The circlet transform A robust tool for detecting feat - PDF document

Thecirclettransform:ArobusttoolfordetectingfeatureswithcircularshapesH.Chauris,I.Karoui,P.Garreau,H.Wackernagel,P.Craneguy,L.BertinoCentredeGe Correspondingauthorat:CentredeGeosciences,MinesParisTech,35rue,77300Fontainebleau,France.E-mailaddress:herve.chauris@mines-paristech.fr(H.Chauris). Computers&Geosciences37(2011)331…342 extensionofwaveletswheretheelementsusedforthedecom-positionareelongatedalongoneaxisandoscillatingintheperpendiculardirection.WerefertoMaandPlonka(2010)forareviewoncurveletapplications.Boththewaveletandcurveletanalysisdonotrevealtheopticdisk.Theproposedherehasbeendesignedtodetectobjectswithcircularshapes.Asforwaveletsandcurvelets,itpreservesthemulti-scaleaspect.However,italsohasthepossibilitytofollowglobalstructuresasfortheHT.Theobjectiveofthenewtransformfordetectingcircularshapesshouldaddressthefollowingissues:(1)theusershouldbeabletospecifyarangeofradiifortheselection;(2)thesegmentationpartshouldbeavoidedformorerobustness;(3)thecircularshapeshouldbeanannulusofacertainwidthinordertohandlenon-perfectlycircularshapes.Thisisalsoawaytotakeintoaccounttheband-limitedaspectofthedata;(4)thetransformshouldbeperfectlyinvertible:ifallcoef“cientsareselected,thentheinvertedimageshouldbethesameastheoriginalimage;(5)thetransformshouldbefastenoughtoenableiterativeprocessessuchasthesoft-thresholdingtechniqueforreducingthenumberofkeyelementsneededtorepresentthecircularstructuresintheimage(Donoho,1995Theoutlineofthepaperisthefollowing.We“rstdescribetransform.Wethenpresentthealgorithmusedtoselectonlyafewrepresentativecoef“cients.Weshowapplicationsinthreedifferent“elds:ophthalmology,moonexplorationandcoastaloceanography,onmainlyrealimages.Theapplicationsrangefromthedetectionofasingleandclearcircularshapetothe“ndingofaseriesofcircularshapeswithmorediffusecontours.Finally,wediscussthecurrentlimitationsofthemethodandwhatcouldbeimproved,inparticulartoremovesomespuriousdetections.2.TheIntheproposedapproach,thereisnoneedforbinaryimagesegmentation.Themethodconsistsofdecomposinganyimageintocircleswithdifferentradiiandacertainwidth,viaaseriesoffastFouriertransforms(FFTs).Thesecirclesarecalledtheycanbeseenastheconvolutionofacirclewitha1Doscillatoryfunction,possiblyawavelet,inthesamewayasrelatetowaves.However,thestrategyforcomputingthecoef“cientsisratherdifferentfromtheclassicalapproachusedforwaveletdecomposition.Usually,waveletcoef“cientsareobtainedviaaseriesofcascadedconvolutionsanddown-sampling(Daubechies,1992).Asexplainedbelow,thedecompositionisformulatedintheFourierdomainwiththede“nitionofspeci“c“lters,followingasimilarapproachastheoneproposedbyCandesetal.(2005)forcurvelets. Fig.1.Retinographyimage(left)andassociatedimagegradientmaps(right),with(A)opticdisk,(B)retinalbloodvessels,and(C)exudates.Localgradientvaluesarehigheraround(B)and(C).Onlytheopticdiskhasacircularshape. 400 600 800 1400 400 600 800 400 600 800 1400 400 600 800 Fig.2.ReconstructedimageafterMeyerwavelet(left)andcurvelet(right)transforms.Ineachcase,onlycoef“cientswithvalueshigherthan40%ofthemaximumvalueareselected.Othercoef“cientsaresettozero.H.Chaurisetal./Computers&Geosciences37(2011)331…342 2.1.Generalframeworkelementsarecharacterizedbyacentralposition),aradiusandacentralfrequencycontentFig.3).This“nitefrequencyprovidesacertainwidthtotheinthespatialdomain.ThisisthemaindifferencewiththeHoughtransform,beyondtheimplementationaspects.AllcanbededucedfromareferencecircletceitherbyashiftorbymodifyingtheradiusorthecentralfrequencycontentoftheFig.4,Eq.(1)).Theparametersfullycharacterizeeach.Formally,thefunctioncanbewrittenasistypicallyanoscillatoryfunc-tion,possiblyawaveletfunction,designedtodetectdisconti-nuities(Daubechies,1992).Fromapracticalpointofview,beexplicitlyde“nedinthe2DFourierdomain.2.2.ThecircletdecompositionTheforwardandinversetransformissimilarinessencetothecurvelettransform(CandesandDonoho,2004;Candesetal., y coordinate 50 100 150 200 250 300 50 100 150 200 250 300 x coordinatey coordinate 50 150 200 250 300 150 200 250 300 Fig.3.Representationofasingle(left)andits2DFouriertransform(right).Byconstruction,itiswell-localizedintheFourierdomain. x coordinate 100 200 300 50 100 150 200 250 300 x coordinatey coordinate 100 200 300 50 100 150 200 250 300 x coordinatey coordinate 100 200 300 50 100 150 200 250 300 Fig.4.Representationofthereference(left)andothercorrespondingtoadifferentradius(middle)andtoadifferentfrequencycontent(right).Artifactsvisibleontheimageinthemiddleareduetoanabrupttruncationintheshapeof“lters(Fig.5 3210123 Frequency (rad)Amplitude Fig.5.Representationofasingle“lter2determiningthefrequencycontentoftheH.Chaurisetal./Computers&Geosciences37(2011)331…342 ).Theobjectiveistodecomposeany2Dimage)intoasumofbasicfunctionsForcurvelets,thebasicelementshaveelongatedshapes,similartotherepresentationoflocalplanewaves(Candesetal.,).For,thebasicfunctionsarecircular.Theconstructionusesthepropertiesofatightframe,sothattheassociatedamplitudesareobtainedbyascalarproductFromapracticalpointofview,thecoef“cientsarede“nedintheFourierdomain,usingParsevalstheorem y coordinate 100 200 300 50 100 150 200 250 300 x coordinatey coordinate 100 200 300 50 100 150 200 250 300 x coordinatey coordinate 100 200 300 50 100 150 200 250 300 Fig.6.Simpleinputimage(left),imagereconstructedafterselectionofallcoef“cientscorrespondingtoasingle2value,andradiusvaluesin[10,55]pixels(middle),andimageafterselectionof0.01%ofcoef“cientswiththehighestamplitudes. 200400600800100012001400 200400600800100012001400 200400600800100012001400 200400600800100012001400 Fig.7.Imagesofeyefundus.Fordetection,radiusrangevariesfrom30to80pixels.H.Chaurisetal./Computers&Geosciences37(2011)331…342 denotesthe2DFouriertransformoftheconjugate.Withthisformulation,thetransformisconstructedinthe2DFourierdomain.Thekeystepconsistsofde“ning,theFouriertransformof.Thisisobtainedbydevelopingappropriate“lterstoensurethatthebasicfunctionshavecircularshapes.2.3.De“nitionof“ltersThe“lterconstructionisobtainedinatwo-stepprocess:“rstwederive1D“ltersandthen2D“lters.Both“ltersarede“nedinthefrequencydomainandformapartitionofit:for,wehaveThisconditionisimportanttoensureaperfectreconstructionscheme(Candesetal.,2005).Firstde“neisthenumberof“lters.For,otherwise0.Notethatthe“ltersaresymmetric(Fig.5).Onecaneasilycheckthatthe“ltersformapartitionofthe1Dfrequencydomain.The“ltersarede“nedfromthe“ltersbyintroducingaphasedelayinordertocreateacircularshapeinthespacedomain,.Oncethe“ltersde“ned,Eq.(8)providesanexplicitformulationfortheFouriertransformofa)isthecentralpositionandwheretheradiusofthe.Byde“nition,denotesthescalarproduct.Wethushave.TheindexcontrolsthefrequencycontentoftheFig.4).WithEqs.(5)and(7),itisalsoeasytoseethatforanygivenvalues,the“ltersalsoformapartitionofthe2Dfrequencydomain.Byusingpolarcoordinates,weshowinAppendixAthatthe2DinverseFouriertransformofiscircular,meaningthatthebasisfunctionshavecircularshapes.2.4.PracticalimplementationTheforwardtransformconsistsof(1)a2DFouriertransformoftheoriginalimage)toobtain;(2)forall“ltersandforallselectedvalues,amultiplicationof;(3)theinverseFouriertransformoftheproductthatprovidesallthecoef“cientsrelatedtoscaleandradius 200400600800100012001400 200400600800100012001400 200400600800100012001400 200400600800100012001400 Fig.8.Imagesofaneyefundus(top)andassociatedgradientmaps(bottom).Bloodvesselsandexudates(seeFig.1)havelargegradients.Forthedetection,radiusrangevariesfrom30to80pixels.H.Chaurisetal./Computers&Geosciences37(2011)331…342 Theinversetransformfollowsthesamerule.Firstapplya2DFouriertransformforallscalesandselectedradii,multiplybytheconjugateofandsumallresults.The“nalimageisobtainedbyapplyinga2DinverseFouriertransform.Becauseoftheconditionon(Eq.(6)),wehaveaperfectreconstructionschemeifallcircletcoef“cientsarepreserved.Formoredetailsontheforwardandinversetransforms,werefertothecurvelettransform(Candesetal.,).Themaindifferencewiththecircletconstructionisthechoiceofthe“lters.Fromapracticalpointofview,weratherselectasinglescale(i.e.single“lter)andaseriesofradii,withexpectedvaluesfrommin0totopotentiallyemphasizecircularformswithsomespeci“cspatialsizes.3.ApplicationsWe“rstindicatehowtochoosethekeycoef“cientsandthenpresentapplicationstodifferent“elds. 300 400 100 200 300 300 400 100 200 300 Fig.10.Zoomfromtheprevious“gure,whereonlyconsideredasartifactsarerepresented,superimposedontheimage(left)anditsgradient(right).Thesearetangenttoarclinesduetothecombinationofthesunshadowandthetopography. 200 300 400 500 100 200 300 400 500 200 300 400 500 100 200 300 400 500 Fig.9.Satelliteimagefromthemoon(left),withthe“rst13selected(right).Radiusrangevariesfrom25to60pixels.H.Chaurisetal./Computers&Geosciences37(2011)331…342 3.1.SelectionofrepresentativecircletsFig.6illustratesaverysimpleexample.Theinputimageconsistsofrectangularandcircularshapes.Thecoef“cientswiththehighestabsoluteamplitudesareassociatedwithcircularshapes.Theselectionofthesecoef“cientsleadstoanimagefreedfromtherectangles.Inthecircletdomain,thecoordinatesofthecoef“cientsdirectlyindicatethecentersofcirclesandtheirassociatedradii.Formorecomplicatedcaseswherethecontoursarediffuse,weapplythefollowingstrategy.Asapre-processingstep,performaspatialgradient(discreteLaplacianoperator)totheoriginalimageinordertoemphasizethediscontinuitiesinthedata.Inthecaseofsatelliteimages,theyclassicallysufferfrommissinginforma-tionduetothepresenceofclouds.Wetheninterpolatethedatabyageostatistical“lteringmethod(kriging)thatprovidesresultsspatiallyconsistentwiththeoriginaldata(Wackernagel,2003Forthedetectionofasinglecircularstructure,wesimplyselectthehighestcoef“cient.Formorecircles,weusedthesoft-thresholdingapproachinaniterativeprocess(Donoho,1995First,weselectthehighestcoef“cient.Thetransformisredundant,meaningthatthenumberofcoef“cientsislargerthanthesizeoftheinputdata.Thenextcoef“cientswithhighvalueshaveapproximatelythesameradiusandcentralpositions.Forthatreason,wesettozeroallcoef“cientsassociatedtothesameradiusandspatiallyclosetotheselectedcoef“cient.Byclose,wemeanadistancelowerthanhalfoftheradius.Itdoesnotpreventfromselectingacoef“cientwithadifferentradiusaroundthesamecentralposition.Wethenrecompose/decomposetheimagetoobtainnewcoef“cients.Theoperationisrepeateduntilthenumberofkeycoef“cientsspeci“edbytheuserisreached.Thecombinationofreconstructionanddecompositionisnotnecessa-rilyneeded.Itisusedherebecausethetransformisredundant, 100 150 200 50 100 150 150 100 150 Fig.12.SyntheticSSTmap.Radiusrangevariesfrom10to30pixels. 200 300 400 500 100 200 300 400 500 200 300 400 500 100 200 300 400 500 Fig.11.Othersatelliteimagefromthemoon(left),withthe“rst16selected(right).Radiusrangevariesfrom15to75pixels.H.Chaurisetal./Computers&Geosciences37(2011)331…342 differentcombinationsofcoef“cientsmayrepresentthesameimage.ThisiterativeapproachisfeasiblewiththeuseofFFTs.Forthedifferentapplications,wespecifytheradiusrangesinthecaptionofthe“gures.3.2.OphthalmologyThemotivationshavebeenpresentedintheintroductionpart.Inthe“rstexample(Fig.7),theopticdiskisclearlyde“nedandthealgorithmeasilydetectsit.InFig.7(bottom),thebrightzonewithintheopticdiskisnotselectedastheminimumradiusissetto30px.Inaslightlymorecomplicatedexamplewiththepresenceofexudates(Fig.8),thealgorithmisstillabletodetecttheopticdisk.Itscircularshapecompensatesforalowervalueofthegradient.Weconcludeonthisexampletheabilityforthetodetectatleastasinglecircle.3.3.AstronomyCountingcratersisamethodforestimatingtheageofaplanetssurface(Kerr,2006).WeselectedtwozonesfromtheMoonimageanddetectedanumberofcratersinaspeci“edradiusrange(Figs.9and11).Thesoft-thresholdingprocessas 100 150 200 50 100 150 200 50 100 150 200 50 100 150 200 50 100 150 200 50 100 150 200 50 100 150 200 50 100 150 200 Fig.13.SyntheticSSTmap(top)anditsgradient(bottom).Radiusrangevariesfrom15to40pixels.H.Chaurisetal./Computers&Geosciences37(2011)331…342 proposedhereallowstoremovecircleswiththesameradiiandslightlydifferentpositions.Withinthespeci“edrange,thecratersareindeedwelldetected(Fig.9,right).However,somespuriouseventsarealsoselected(Fig.10).Theseareduetothecombinationoftheshallowsunandthetopography.Weindicateinthediscussionparthowtheartifactscouldberemoved.Theapplicationonasecondmoonimageleadstosimilarconclusions(Fig.11).Onthecrateronthebottomleftside,theshapeisratherelliptic.Inthatcase,twosigni“cantweredetected.3.4.OceanographyIncoastaloceanography,thedetectionofeddieswithasub-mesoscalestructure(20…100km)isakeyelementforabetterunderstandingofthesurfacecirculation.Onremotesensingimages,tracerssuchasthesea…surfacetemperature(SST)orthechlorophyllmaps,areusedtorevealtheoceancirculation.Theunderlyingassumptionisastrongcorrelationbetweenthetracerandthevelocity“eld(Sugimuraetal.,1984;BorisovandMonin,1989;Essen,1995).Theeddiesaregenerallydetectedeitheronspatialortemporalgradientmaps(AleksaninandAleksanina,2001;Yangetal.,2000).WerefertoCastellani(2005)andNascimento(2006)Haietal.(2008);DAlimonte(2009)foramorecompletereview.Classically,theHoughtransformandseveralothercircleorellipse“ttingalgorithmsareappliedtodeterminetheradiusandthecentralpositionofeddiesonbinaryedgemaps(PeckinpaughandHolyer,1994;FernandesandNascimento,2006Themaindif“cultyfordetectingeddiesiscertainlyduetotheweakandblurrycontours.Weapplythetransformonbothsyntheticandsatelliteimages.Inthe“rstexample,themaincircularfeatureisextractedfromanSSTimagecomputedfromamodelinaturbulent”ow(Fig.12limitthespiralshape.Onalargerimage,thesoft-thresholdingprocesscoupledwiththetransformisabletodetectaseriesofturbulentstructures(Fig.13).Furtherre“ne-mentsareindicatedinthediscussionpart.Remotesensingimagesareusuallymoreblurry(Figs.14…16transformisabletodetectthemaineddyonSSTimages.Forthechlorophyllmap,the“rstfourelementsarerelatedtospiralfeatures:thechlorophyll“lamentistrappedinacycloniceddy(Fig.16).Theuserhastospecifythenumberof.Abroaderselectionsimplyconsistsofresumingtheiterativethresholdingprocess.4.DiscussionThroughaseriesofapplicationsonsyntheticandrealimages,wehaveshownthecapabilityforthetransformtodetect 100 150 50 100 50 100 150 50 100 50 100 150 50 100 50 100 150 50 100 Fig.14.RemotesensingSSTimageintheGulfofLion,NorthWesternMediterraneanSea(top),andimagegradientmaps(bottom).Radiusrangevariesfrom10to20H.Chaurisetal./Computers&Geosciences37(2011)331…342 50 100 150 200 50 100 50 100 150 200 50 100 150 200 50 100 50 100 150 200 Fig.15.RemovesensingSSTimage(top)anditsgradient(bottom),stillintheGulfofLion.Radiusrangevariesfrom10to20pixels. 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300 Fig.16.Gradientimageofthechlorophyllmap(left),withthe“rstfourmoresigni“cantcircularstructures(right).Fortheselection,theradiusrangevariesfrom5to20pixels.ThespiralshapeonthebottomleftisdetectedbytwoH.Chaurisetal./Computers&Geosciences37(2011)331…342 circularshapes.Theapplicationsarecertainlynotrestrictedtothe“eldspresentedhere.Thewidthoftheisimportanttohandleblurryandweakcontrasts.Theusermayeasilychangetheshapeofthe1DwaveletundertheconditionthatEq.(5)issatis“ed.The”exibilityalsocomesfromthepossibilitytospecifyradiusranges.Thelargeredundancyofthetransformisduetotheselectionofmanyradii.Withthesoft-thresholdingapproachpresentedhere,thisisnotreallyanissue,exceptforapplicationsonverylargeimages.Inpractice,theseimagescouldbesplitintosmallerimages.Thesizeoftheoverlappingzoneshouldbetwicethemaximumspeci“edradiustoavoidedgeeffects.Wehaveobservedsomespuriousdetections(e.g.Fig.10The“rstpossibilitywouldbetomeasurethelocalcoherencyalongarcsfortheselected.Thiswouldremoveartifactsrelatedtotheshadowonthemoonimage.Theotherpossibilitywouldbetodetectcircularstructuresonasequenceofimagesandtrackthem.Applicationsincoastaloceanographywouldbeinteresting,inparticularforremotesensingimageswithweakcontrasts(Maetal.,2006).Otherinvestigationscoulddealwiththemulti-scaleapproachprovidedby.Currently,wehaveselectedasinglescaleandvariousradii.Thesameanalysiscouldberepeatedfordifferentscales.Thiscoulddeliveramorerobustapproachifcircularshapesarestableatdifferent5.ConclusionWehavepresenteda”exiblemethodfordetectingdisconti-nuitieswithcircularshapeson2Dimages.Thekeypropertyofthetransformiscertainlythe“nitefrequencyaspectofthebasisfunctions.Thetransformisef“cientduetoitsimplementationintheFourierdomain.Combinedwithasoft-thresholdingalgorithm,itappearstosuccessfullydetectcircularstructuresonaseriesofimagesfromdifferent“elds.Thenextstepistotrackthesestructuresonasequenceofimages.Thisisalsopotentiallyawaytoremovesomespuriousdetections.Theauthorsaregratefultothetworeviewersforconstructivecomments.TheyacknowledgepartialfundingfromthePRECOCprojectfromtheFranco-NorwegianFoundation.TheythankJianweiMa(TsinghuaUniversity,China)fortheapplicationontheimagefromthemoon.TheyalsothankEtienneDecenciereandJean-ClaudeKlein(MinesParistech)forthemedicalimages.TheeyeimageswerekindlyprovidedbytheMessidorprogrampartners(seeAppendixA.CircularshapesWeaimatprovingthatwiththede“nitionofEq.(7),thehavecircularshapes.Inotherterms,thefunctionsexpressedinpolarcoordinatesshouldnotdependon.Byde“nition,wehave.TheinverseFouriertransformof 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