 IV.1 RegularisationwithNormsandClosedOperators . . . . . . . . . . . . . . . . . . 63 IV.2 TotalDeformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 IV.2.1 SymmetricTensorFields . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 IV.2.2 TensorFieldsofBoundedDeformation. . . . . . . . . . . . . . . . . . . . 72 IV.3 TotalGeneralisedVariation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 IV.3.1 BasicProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 IV.3.2 TopologicalProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 IV.3.3 TotalGeneralisedVariationofVector-ValuedFunctions . . . . . . . . . . 82 viii Contents IV.4 TGVRegularisationinaLinearSetting . . . . . . . . . . . . . . . . . . . . . . . 86 V ApplicationtoSTEMTomographyReconstruction 89 V.1 TheRadonTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 V.1.1 DerivingtheRadonTransform . . . . . . . . . . . . . . . . . . . . . . . . 89 V.1.2 AnalyticalProperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 V.1.3 FilteredBackprojection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 V.2 TikhonovApproachtoMulti-SpectraSTEMTomographyReconstruction . . . . 98 V.2.1 ContinuousTikhonovProblemforSTEMTomographyReconstruction . . 98 V.2.2 DiscretisationScheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 V.2.3 Primal-DualOptimisationAlgorithm. . . . . . . . . . . . . . . . . . . . . 105 V.2.4 STEMTomographyReconstructionAlgorithm . . . . . . . . . . . . . . . 106 V.3 DiscussionofNumericalResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 V.3.1 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 V.3.2 SyntheticExperiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 V.3.3 ReconstructionofSingle-DataHAADFSignals . . . . . . . . . . . . . . . 120 V.3.4 STEMMulti-SpectralReconstructions . . . . . . . . . . . . . . . . . . . . 124 Summary 129 Bibliography 131 List of Figures IV.1 IllustrationforExampleIV.32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 IV.2 IllustrationforExampleIV.33. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 V.1 LineL(s,φ,z)forfixedzin2Dcross-section. . . . . . . . . . . . . . . . . . . . . 90 V.2 Phantomdensitydistributionandcorrespondingsinogram-data. . . . . . . . . . 92 V.3 Supportofafunctionvanishinginsideacircle. . . . . . . . . . . . . . . . . . . . 93 V.4 DiscretisationΩ˜RinΩRforfixedz.. . . . . . . . . . . . . . . . . . . . . . . . . . 100 V.5 Adjacencyrelationbetweenlinesandpointsforfixedφ. . . . . . . . . . . . . . . 103 V.6 TruesinogramdataandreconstructionsforSectionV.3.1. . . . . . . . . . . . . . 112 V.7 Brightnessfluctuationsanditseffectsonreconstructions. . . . . . . . . . . . . . 114 V.8 ModifiedbasevalueandaddedGaussiannoise,andtheeffectsonreconstruction. 115 V.9 Misalignmentsandthecorrespondingreconstructions. . . . . . . . . . . . . . . . 116 V.10SyntheticsinogramdataforSectionV.3.2.. . . . . . . . . . . . . . . . . . . . . . 117 V.11TruedensitydistributionandFilteredBackprojectionreconstructionsofsynthetic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 V.12Reconstructionofnoisysyntheticdatawithvaryingmethodsandregularisation parameter.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 V.13RealHAADFprojectionsofaspecimenfromvariousangles.. . . . . . . . . . . . 120 V.14HAADFreconstructionsbycommonlyusedalgorithms. . . . . . . . . . . . . . . 121 V.15HAADFreconstructionsshowingeffectsofpreprocessingand2Dvs3D . . . . . 122 V.16SeriesofslicesfromreconstructionswithTGVandKL. . . . . . . . . . . . . . . 123 V.17Spectralprojectionsofaspecimenfromdifferentangles. . . . . . . . . . . . . . . 125 V.18UncoupledTGV-KLreconstructionofspectraldata. . . . . . . . . . . . . . . . . 126 V.19JointTGV-KLreconstructionofspectraldatausingFrobeniusnorms. . . . . . . 127 I. Introduction I.1. Motivation Manymathematicalproblemsoccurringinindustrialandscientificapplicationscannotbesolved bydirectcomputation,butratherrequiretheinversionofaprocess. Thismeansconsideringa processT,whichtakesaninputu,transformingitintoanoutputf,buttryingtofindasuitable inputleadingtogivenoutputf. Hence,oneconsidersproblemsoftheform Tu=f typicallyfeaturinginfinite-dimensionalspaces,wheretheoutputf andtheprocessT:X →Y aregiven,andoneaimstosolvethisequationforthenecessaryinputu. Suchproblemsoften occurwhenoneobtainsdataf throughawell-understoodprocessmodeledbyT,butaimsto reconstructthecauseuwhichwouldhaveledtof. Asimpleexampleofsuchproblems,whichnicelyillustratesthedifferenceinthoughtbetween directandinverseproblems,isdeconvolution,whereonetriestosolve (cid:2) k∗u=f, where k∗u(·)= u(·−y)k(y)dy, Rn withgivenfandkforu,meaningonetriestoinverttheconvolutionoperation. Thistheoretical settingcanforexamplebefoundinthebackwardheatequationforpracticalpurposes. Imagine arodofironwhoseheatdistributionovertimeisobserved. Obtainingtheheatdistributionat somepointintimefromgivenstartingdistributionisabasicPDEproblemonecansolvevia convolutionh∗u0=u1,whereu0 denotestheinitialdistributionandhtheheat-kernel. This would represent the direct problem, while obtaining a starting distribution which would have led to a given distribution later in time represents the inverse problem. Therefore, for given heat-distributionu1attimet=1onesolvesh∗u0=u1foru0. AproblemwefocusonmoreheavilyinlatersectionsistheinversionoftheRadontransform usedinCT[3]. TheComputedTomography(CT)methodisusedinmedicalpracticetoobtain 3-dimensionaldensitydistributionsofpatients. Inordertodoso, asequenceofX-rayimages of the patient from different angles is taken. Through mathematical modelling it is easy to understand how to obtain the sinograms (X-ray data) from given density distribution, while the converse is not so obvious. Hence, in order to obtain suitable CT reconstructions for the physiciantoanalyse,onesolves Tu=f (I.1) whereT modelstheforwardoperator,i.e. theprocedureofobtainingthesinogramsdatafrom thedensitydistribution,andf thesinogramdataobtainedfromtheexaminationofapatient. Suchmeasurementsfhowevertypicallysufferfromnoise,inparticularPoissondistributednoise sincethemeasurementismadeviadetectioncountsofthetransmittedphotonswhichtypically ismodelledtobePoissondistributed,seee.g. [50]. Unfortunately, solving an inverse problem is often not stable (in particular the ones men- tionedabove),i.e. smallaberrationindatafmightresultinmassivechangesinthecorrespond- ing solution u. Thus, regularisation methods are required in order to overcome this stability issue. Specifically,theTikhonovregularisation[47,48]isacommonlyusedmethod,andsince thisthesiswillfocusonTikhonovregularisation,wequicklymotivateitsuse: © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 R. Huber, Variational Regularization for Systems of Inverse Problems, BestMasters, https://doi.org/10.1007/978-3-658-25390-5_1
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