Level Set and PDE Based Reconstruction Methods in Imaging: Cetraro, Italy 2008, Editors: Martin Burger, Stanley Osher PDF
Preview Level Set and PDE Based Reconstruction Methods in Imaging: Cetraro, Italy 2008, Editors: Martin Burger, Stanley Osher
Lecture Notes in Mathematics 2090 CIME Foundation Subseries Martin Burger Andrea C.G. Mennucci Stanley Osher Martin Rumpf Level Set and PDE Based Reconstruction Methods in Imaging Cetraro, Italy 2008 Editors: Martin Burger, Stanley Osher Lecture Notes in Mathematics 2090 Editors: J.-M.Morel,Cachan B.Teissier,Paris Forfurthervolumes: http://www.springer.com/series/304 FondazioneC.I.M.E.,Firenze C.I.M.E. stands for Centro Internazionale Matematico Estivo, that is, International MathematicalSummerCentre.Conceivedintheearlyfifties,itwasbornin1954inFlorence, Italy,andwelcomedbytheworldmathematical community:itcontinues successfully,year foryear,tothisday. Many mathematicians from all over the world have been involved in a way oranother in C.I.M.E.’sactivitiesovertheyears.ThemainpurposeandmodeoffunctioningoftheCentre maybesummarisedasfollows:everyyear,duringthesummer,sessionsondifferentthemes from pure and applied mathematics are offered byapplication to mathematicians from all countries. ASessionis generally basedonthree orfourmaincourses given byspecialists ofinternationalrenown,plusacertainnumberofseminars,andisheldinanattractiverural locationinItaly. TheaimofaC.I.M.E.sessionistobringtotheattentionofyoungerresearcherstheorigins, development, and perspectives of some very active branch of mathematical research. The topicsofthecoursesaregenerallyofinternationalresonance.Thefullimmersionatmosphere ofthecoursesandthedailyexchangeamongparticipantsarethusaninitiationtointernational collaborationinmathematicalresearch. C.I.M.E.Director C.I.M.E.Secretary PietroZECCA ElviraMASCOLO DipartimentodiEnergetica“S.Stecco” DipartimentodiMatematica“U.Dini” UniversitàdiFirenze UniversitàdiFirenze ViaS.Marta,3 vialeG.B.Morgagni67/A 50139Florence 50134Florence Italy Italy e-mail:zecca@unifi.it e-mail:[email protected]fi.it FormoreinformationseeCIME’shomepage:http://www.cime.unifi.it CIMEactivityiscarriedoutwiththecollaborationandfinancialsupportof: -INdAM(IstitutoNazionalediAltaMatematica) -MIUR(Ministerodell’UniversitàedellaRicerca) Martin Burger Andrea C.G. Mennucci (cid:2) Stanley Osher Martin Rumpf (cid:2) Level Set and PDE Based Reconstruction Methods in Imaging Cetraro, Italy 2008 Editors: Martin Burger Stanley Osher 123 MartinBurger AndreaC.G.Mennucci InstituteforComputational ScuolaNormaleSuperiore andAppliedMathematics Pisa,Italy UniversityofMünster Münster,Germany StanleyOsher MartinRumpf DepartmentofMathematics InstituteforNumericalSimulation UniversityofCalifornia UniversityofBonn LosAngeles,CA,USA Bonn,Germany ISBN978-3-319-01711-2 ISBN978-3-319-01712-9(eBook) DOI10.1007/978-3-319-01712-9 SpringerChamHeidelbergNewYorkDordrechtLondon LectureNotesinMathematicsISSNprintedition:0075-8434 ISSNelectronicedition:1617-9692 LibraryofCongressControlNumber:2013950139 MathematicsSubjectClassification(2010):94A08,35Q94,68U10,65K10,35J20,45Q05,53C44 (cid:2)c SpringerInternationalPublishingSwitzerland2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface The present volume collects the lecture notes of courses given at the CIME SummerSchoolinCetraroinSeptember2008.Theschoolappearedtobeahighly successful event with around 50 participants from all over the world, enjoying boththescientificcontentofthecoursesandtheatmosphereandopportunitiesfor discussionsamongparticipantsandwiththespeakers. Mathematical imaging and inverse problems are two of the fastest growing disciplinesinappliedandinterdisciplinarymathematics,withastrongoverlapsince many image analysis and reconstruction problems are effectively formulated as inverse problems. A key issue in such problems is to preserve and analyze edges orshapesintheimage.Thosearetakencarebythemethodscoveredinthesummer school, which had the particular aim of providing a unified picture to methods of image reconstruction and analysis, on the one hand, and techniques for shape reconstruction and analysis, on the other hand. The key steps connecting those are methods based on level set representations and geometric partial differential equations, which effectively operate on the level sets (respectively, discontinuity set)ofanimageaswellasonasingleshape.Thesubjectoflectureswerechosento covertheseaspectsandprovideanoutlooktootherrelevanttopics: • Computational methods for nonlinear PDE and L1 techniques: Stanley Osher (UCLA). • Totalvariationandrelatedmethods:MartinBurger(Münster). • Theuseoflevelsetmethodsininverseproblems:OliverDorn(Manchester). • Variational methods in image matching and motion extraction: Medical and biologicalapplications:MartinRumpf(Bonn). • Metrics of curves in shape optimization and analysis: Andrea C.G. Mennucci (SNSPisa). In addition, few young participants already involved in research related to the coursetopicsweregivenanopportunitytopresenttheirworkinshortertalks. ThelectureseriesbyStanleyOshergaveageneraloverviewtomoderncomputa- tionalmethodsforimagereconstructionandanalysis,includingBregmaniterations, v vi Preface sparsity-based methods with `1-type functionals, and local and nonlocal total variationmethodsforimagedenoising. MartinBurgerfurtherexpandedontotalvariationtechniquesanddiscussedthe analysisoftotalvariationmethods,furtherdetailsonBregmaniterationsandinverse scalespacemethods,andalsoapplicationsinbiomedicalimaging. The lectures by Oliver Dorn gave an overview of level set methods for the reconstruction of shapes in inverse problems. He discussed basics about level set representation of shapes and the construction of computational methods to solve appropriate variational formulations, including the necessary shape calculus. A major part of the lecture series was devoted to several applications modeled by inverseproblemsforpartialdifferentialequations. MartinRumpfmovedfromsingleimagestopairsofimages.Themajortasksthen consistoffindingdeformationsoftheimagedomainsthatmaketheimageslookas similar as possible. The correspondingobjectivesare either matchingif interested inresultingimages,andmotionestimationifinterestedinthedeformation.Exten- sions of the framework to shapes and their matching respectively averaging were presentedaswell. Andrea C.G. Mennucci provided an overview of metrics on curves, linking this topic originating in differential geometry to very applied problems in image processing and shape optimization. He showed that by choosing differentmetrics methodsfortaskssuchasobjectsegmentationinimagescanbeimproved. Inthelecturenotesmosttopicsofthefirsttwolectureswereunifiedinachapter providing a comprehensive introduction to the zoo of total variation and related methods.Morespecializedtopicsrelatedtophotoncountdatafrequentlyarisingin imagingwereunifiedwiththecontentsofparticipanttalksbyChristophBruneand AlexSawatzkyinthesecondchapter.ThethirdchaptercoversthecontentsofMartin Rumpf’slectureseries,thefourthchaptertheonesofAndreaC.G.Mennucci’s.We hopethatreadersfindthematerialasinterestingaswedoandenjoythebreadthof coveredtopicsaswellasthedepthofdetailinthesinglechapters. Münster,Germany MartinBurger LosAngeles,CA StanleyOsher Contents AGuidetotheTVZoo .......................................................... 1 MartinBurgerandStanleyOsher EM-TVMethodsforInverseProblemswithPoissonNoise ................. 71 Alex Sawatzky, Christoph Brune, Thomas Kösters, FrankWübbeling,andMartinBurger VariationalMethodsinImageMatchingandMotionExtraction .......... 143 MartinRumpf MetricsofCurvesinShapeOptimizationandAnalysis..................... 205 AndreaC.G.Mennucci vii A Guide to the TV Zoo MartinBurgerandStanleyOsher Abstract Totalvariationmethodsandsimilarapproachesbasedonregularizations with `1-type norms (and seminorms) have become a very popular tool in image processing and inverse problems due to peculiar features that cannot be realized with smooth regularizations.In particulartotal variation techniqueshad particular successduetotheirabilitytorealizecartoon-typereconstructionswithsharpedges. Duetoanexplosionofnewdevelopmentsinthisfieldwithinthelastdecadeitisa difficulttasktokeepanoverviewofthemajorresultsinanalysis,thecomputational schemes,andtheapplicationfields.Withtheselecturesweattempttoprovidesuch anoverview,ofcoursebiasedbyourmajorlinesofresearch. Wearefocusingonthebasicanalysisoftotalvariationmethodsandtheextension of the original ROF-denoising model due various application fields. Furthermore weprovideabriefdiscussionofstate-of-theartcomputationalmethodsandgivean outlooktoapplicationsindifferentdisciplines. 1 Introduction Reconstructingand processingimageswith appropriateedgesis of centralimpor- tance in modern imaging. The development of mathematical techniques that preserve or even favour sharp edges has become a necessity and created various interesting approaches. The two most succesful frameworks are two variational approaches:totalvariationmodelsontheonehandandmodelswithexplicitedges M.Burger((cid:2)) InstituteforComputationalandAppliedMathematics,UniversityofMünster, Münster,Germany e-mail:[email protected] S.Osher DepartmentofMathematics,UCLA,LosAngeles,USA M.Burgeretal.(eds.),LevelSetandPDEBasedReconstructionMethodsinImaging, 1 LectureNotesinMathematics2090,DOI10.1007/978-3-319-01712-9__1, ©SpringerInternationalPublishingSwitzerland2013 2 M.BurgerandS.Osher intheframeworkofMumfordandShah[140]ontheother.Thelatterusuallylead to various difficulties in the analysis (cf. [138]) and numerical realization due to the explicit treatment of edges and arising nonconvexity, consequently they have foundlimitedimpactinpracticalapplicationsbeyondimagesegmentation.Inthese lectures notes we will discuss various developments for total variation methods, whichcanbeformulatedasconvexvariationalmethods.Awholezooofapproaches tothemodelling,analysis,numericalsolution,andapplicationshasdevelopedinthe lasttwodecades,throughwhichweshalltrytodevelopaguide. The starting pointof total variation(TV) methodshas been the introductionof avariationaldenoisingmodelbyRudinetal.[163],consistinginminimizingtotal variationamongallfunctionswithinavariancebound Z TV.u/!min subjectto .u(cid:3)f/2dx (cid:4)(cid:3)2: (1) u (cid:2) IntroducingaLagrangemultiplier(cid:4)itcanbeshownthatthisapproachisequivalent totheunconstrainedproblemofminimizing Z (cid:4) E .u/WD .u(cid:3)f/2dxCTV.u/; (2) ROF 2 (cid:2) in the following often referred to as the ROF model. In subsequent years this model was generalized for many imaging tasks and inverse problems and found applicationsindifferentareas. We will providea more detailed motivationfortotal variationregularizationin a rather general setup in Sect.2. In Sects.3–5 we provide an overview of various aspects in the analysis of total variation regularization. In Sect.6 we discuss the concepts of Bregman iterations and inverse scale space methods, which allow to compensatesystematic errorsof variationalmethods,e.g. contrastloss in the case ofTVregularization,andgaveanotherboosttoresearchinthisareainrecentyears. InSects.7and8wediscusssomevariantsofthemodels,withchangesconcerning thefidelityterminSect.7andtheregularizationterminSect.8.InSect.9wediscuss someapproachesforthenumericalsolutionofthevariationalproblems.Section10 is devoted to geometric aspects of total variation minimization and the relaxation of segmentationproblemsinto convexmodelsfor functionsof boundedvariation. We then proceed to applications, which we mainly incorporate as further links to literatureinSect.11.Finallywepresentsomeopenquestionsinthemodellingand analysisofTVmethodsinSect.12. 2 TheMotivation for TVandRelated Methods InthefollowingweprovidebasicmotivationsforthegeneralsetupofTVmethods withrespecttoforwardoperators,datafidelityterms,andregularization.
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