Saturday Sunday Monday Tuesday Wednesday Thursday 700-815 X 700-815 Continental Breakfast,

MathematicalImageAnalysisandProcessing

October23-28,2004

SCHEDULE

Oct23SaturdayXXXXXXXXXXXXXXRumpfWolfOct24Oct25Oct26Oct27Oct28SundayMondayTuesdayWednesdayThursday7:00-8:15ContinentalBreakfast,2ndfloorlounge,CorbettHallOrientation1VemuriLevineWhitakerVeseTaiSiddiqiVogelChanXXXXXXXXXXXXXXX7:00-8:158:15-9:009:00-9:459:45-10:3010:30-11:0011:00-11:4511:45-12:3012:30-1:301:30-2:152:15-3:003:00-3:303:30-4:154:15-5:005:00-5:455:45-7:308:00-10:00LucierMalgouyres10:30-11:00CoffeeBreak,2ndfloorlounge,CorbettHallScherzerSpiraAllardVixieMetaxas2GroupPhoto3SantosaGunturk12:30-1:30BuffetLunch,DonaldCameronHallShahMarchfreeafternoonfreeafternoonBertozziGreer3:00-3:30CoffeeBreak,2ndfloorlounge,CorbettHall(exceptTues.)NikolovaBuadesCapoMajavaYezziZhouTsaifreeafternoonfreeafternoonfreeafternoonBoutinGarc´ıaAlmeidaSchmidt5:45-7:30BuffetDinner,DonaldCameronHallXdiscussionsdiscussionsMEALS

Breakfast(Continental):7:00-9:00am,2ndfloorlounge,CorbettHall,Sunday-Thursday*Lunch(Buffet):11:30am-1:30pm,DonaldCameronHall,Sunday-Thursday*Dinner(Buffet):5:30-7:30pm,DonaldCameronHall,Saturday-WednesdayCoffeeBreaks:Asperdailyschedule,2ndfloorlounge,CorbettHall

*Pleaseremembertoscanyourmealcardatthehost/hostessstationinthediningroomforeachlunchanddinner.

At8:45am.Until12:00pm.3

AgroupphotowillbetakenonTuesdayat12:00pm,directlyafterthelastlectureofthemorning.PleasemeetonthefrontstepsofCorbettHall.

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MEETINGROOMS

Alllecturesareheldinthemainlecturehall,MaxBell159.PleasenotethatthemeetingspacedesignatedforBIRSisthelowerlevelofMaxBell,Rooms155-159.PleaserespectthatallotherspacehasbeencontractedtootherBanffCentreguests,includinganyFoodandBeverageinthoseareas.

ABSTRACTS

(inalphabeticorderbyspeakersurname)

Speaker:WilliamK.Allard(Mathematics,DukeUniversity)

Title:OntheregularityoflevelsetsofminimizersofdenoisingmodelsbasedontotalvariationregularizationAbstract:LetΩbeanopensubsetofRn,n≥2;letβ,γ:[0,∞)→[0,∞)beincreasing,Lipschitzianonboundedsetsandzeroatzero;andletfbearealvaluedLebesguemeasurablefunctiononΩsuchthat󰀂nnn

Ωγ(|f(x)|)dLwhereLisLebesguemeasureonR.Foreach󰀃>0welet

󰀄󰀃󰀁

U󰀅(u)=󰀃TV(u)+βγ(|u(x)−f(x)|)dLnx

Ω

foreachrealvaluedLebesguemeasurablefunctionuonΩ;hereTV(u)isthetotalvariationofu.

Rudin,OsherandFatemistudiedminimizersofthisfunctionalwithβ(y)=yandγ(y)=y2,y∈[0,∞),andChanandEsedoglustudiedminimizersofthisfunctionalwithβ(y)=yandγ(y)=y,y∈[0,∞);inbothcasesthefunctionalwasusedasadenoisingmodel.

Letusnowassumethatn≤7;incasen>7muchcanbesaidbutwewillnottreatthatcasehere.Ourmainresult,inwhichwehavetoassumefisessentiallyboundedifγgrowsmorethanlinearlyatinfinity,isthatifuisaminimizerofU󰀅andy∈Rthen{u≥y}isasetoffiniteperimeterwhosereducedboundaryisaC1hypersurfaceinΩwhosenormalislocallyH¨oldercontinuouswithrespecttoanyexponentµ∈(0,1);incasen=2thisnormalislocallyLipschitzian.Inaddition,givenabitmoreregularityofβandγ,weobtainaveryusefulEuler-Lagrangetypeequationwhichminimizersmustsatisfywhichinturngivesdetailedlocalinformationonthestructureofu,particularlyincasen=2.

Speaker:AndreaBertozzi(Mathematics,UniversityofCaliforniaatLosAngelesandDukeUniversity)Title:FourthOrderPDEinimageprocessingAbstract:

Speaker:MireilleBoutin(MathematicsandElectrical&ComputerEngineering,PurdueUniversity)Title:SceneReconstructionfromImages:anovelapproachbasedonmovingframes

Abstract:Theproblemofreconstructingascenefromasetofimagestakenfromunknownviewpointsinvolvesalotof”superflous”unknowns.Forexample,thecameraparametersusedfortakingeachpictureareirrelevantsinceweareonlyinterestedinthestructureofthescene.However,withthetraditionalapproaches,westillneedtosolvefortheseparametersbecausetheyareincludedintheintermediatestepsofthesolutionprocess.Inthistalk,wewillshowthatmanyofthesesuperflousparameterscanbeseenasgroupparametersactingontheotherunknownsoftheproblems.Usingthemovingframemethod,weobtainasetofinvariantsofthisgroupaction.Theinvariantsprovideanewformulationtothereconstructionproblemwherethesuperfluousunknownsdonotappearanymore.Inparticular,thecameraangles,whicharedifficulttosolvefor,donotappearintheequations.Asaresult,amuchsimpler/robustformulationisobtained.

Speaker:ToniBuades-Capo(EcoleNormaleSuperioredeCachan)Title:OnImagedenoisingmethods

Abstract:Thesearchforefficientimagedenoisingmethodsstillisavalidchallenge,atthecrossingpoint

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offunctionalanalysisandstatistics.Inspiteofthesophisticatedmethodsrecentlyproposed(empiricalWienerfilters,totalvariationminimization,anisotropicdiffusion,waveletthresholdings,image+texture+noisemodels),mostalgorithmshavenotyetattainedadesirablelevelofapplicability.Allshowanoutstandingperformancewhentheimagesmoothnessmodelcorrespondstothealgorithmassumption,butfailatotherlocationsandcreateartifactsorremoveimagefinestructures.Themainfocusofthetalkistodefineageneralmathematicalandexperimentalmethodologytocompareandclassifyclassicalimagedenoisingalgorithms.Themathematicalanalysisisbasedonthestructureanalysisofwhatwecallthe“methodnoise”,namelythedifferencebetweenthe(alwaysslightlynoisy)digitalimageanditsdenoisedversion.Adenoisingalgorithmisconsistentwhenithasalow,orevenazeromethodnoiseforfunctionswiththerightregularity.Wealsointroduceanewkindofalgorithm,whichseemstoseparatebetternoisefromimagedetails,thenonlocalmeansalgorithm(NL-means).Weshowwhythisalgorithminheritsofmostconsistencypropertiesoftheformermethodsandproveitsasymptoticoptimalityunderarathergenericstatisticalimagemodel.

Speaker:TonyChan(Mathematics,UniversityofCaliforniaatLosAngeles)

Title:Alogicframeworkfomulti-channelimagesegmentationandapplicationtotrackinginvideos

Abstract:I’llpresentageneralframeworkfordefiningmeaningfulsegmentationofmulti-channelimagesbyallowingarbitrarylogicalcombinationofobjectinformationfromthedifferentchannels.Underthisframework,I’lldevelopspecificactivecontourandsegmentationmodelsandalgorithmsbasedonextensionsofthescalarregion-basedChan-Vesesegmentationmodel.Finally,I’llshowanapplicationofthislogicframeworktotrackingofobjectsinlowframe-ratevideosequences.ThisisjointworkwithMarkMoelichandBertaSandberg.

Speaker:GerardoEmilioGarc´ıaAlmeida(Mathematics,UniversidadAutonomadeYucatan)Title:Estimatesforconvolutionsinfunctionspaceswithfractionalorderofsmoothness

Abstract:Inthistalkanintegralequationofthefirstkindofconvolutiontypeisconsidered.TheTikhonovregularisationmethodisusedtoconstructasequenceofapproximatesolutionsthatconvergestotheexactsolutionofthisill-posedproblem.Theelementsofthissequencearecalledregularisedsolutions.

IntheTikhonovregularisationmethoditisassumedthattheexactsolution,therighthandsideoftheintegralequation,andtheerrorbelongtosuitablefunctionspaces.

Itisknownthatthechoiceoftheparticularspacesisveryimportant.Ononehandtoostrongapriorismoothnessassumptionsontheexactsolutionandtheerrorwhicharefarfromtheiractualsmoothnesspropertiescausegreatdifficultiesinthenumericalimplementationofthealgorithm.Ontheotherhandtooweaksmoothnessassumptionsleadtoaveryslowconvergence.

Insomecasestheassumption,whichisoftenused,thattheexactsolutionbelongstotheSobolevspaceoforderofsmoothnessoneistoostrong.IninvestigationsofV.I.Burenkov,I.F.DorofeevandA.S.Pankratov,relatedtotheisotropiccase,itwasshownthattheapplicationoftheisotropicNikol’ski˘ı–Besovspacesoffunctionspossessingsomecommonsmoothnessoffractionalordergavemoreflexibilityincharacterisingthesmoothnesspropertiesoftheexactsolutionandtheerrorandbetterfittedtotheappliedproblems.Inparticular,thesmoothnessparametermaybechosentobesufficientlysmall,thusimposingconsiderablyweaksmoothnessassumptionsontheexactsolutionwhichmaybepossessedevenbyunboundedsolutionswithpowergrowth,whilsttheconvergenceoftheregularisedsolutionstotheexactoneisstillreasonablyquick.

ThemainaimofthepresenttalkistoobtainsimilarresultsintheanisotropiccasebyusinganisotropicNikol’ski˘ı–Besovspaces,thusgivingmorepossibilitiesfortheapplicationofthismethod.Speaker:JohnGreer(CourantInstitute,NewYorkUniversity)Title:Fourthorderequationsforimageprocessing

Abstract:Anumberoffourthorderdiffusionequationshaverecentlybeenintroducedforimagesmoothinganddenoising.Althoughnumericalimplementationsofthesemethodsproduceimpressiveresults,verylittleisknownaboutthemathematicalpropertiesoftheequationsthemselves.Iwilldiscusssomeofthe

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firstresultsregardingafewofthesenonlineardiffusions.Inparticular,Iwilldescribetheuseofenergymethodstoprovethewell-posednessofaclassofH1diffusionsforimageprocessing,includingthe‘LowCurvatureImageSimplifier’(LCIS)equationofTumblinandTurk(SIGGRAPH,August,1999).IwilldemonstrateimplementationsofanewfinitedifferencediscretizationoftheLCISequationthatensuresthediscreteLaplacianoftheimageintensityremainsbounded.TheseresultswillbecomparedtosecondordermethodssuchasthePerona-MalikequationandTotalVariationflow.

Speaker:SinanGunturk(CourantInstitute,NewYorkUniversity)Title:Someoldandnewideasondigitalhalftoning

Abstract:Inthefirstpartofthistalk,wewillrevisitawell-knownclassofdigitalhalftoningalgorithmscallederrordiffusion.Wewillargueinatheoreticalsettingthatexistingerrordiffusionschemeshaveloworderofapproximationandpresentawaytoimprovethis.

Inthesecondpartofthetalk,acompletelydifferenthalftoningalgorithm,whichisbasedonmultiscaleideas,willbeintroduced.Theapproximationorderofthisnewalgorithm,alongwithsomegeneralizationsofit,willfollow.

Speaker:StaceyLevine(MathandComputerScience,DuquesneUniversity)Title:Noiseremovalandtextureextractionusingnonstandardgrowthfunctionals

Abstract:Wepresentavariationalformulationforprocessingimagesusingfunctionalswithp(x)growth(p(x)≥1).Theminimizationproblemprovidesamodelforremovingnoisewhilepreservingandenhancingedges.Inparticular,itsignificantlyreducesthe’staircasingeffect’whichcanresultinthedetectionoffalseartifacts.Furthermore,basedonthenovelimagedecompositionmodelsrecentlyintroducedbyMeyerandmodifiedbyVeseandOsher,thefunctionalcanbemodifiedtoaddresstheproblemofimagedenoisingwithtextureextraction;thisisparticularlyusefulinremotesensingimageswheretexturesareobscuringtheboundaryofanobjectofinterest.Inthistalk,themathematicalvalidityofthemodelisestablishedandnumericalresultsdemonstrateitseffectivenessinbothnoiseremoval,andnoiseremovalwithtextureextraction.Directapplicationstoboundarydetectioninremotesensingimagesarealsopresented.Speaker:BradLucier(MathematicsandComputerScience,PurdueUniversity)Title:YAWTSI:YetAnotherWayToSmoothImages(andkeepedges)

1(L)variationalsmoothing,onebasedonwaveletsAbstract:WeshowtheresultsoftwoalgorithmsforB∞1

1(similartoWaveletShrinkageforB1(L1)smoothing)andonebasedonpixelvalues(similartoChambolle’s

methodforBVsmoothing).Experimentsshowthatthequalitativepropertiesofthesmoothedimages

1(L)seminormthatoneusesinthevariationalproblem.ThisisdependcriticallyontheformoftheB∞1

jointworkwithAntoninChambolle.

Speaker:FrancoisMalgouyres(UniversiteParis13,France)Title:Imagecompressionthroughaprojectiononapolyhedralset

Abstract:Fewyearsagoanewimagerestorationmodelhasbeenproposed.Itconsistsinminimizingaregularitycriterion(inpracticethetotalvariation)amongpointsofapolyhedron.Thispaperproposetoadaptthismodelforimagecompression.Theresultsareratherconvincingwhilealotofworkstillneedstobeperformedtomakethemodelusable.Thisadaptationshowshowtotranslatetheusual”waveletcoefficientmodulusdecayrate”argument,forsuchoptimizationmodels.

Speaker:KirsiMajava(MathematicalInformationTechnology,UniversityofJyv¨askyl¨a)Title:Active-setalgorithmsforsolvingnonsmoothimagedenoisingproblems

Abstract:Inthispresentation,wediscussactive-setmethodsforsolvingnonsmoothoptimizationproblemsappearinginimagedenoising.Thebasicoptimization-basedformulationforimagedenoisingproblemcanbegiveninthefollowingform,󰀃󰀃

min

u

|u−z|sdx+β|∇u|rdx,

(1)

ΩΩ

4

wherezdenotesthenoisydataandβisaregularizationparameter.Obviously,r=2correspondstotheclassicalsmoothregularizationandr=1yieldsthetotalvariation(TV)regularization.Moreover,theusualchoices=2yieldsL2-fitting,whichassumesGaussian(normallydistributed)noisewithzeromean,whereass=1allowsheavytails(e.g.outliers)inthenoisedistribution.

Thebasicdifficultyconcerningtheimagedenoisingproblemswiths=1orr=1istheirnondiffer-entiabilityintheclassicalsense,whichexcludestheusageofcommongradient-basedsolutionmethods,suchastheconjugategradientmethod,forsolvingtheseproblems.ItoandKunisch(1999)firstpro-posedaso-calledactive-setmethod(basedontheaugmentedLagrangianregularizationofthenonsmoothoptimizationproblem)forsolvingproblem(??)withs=2,r=1.Basedontheseideas,wepresented(K¨arkk¨ainen&Majava,2000)efficientimplementationofactive-setalgorithmstosolvethesameproblem.Encouragedbythegoodperformanceofthesealgorithms,wehavedevelopedactive-setalgorithmsalsoforothercombinationsofs,r(s=1,r=2ands=r=1).Inthispresentation,wedescribethebasicideasofanactive-setmethodanddiscussthealgorithmsdeveloped.Numericalexperimentsarepresentedtotesttheefficiencyofthepresentedalgorithmsandtoillustratetherestorationcapabilityoftheformulations.ThisisjointworkwithTommiK¨arkk¨ainen(UniversityofJyv¨askyl¨a)andKarlKunisch(Karl-FranzensUniversityofGraz,Austria).

Speaker:RiccardoMarch(IstitutoperleApplicazionidelCalcolo,Roma)Title:Variationalapproximationofacurvaturedependingfunctional

Abstract:Weconsiderafunctionalforimagesegmentationwhichisdefinedonfamiliesofcurves.ThefunctionalisoftheMumford-Shahtypeanditpenalizescurvature,length,andnumberofendpointsofthecurves.AΓ-convergencetheoremispresentedfortheapproximationofsuchafunctionalbymeansofellipticfunctionals.TheapproximationiscloseinspirittotheAmbrosioandTortorelliapproximationoftheMumford-Shahfunctional.JointworkwithA.Braides.

Speaker:DimitrisMetaxas(ComputerScience,RutgersUniversity)

Title:AnimationandControlofBreakingWavesAbstract:Controllingfluidsisstillanopenandchal-lengingprobleminfluidanimation.Wehaverecentlydevelopedanovelfluidanimationcontrolapproachandwepresentitsapplicationtocontrollingbreakingwaves.InourSliceMethodframeworkananimatordefinestheshapeofabreakingwaveatadesiredmomentinitsevolutionbasedonalibraryofbreakingwaves.Oursystemcomputesthenthesubsequentdynamicswiththeaidofa3DNavier-Stokessolver.Thewavedynamicsprevioustothemomenttheanimatorexertscontrolcanalsobegeneratedbasedonthewavelibrary.Theanimatoristhusenabledtoobtainafullanimationofabreakingwavewhilecontrollingtheshapeandthetimingofthebreaking.Anadditionaladvantageofthemethodisthatitprovidesasignificantlyfastermethodforobtainingthefull3Dbreakingwaveevolutioncomparedtostartingthesimulationatanearlystageandusingsolelythe3DNavier-Stokesequations.Wepresentaseriesof2Dand3Dbreakingwaveanimationstodemonstratethepowerofthemethod.

Speaker:DimitrisMetaxas(ComputerScience,RutgersUniversity)Title:Metamorphs:DeformableShapeandTextureModels

Abstract:WepresentanewclassofdeformablemodelstermedMetamorphswhoseformulationintegratesbothshapeandinteriortexture.Themodeldeformationsarederivedfrombothboundaryandregioninformationbasedonavariationalframework.Thisframeworkrepresentsageneralizationofpreviousparametricandimplicitgeometricdeformablemodels,byincorporatingmodelinteriortextureinformation.Theshapeofthenewmodelisrepresentedimplicitlyasan“image”inthehigherdimensionalspaceofdistancetransforms.Theinteriortextureiscapturedusinganonparametrickernel-basedapproximationoftheintensityprobabilitydensityfunction(p.d.f.)insidethemodel.Thedeformationsthatthemodelcanundergoaredefinedusingaspacewarpingtechnique-thecubicB-splinebasedFreeFormDeformations(FFD).Whenusingthemodelsforboundaryfindinginimages,wederivethemodeldynamicsfromanenergyfunctionalconsistingofbothedgeenergytermsandtextureenergyterms.Thisway,themodelsdeformundertheinfluenceofforcesderivedfrombothboundaryandregioninformation.AMetaMorph

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modelcanbeinitializedfar-awayfromtheobjectboundaryandeffi-cientlyconvergetoanoptimalsolution.Theproposedenergyfunctionalenablesthemodeltopasssmallspuriousedgesandpreventsitfromleakingthroughlargeboundarygaps,hencemakestheboundaryfindingrobusttoimagenoiseandinhomogeneity.Wedemonstratethepowerofournewmodelstosegmentationapplications,andvariousexamplesonfindingobjectboundariesinnoisyimageswithcomplextexturesdemonstratethepotentialoftheproposedtechnique.

Speaker:MilaNikolova(EcoleNormaleSuperioredeCachan)

Title:Recoveryofedgesinsignalsandimagesbyminimizingnonconvexregularizedleast-squares

Abstract:Weanalyzethepropertiesofimagesandsignalsrestoredbyminimizingregularizedleast-squaresaccordingtotheshapeoftheregularizationfunction.Thisquestionisofparamountimportanceforarel-evantchoiceofregularizationterm.Wegiveboundswhichcharacterizethesmoothingincurredby(local)minimizers.Themainpointofinterestistherestorationofedges.Weshowthatundernonconvexregular-ization,thedifferencesbetweenneighboringsamplesataminimizerareeithershrinked,orenhanced.Thisnaturallyentailsaneatclassificationofdifferencesassmoothorasedges.(Thiseffectisessentiallydifferentfromedge-preservationusingconvexregularizationfunctions.)Wealsogiveconditionsforexactrestora-tionofedges.Explicitexpressionsarederivedforthetruncatedquadraticandthe“0-1”regularizationfunction.Theseresultsareillustratedusingnumericalexamples.

Speaker:MartinRumpf(Mathematics,Gerhard-Mercator-Universit¨at-GesamthochschuleDuisburg)Title:Onhigherordergeometricflowsinimageandsurfaceprocessing

Abstract:Willmoreflowiswellestablishedasamimportanttoolinmanyimageandsurfaceprocessingapplications.Afterageneraldiscussionoftheperspectivesofsuchhigherorderflowsandinparticularofthecorrespondinganisotropicversions,wewillfocusontheirappropriateformulationinlevelsetform.Thus,ageneralapproachfortheintegrationofgeometricgradientflowsoverlevelsetsensemblesispresented.Itenablestoderiveavariationalformulationforthelevelsetsolutionofvarioussecondandfourthedorderevolutionproblems,inparticulartheabovefourthorderflows.Furthermore,spatialandtemporaldiscretizationarediscussedandnumericalsimulationsarepresented.Finally,wediscussageneralimplicitnarrowbandmethodforsecondandfourthordergeometricflows.

Speaker:FadilSantosa(Mathematics,UniversityofMinnesotaatMinneapolis)Title:Aninverseprobleminnondestructiveevaluationofspotwelds

Abstract:Spotweldsareusedinattachingmetalsheetstogether.Thismethodofjoiningsheetmetalisespeciallycommonintheautoindustry.Therearemorethan20thousandspotweldsinatypicalcar,andtheyplayacrucialroleinthestructuralintegrityofthevehicle.Athermalimagingmethodfornondestructiveevaluationofspotweldshavebeenproposed.Inthismethod,atransducerisemployedtogenerateheatneartheweldwhileskintemperatureofthemetalsheetismeasured.Theinverseproblemistoaccessthequalityoftheweldfromthetemperaturereading.

Inthispresentation,wedevelopasimplemodelforthethermaldiffusionproblem.Theinverseproblemweseektosolveamountstofindingaheatsourceina2-Ddomaingiventemperatureasafunctionofspaceandtime.Wesolvethisclassicallyillposedproblembydevisingatime-steppingalgorithmwhichsolvesaregularizedproblemineachtimestep.Severalregularizationstrategiesareconsidered.Weillustratethemainideasofourworkinnumericalexamples.

Speaker:OtmarScherzer(ComputerScience,UniversityofInnsbruck)

Title:Linear,Non-linear,Non-differentiable,Non-convexregularizationinvolvingUnboundedOperatorsAbstract:Tikhonovinitiatedtheresearchonstablemethodforthenumericalsolutionofinverseandill-posedproblems.Tikhonov’sapproachconsistsinapproximatingasolutionofanoperatorequation

F(x)=y

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byaminimizerofthepenalizedfunctional

󰀊F(x)−y󰀊2+α󰀊x󰀊2

(α>0).

Inthebeginningmainlylinearill–posedproblems(i.e.Fislinear)suchascomputerizedtomographyhavebeensolvedwiththesemethods.ThetheoryofTikhonovregularizationmethodsdevelopedsystematically.Untilaround1980therehasbeensuccessinarigorousandrathercompleteanalysisofregularizationmethodsforlinearill-posedproblems.WementionthebooksofTikhonov&Arsenin,Nashed,Engl&Groetsch,Groetsch,Morozov,LouisNatterer,Bertero&Boccacci,Kirsch,Colton&Kress...In1989Engl&Kunisch&NeubauerandSeidman&Vogeldevelopedaregularizationtheoryfornon–linearinverseproblemswhereFisanon-linear,differentiableoperator.AboutthesametimeOsher&Rudinusedboundedvariationregularizationfordenoisinganddeblurring,whichconsistsinminimizationofthefunctional󰀃

󰀊F(x)−y󰀊2+α|∇x|.Thismethodishighlysuccessfulinrestoringdiscontinuities.Theanalysisofboundedvariationregular-izationissignificantlymoreinvolvedsincethepenalizationfunctionalisnotdifferentiable.Overthepastyearsthisconcepthasattractedmanymathematicalresearch.Thenextsteptowardgeneralizationofregularizationmethodsisnon-convexregularization.Herethegeneralgoalistominimizefunctionalsoftheform󰀃

g(F(x)−y,x,∇x),whichmaybenonconvexwithrespecttothethirdcomponent∇x.

AnothercomplicationsisintroducedintheanalysisofregularizationfunctionalsifforinstancetheoperatorFcanbedecomposedintoacontinuousandadiscontinuousoperator.Suchmodelshavebecomepopularforlevelsetregularizationrecently.

Speaker:VolkerSchmidt(AbteilungStochastik,Universit¨atUlm)

Title:SimultaneousnonparametricestimationofthespecificintrinsicvolumesofstationaryrandomsetsAbstract:Anactualquestioninstatisticalanalysisofimagedataisthedevelopmentofmethodsbymeansofwhichonecanautomaticallydistinguishbetweentwo(ormore)imagesofsimilarstructure.Examples,wherethistypeofdecisionproblemsappear,rangefromtestingthespatialstructureofbiologicalcellsortissuesincomputer-aidedcancerdiagnostics,viaspace-timeanalysisofcoverageandconnectivitypropertiesinmobilecommunicationsystems,tointelligentmanagementofcomplextransportationsystems.

Inthistalk,wepresentanewmethodforstatisticalanalysisofthespatialstructureofbinaryimagedata,whichcancontributetosolvetheproblemsmentionedabove.Theideabehindthismethodisanewapproachto(indirect)statisticalestimationofmorphologicalimagecharacteristics,usingtoolsofconvexandstochasticgeometry.

Moreprecisely,weinterpretbinaryimagesinthed-dimensionalEuclideanspaceasrealizationsofspatiallyhomogeneousrandomclosedsets,assumingthattheybelongtotheextendedconvexring.Then,weconstructnonparametricjointestimatorsforthed+1specificintrinsicvolumes(or,equivalently,thespecificMinkowskifunctionals)oftheserandomclosedsets,includingestimatorsforthespecificEuler-Poincarcharacteristic(or,equivalently,thespecificconnectivitynumber),thespecificsurfacearea,andthevolumefractionitself.TheestimatorsarebasedonanexplicitextensionoftheclassicalSteinerformulatotheconvexringandtheycanberepresentedbyintegralsofsomestationaryrandomfields.Thisimpliesinparticularthattheestimatorsareunbiased.Moreover,conditionsarederivedunderwhichtheyaremean-squareconsistent,andapositive-definiteandconsistentestimatorfortheirasymptoticcovariancematrixisgiven.Someissuesoftheirefficientnumericalimplementationarealsodiscussed.References:

S.Klenk,V.Schmidt,E.Spodarev(2004)AnewalgorithmicapproachtothecomputationofMinkowskifunctionalsofpolyconvexsets.Preprint(availableathttp://www.geostoch.de)

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V.Schmidt,E.Spodarev(2004)Jointestimatorsforthespecificintrinsicvolumesofstationaryrandomsets.Preprint(availableathttp://www.geostoch.de)

E.Spodarev,V.Schmidt(2004)Onthelocalconnectivitynumberofstationaryrandomclosedsets.Workingpaper(underpreparation)

Speaker:JayantShah(Mathematics,NorthesternUniversity)Title:GrayskeletonsandsegmentationofshapesAbstract:

Speaker:KaleemSiddiqi(ComputerScience&CentreForIntelligentMachines,McGillUniversity)Title:Medialintegralsforshapeanalysis

Abstract:InthistalkwediscussaverysimpletypeofintegralperformedonavectorfielddefinedasthegradientoftheEuclideandistancefunctiontotheboundingcurve(orsurface)ofabinaryobject.Thelimitingbehaviorofthisintegralastheenclosedarea(orvolume)shrinkstozerorevealsaveryusefulinvariantwhichcanbeusedtocomputetheBlumskeletonaswellastorevealthegeometryoftheobjectthatitdescribes.JointworkwithPavelDimitrovandJamesN.Damon.

Speaker:AlonSpira(ComputerScience,Technion)

Title:Geometricimageandcurveevolutiononparametricmanifolds

Abstract:ThemotionofimagesandcurvesinR2hasbeenresearchedextensively.Applicationsinimageprocessingandcomputervisionincludeimageenhancementthroughanisotropicdiffusion,imagesegmen-tationbyactivecontours,andmanyothers.Extendingthesemotionstomanifoldsembeddedinspacesofhigherdimensionscanbemostbeneficial.

Inthistalkwepresentnumericalschemesforimplementinggeometricflowsonparametricmanifolds.Weconsidera2DparameterizationplanethatismappedtoanN-dimensionalspace.OurapproachindevisingtheschemesistoimplementthemontheuniformCartesiangridoftheparameterizationplaneinsteadofdoingsointheN-dimensionalspace.Thisenhancestheefficiencyandrobustnessoftheresultingnumericalschemes.

Thefirstnumericalschemeisanefficientsolutiontotheeikonalequationonparametricmanifolds.TheschemeisbasedonKimmelandSethian’ssolutionfortriangulatedmanifolds,butusesthemetrictensoroftheparametricmanifoldinordertoimplementtheschemeontheparameterizationplane.TheschemeisusedtodeviseashorttimekernelfortheBeltramiimageenhancingflow.Thekernelenablesanarbitrarytimestepfortheflowforregularimagesaswellasimagespaintedonmanifolds,suchasfaceimages.Thenumericalschemeisfurtherusedforfacerecognitionbyconstructinganinvariantfacesignaturefromdistancescalculatedonthefacemanifold.

Anothernumericalschemeimplementscurveevolutionbygeodesiccurvatureflowonparametricman-ifolds.Theflowisimplementedbybackprojectingthecurvefromthemanifoldtotheparameterizationplane,calculatingtheflowontheplanebythelevelsetsmethodandthenmappingitbacktothemanifold.Combiningthisflowwithgeodesicconstantflowenablestheimplementationofgeodesicactivecontoursforimagespaintedonparametricmanifolds.

Thenumericalschemespresentedinthistalkenableaproperimplementationofimageprocessing,computervision,andcomputergraphicsapplicationsforimagespaintedonparametricmanifolds.Takingintoaccountthegeometryofthemanifoldspromisessuperiorresultstotheconventionalprocessingoftheimagesasregular2Dimages.

ThisisjointworkwithRonKimmel.Speaker:Xue-ChengTai(Mathematics,UniversityofBergen)

Title:Piecewiseconstantlevelsetmethodsandtheirfastsolutionsforimagesegmentation

Abstract:InthisworkwediscussvariantsofthePDEbasedlevelsetmethodproposedearlierbyOsherandSethian.Traditionallyinterfacesarerepresentedbythezerolevelsetofcontinuousfunctions.Weinsteadusepiecewiseconstantlevelset(PCLS)functions,i.e.thelevelsetfunctionequalstoaconstantineach

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oftheregionsthatwewanttoidentify.Usingthemethodsforinterfaceproblems,weneedtominimizeasmoothconvexfunctionalunderaconstraint.Thelevelsetfunctionsarediscontinuousatconvergence,buttheminimizationfunctionalissmoothandlocallyconvex.Themethodsaretrulyvariational,i.e.alltheequationsweneedtosolvearetheEuler-Lagrangianequationsfromtheminimizationfunctionals.Thus,thefastNewtontypeofmethodcanbeeasilyused.Weshownumericalresultsusingthemethodsforsegmentationofdigitalimages.Applicationtoinverseproblemswillbebrieflydiscussed.

Weshallpresenttwovariantsofthepiecewiseconstantlevelsetmethods(PCLSM).Oneofthemisabletousejustonelevelsetfunctionforidentifyingmultiphaseproblemswitharbitrarynumberofphases.Anothervariant,whichwecallthebinarylevelsetmethod,onlyrequiresthelevelsetfunctionequals1or-1.Thegeometricalquantitiesliketheboundarylengthandareaofthesubdomaincanbeeasilyexpressedasfunctionsofthenewlevelsetfunctions.

ThisisajointworkwithJ.LieandM.Lysaker.ThepaperscanbedownloadedfromtheUCLAcam-reportwebpage:http://www.math.ucla.edu/applied/cam/

Speaker:RichardTsai(Mathematics,UniversityofTexasatAustin)Title:ThresholdDynamicsfortheMumford-ShahFunctional

Abstract:WeproposedafastalgorithmforconstructingminimizingsequencesfortheMumford-Shahfunctionalforimagesegmentationapplications.TheproposedmethodismotivatedbythethresholddynamicsoftheMerriman-Bence-Osherschemethatisproposedformeancurvaturemotion.Wepresentseveralnumericalexamplesthatarecarefullydesignedandtestedtoreflectthebehavioroftheproposedscheme.ThecomplexityofourschemeisO(N),whereNisthenumberofpixelsinagivenimage.

ThisisajointworkwithSelimEsedoglu.Speaker:BabaVemuri(ComputerandInformationScienceandEngineering,Univ.ofFlorida)Title:AnAffineInvariant“Distance”MeasureforDiffusionTensorMRISegmentation

Abstract:Diffusiontensorimages(DTI),whicharematrixvalueddatasets,haverecentlyattractedin-creasedattentioninthefieldsofmedicalimagingandvisualization.Inthistalk,Iwillpresentanoveldefinitionoftensor“distance”groundedinconceptsfrominformationtheoryandincorporateitinthesegmentationofDTI.InaDTI,thesymmetricpositivedefinite(SPD)diffusiontensorateachvoxelcanbeinterpretedasthecovariancematrixofalocalGaussiandistribution.Thus,anaturalmeasureofdissimi-laritybetweenSPDtensorswouldbetheKLdivergenceoritsrelative.ThesquarerootoftheJ-divergence(symmetrizedKL)betweentwoGaussiandistributionscorrespondingtothetensorsbeingcomparedisproposedandthisleadstoanovelclosedformexpressionforthe“distance”aswellasthemeanvalueofaDTI.UnlikethetraditionalFrobeniusnorm-basedtensordistance,our“distance”isaffineinvariant,adesirablepropertyinmanyapplications.Thisnewtensor“distance”isthenincorporatedinaregionbasedactivecontourmodelforDTIsegmentation.Syntheticandrealdataexperimentsareshowntodepicttheperformanceoftheproposedmodel.

Speaker:LuminataVese(Mathematics,UniversityofCaliforniaatLosAngeles)

Title:Decompositionofimagesintocartoonandtextureusingthetotalvariationanddiv(BMO)

Abstract:Animportantprobleminimageanalysisistheseparationoflargescales(cartoonfeatures)fromsmallerperiodicscales(texture)inimages.YvesMeyersuggestedthatmodelssuchasMumford-ShahorRudin-Osher-Fatemicanbeviewedasdecompositionmodelsintocartoonandtexture,andnotonlyasimagesegmentationandrestorationmodels.Inthesetwomodels,thetexturecomponentismodeledbyasquare-integrablefunction.FollowingY.Meyer,weproposeandanalyzeamodelwherethetexturedcomponentbelongstothespacediv(BMO)insteadofL2,whilethecartooncomponentisafunctionofboundedvariation.Theoretical,approximationsandnumericalresultsofimagedecompositionwillbepresented.

ThisisjointworkwithTrietLe,UCLA.Speaker:KevinVixie(MathematicalModelingandAnalysis(T-7),LosAlamosNationalLab)

Title:ExactsolutionsfortheTotalVariationMinimizationwithanL1DataFidelityTerm:Containment,

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Convexity,andCurvature

Abstract:TheROFmodel,whichhasseenquiteabitofdevelopmentandstudyhas,asadatafidelityterm,thesquareoftheL2distancebetweenthemeasuredimageandtheproposedsolutionimage.Whilethistermiseasytodealwithanalytically,italsohaslessdesirableeffectssuchaslossofcontrast.IfthedatafidelityischangedtotheL1distance,weobtainadifferent,L1TVfunctional.

InthistalkIwillpresentjointworkwithSelimEsedogluonexactsolutionstoL1TVfunctionalminimizationwhenthatinitial“measured”imageisacharacteristicfunctionofsomeboundedset(actuallyaCaccioppoliset).Ourthreeresultshowthatinfactthesomeexactsolutionscanbecharacterizedintermsofdiskswithradii1/λand2/λ

Speaker:CurtVogel(MathematicalSciences,MontanaStateUniversity)Title:TrackingEyeMotionfromRetinalScanData

Abstract:Thistalkdealswithseveraldifferentaspectsofimageanalysis.Theproblembeingaddressed—trackingthemotionoftheretinaoftheeye—isimportantinunderstandinghowthehumanvisualsystemworks.Ourdatacomesfromanadaptiveopticsscanninglaseropthalomoscope(AOSLO),adevicewhichyieldshighresolutionscansofthelivingretina.Difficultiesarisewhentheeyemovesasthescansaretaken.Ourproblembecomesthefollowing:Givenaseriesofscansofroughlythesameobject,determineboththemotionoftheobjectandtheopticalpropertiesoftheobject.Tosolvethisproblem,weapplyavariantofDavidArathorn’smap-seekingcircuit(MSC)algorithm.Inadditiontobeingofmathematicalinterestinitsownright,MSCprovidesaplausiblemechanismforbiologicalvision.

ThisisjointworkwithDavidArathorn,CenterforComputationalBiologyatMontanaStateUniversityandAustinRoorda,SchoolofOptometryattheUniversityofHouston,theinventoroftheAOSLO.Speaker:RossWhitaker(ComputerScience,UniversityofUtah)Title:UINTA:Unsupervised,Information-Theoretic,AdaptiveFiltering

Abstract:Theproblemofdenoisingimagesisoneofthemostimportantandwidelystudiedproblemsinimageprocessingandcomputervision.Variousimagefilteringstrategiesbasedonlinearsystems,statistics,informationtheory,andvariationalcalculus,havebeeneffective,butinvariablymakestrongassumptionsaboutthepropertiesofthesignaland/ornoise.Thereforetheylackthegeneralitytobeeasilyappliedtonewapplicationsordiverseimagecollections.Thispaperdescribesanovelunsupervised,information-theoretic,adaptivefilter(UINTA)whichimprovesthepredictabilityofimagepixelsfromtheirneighborhoodsbyminimizinganinformation-theoreticmeasureofgoodness.InthiswayUINTAautomaticallydiscoversthestatisticalpropertiesofthesignalandcantherebyreduceimagenoiseinawidespectrumofimagesandapplications.Thetalkdescribestheformulationrequiredtominimizethejointentropymeasure,presentsseveralimportantpracticalconsiderationsinestimatingimage-regionstatistics,andthenpresentsaseriesofresultsandcomparisonsonbothrealandsyntheticdata.

Speaker:LiorWolf(CenterforBiologicalandComputationalLearning,MIT)Title:LearningusingtheBornRule

Abstract:InQuantumMechanicsthetransitionfromadeterministicdescriptiontoaprobabilisticoneisdoneusingasimpleruletermedtheBornrule.Thisrulestatesthattheprobabilityofanoutcome(a)givenastate(Ψ)isthesquareoftheirinnerproducts((a󰀁Ψ)2).

Inthistalk,IwillexploretheuseoftheBorn-rule-basedprobabilitiesforclusteringandimageseg-mentation,featureselection,classificationandobjectrecognition,andforimageretrieval.Weshowhowtheseprobabilitiesleadtoexistingandnewalgebraicalgorithmsforwhichnoothercompleteprobabilisticjustificationisknown,formingaconnectionbetweenspectraltheoryandprobabilitytheory.

Speaker:AnthonyYezzi(Systems&ControlandBioengineering,GeorgiaTech)Title:ConformalH0MetricsontheSpaceofCurves

Abstract:EversincetheintroductionofsnakesbyKass,Witkin,andTerzopoulos,activecontourshave

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playedaprominentroleinavarietyofimageprocessingandcomputervisiontasks,mostnotablyseg-mentation.Earlyreasearchonactivecontourssawthetansitionfromparameterizationdependendmodelstogeometricmodelsindependentoftheparameterizationoftheevolvingcurve.Next,thereweremanyeffortstoincorporatedregionbasedimageinformationtomaketheactivecontourdependuponglobalinformationabouttheimageratherthanjustthetraditionallocallycomputededgedescriptors.Inrecentyears,thelatesttrendinactivecontourresearchseemstobethatofincorporatingglobalshapepriorsintotheactivecontourparadigm.Thishasbroughtupnon-trivialquestionssuchashowtodefinean”averageshape”orhowtocharacterize”variationsinshape”.Allofthesequestionsultimatelyleadtoamorebasicandfundamentalquestionofhowtomeasurethedistancebetweentwogivenshapes.Goingeverdeeper,onemightaskwhenaschemeformeasuringdistancesbetweenshapesisassociatedwithatrueRiemannianmetriconthemanifoldofallpossiblesmoothshapesand,ifso,whataretheinterestingpropertiesofthismetric(forexample,whatarethegeodesics).

Inthistalk,wewillfollowthistop-downapproach,startingwithadiscussionofshapepriorsintheactivecontourframework,notionsofaverageshape,distancesbetweenshapes,andultimatelyRiemannianmetricsonthemanifoldofshapes.Wewilldiscussthesurprisinglypathologiesassociatedwithwhatseemstobethemostnaturalmetric(H0)onthespaceofcurvesandshowhowitispossibletoaddressthesepathologieswithoutdrasticallychangingthemetricstructurebytheintroductionofawell-chosenconformalfactor.

Speaker:HaoMinZhou(Mathematics,GeorgiaTech)

Title:ThePDEandVariationalTechniquesinWaveletTransformsandTheirApplicationsinImagePro-cessing

Abstract:Standardwaveletlinearapproximations(truncatinghighfrequencycoefficients)generateoscilla-tions(Gibbs’phenomenon)nearsingularitiesinpiecewisesmoothfunctions.Nonlinearanddatadependentmethodsareoftenusedtoovercomethisproblem.Recently,partialdifferentialequation(PDE)andvari-ationaltechniqueshavebeenintroducedintowavelettransformsforthesamepurpose.

Thistalkwillincludeourworkontwodifferentapproachesinthisdirection.OneistousePDEideastodirectlychangewavelettransformalgorithmssoastogeneratewaveletcoefficientswhichcanavoidoscillationsinreconstructionswhenthehighfrequencycoefficientsaretruncated.WehavedesignedanadaptiveENOwavelettransformbyusingideasfromEssentiallyNon-Oscillatory(ENO)schemesfornumericalshockcapturing.ENO-wavelettransformsretainstheessentialpropertiesandadvantagesofstandardwavelettransformswithoutanyedgeartifacts.Wehaveshownthestabilityandarigorouserrorboundwhichdependsonlyonthesizeofthederivativeofthefunctionawayfromthediscontinuities.

ThesecondoneistostaywithstandardwavelettransformsandusevariationalPDEtechniquestomodifythecoefficientsinthetruncationprocesssothattheoscillationsarereducedinthereconstructionprocesses.Inparticular,weuseminimizationoftotalvariation(TV),toselectandmodifytheretainedstandardwaveletcoefficientssothatthereconstructedimageshavefeweroscillationsnearedges.WehavealsoproposednewTVbasedwaveletmodelsforimagereconstructions.

Examplesinapplicationsincludingimagecompression,denoising,inpaintingarepresented.

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