Variational Regularization for Systems of Inverse Problems: Tikhonov Regularization with Multiple Forward Operators

140 Pages·2019·5.284 MB·English

by  Richard Huber

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