HandbookofMathematicalMethodsinImaging Otmar Scherzer(Ed.) Handbook of Mathematical Methods in Imaging withFiguresandTables 123 Editor: OtmarScherzer ComputationalScienceCenter UniversityofVienna Nordbergstrasse Vienna Austria and RICAM AustrianAcademyofSciences Linz Austria LibraryofCongressControlNumber: ISBN---- Thispublicationisavailablealsoas: ElectronicpublicationunderISBN---- PrintandelectronicbundleunderISBN---- DOI./---- ©SpringerScience+BusinessMedia,LLC Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenper- missionofthepublisher(SpringerScience+BusinessMedia,LLC,SpringStreet,NewYork,NY,USA), exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyform ofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyarenot identifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjectto proprietaryrights. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. SpringerispartofSpringerScience+BusinessMedia www.springer.com Printedonacid-freepaper SPIN:SPi– Preface Today,computerimagingcoversvariousaspectsofdatafiltering,patternrecognition, feature extraction,computeraideddesign,andcomputeraidedinspectionanddiagnosis. Pillarsofthefieldofcomputerimagingareadvanced,stable,andreliablealgorithms.In addition,feasibilityanalysisisrequiredtoevaluatepracticalrelevanceofthemethods.To putthesepillarsonsolidgrounds,asignificantamountofmathematicaltoolsarerequired. Thishandbookmakesahumbleattempttoprovideasurveyofsuchmathematicaltools. We had the vision that this imaging handbook should contain individual chapters whichcanserveastoolboxes,which,whenaligned,formbackgroundmaterialforcomplete appliedimagingproblems.Thereforeitshouldalsogiveanimpressiononthebroadmathe- matical knowledge required to solve industrial and applied research applications: The imageformationprocess,veryfrequently,isassignedtotheinverseproblemscommunity, whichisprominentlyrepresentedinthishandbook.Thesubsequentstepisimageanalysis. Nowadays,advancedImageAnalysis,andImageProcessingingeneral,usessophisticated methodsfromGeometry,DifferentialGeometry,ConvexAnalysis,NumericalAnalysis,to mentionjustafew.Infact,bytherapidadvanceofImaging,themathematical areashave beenpushedforwardheavily,andraisedtheirimpactinapplicationsciences. Myspecialthanksgotoallindividualauthorsfortheirvaluablecontributionsandthe refereesfortheirhelpinimprovingthecontributionsandmakingdetailedcomments.My sincerethanksgototheSpringer’seditorsandstaff,VaishaliDamle,JenniferCarlson,and SaskiaEllisfortheirpatience,andtheirconstantsupportandencouragementoverthelast twoyears. Finally,Iwouldliketoencouragethereaderstosubmitsuggestionsregardingthehand- book’s content. For a projectof this size, itislikely that essential topicsare missed. Ina rapidlyevolvingarealikeImagingitislikelythatnewareaswillappearinaveryshorttime and should beadded tothe handbook, as wellas recent development enforce modifica- tionsofexistingcontributions.Wearecommittedtoissuingperiodicupdatesandwelook forwardtothefeedbackfromthecommunity. OtmarScherzer ComputationalScienceCenter UniversityofVienna,Austria and RICAM AustrianAcademyofSciences Linz,Austria Table of Contents Preface................................................................................................... v AbouttheEditor...................................................................................... xi ListofContributors.................................................................................. xiii Volume InverseProblems–Methods  LinearInverseProblems............................................................  CharlesGroetsch  Large-ScaleInverseProblemsinImaging.....................................  JulianneChung⋅SarahKnepper⋅JamesG.Nagy  RegularizationMethodsforIll-PosedProblems............................  JinCheng⋅BerndHofmann  DistanceMeasuresandApplicationstoMulti-Modal VariationalImaging.................................................................  ChristianePöschl⋅OtmarScherzer  EnergyMinimizationMethods...................................................  MilaNikolova  CompressiveSensing...............................................................  MassimoFornasier⋅HolgerRauhut  DualityandConvexProgramming..............................................  JonathanM.Borwein⋅D.RussellLuke  EMAlgorithms........................................................................  CharlesByrne⋅PaulP.B.Eggermont  IterativeSolutionMethods........................................................  MartinBurger⋅BarbaraKaltenbacher⋅AndreasNeubauer  LevelSetMethodsforStructuralInversionandImage Reconstruction........................................................................  OliverDorn⋅DominiqueLesselier viii TableofContents Volume InverseProblems–CaseExamples  ExpansionMethods..................................................................  HabibAmmari⋅HyeonbaeKang  SamplingMethods...................................................................  MartinHanke⋅AndreasKirsch  InverseScattering....................................................................  DavidColton⋅RainerKress  ElectricalImpedanceTomography..............................................  AndyAdler⋅RominaGaburro·WilliamLionheart  SyntheticApertureRadarImaging..............................................  MargaretCheney⋅BrettBorden  Tomography...........................................................................  GaborT.Herman  OpticalImaging.......................................................................  SimonR.Arridge⋅JariP.Kaipio·VilleKolehmainen⋅TanjaTarvainen  PhotoacousticandThermoacousticTomography: ImageFormationPrinciples.......................................................  KunWang⋅MarkA.Anastasio  MathematicsofPhotoacousticandThermoacousticTomography....  PeterKuchment⋅LeonidKunyansky  WavePhenomena....................................................................  MattiLassas⋅MikkoSalo⋅GuntherUhlmann Volume ImageRestorationandAnalysis  StatisticalMethodsinImaging...................................................  DanielaCalvetti⋅ErkkiSomersalo  SupervisedLearningbySupportVectorMachines........................  GabrieleSteidl TableofContents ix  TotalVariationinImaging.........................................................  V.Caselles⋅A.Chambolle⋅M.Novaga  NumericalMethodsandApplicationsinTotalVariationImage Restoration............................................................................. RaymondChan⋅TonyChan⋅AndyYip  MumfordandShahModelanditsApplicationstoImage SegmentationandImageRestoration......................................... LeahBar⋅TonyF.Chan⋅GinmoChung⋅MiyounJung⋅NahumKiryati⋅ RamiMohieddine⋅NirSochen⋅LuminitaA.Vese  LocalSmoothingNeighborhoodFilters.......................................  Jean-MichelMorel⋅AntoniBuades⋅TomeuColl  NeighborhoodFiltersandtheRecoveryofDInformation.............  JulieDigne⋅MariellaDimiccoli⋅PhilippeSalembier⋅NeusSabater  SplinesandMultiresolutionAnalysis..........................................  BrigitteForster  GaborAnalysisforImaging.......................................................  OleChristensen⋅HansG.Feichtinger⋅StephanPaukner  ShapeSpaces.......................................................................... AlainTrouvé⋅LaurentYounes  VariationalMethodsinShapeAnalysis........................................ MartinRumpf⋅BenediktWirth  ManifoldIntrinsicSimilarity...................................................... AlexanderM.Bronstein⋅MichaelM.Bronstein  ImageSegmentationwithShapePriors:ExplicitVersusImplicit Representations......................................................................  DanielCremers  StarletTransforminAstronomicalDataProcessing....................... Jean-LucStarck⋅FionnMurtagh⋅MarioBertero  DifferentialMethodsforMulti-DimensionalVisualDataAnalysis..... WernerBenger⋅RenéHeinzl⋅DietmarHildenbrand⋅TinoWeinkauf⋅ HolgerTheisel⋅DavidTschumperlé Index..................................................................................................... 