SmoothingMedical/Non-MedicalImageData
JasjitS.Suri,SeniorIEEEMember&DeeWu
MRClinicalResearchDivisionPhilipsMedicalSystems,Inc.,
Cleveland,OH,USA
email:jasjit.suri@cle.philips.comABSTRACT
J.Gao**,SameerSingh***,&SwamyLaxminarayan****
**KLA-Tencor,Milpitas,CA,USA
***PANNResearch,UniversityofExeter,Exeter,UK****Dept.ofBiomedicalEngineering,NJIT,Newark,NJ,USA
Abstract
Partialdifferentialequations(PDE’s)havedominatedimageprocessingresearchrecently(seeSurietal.[1],[2],[3],[4],[5],[6]andHaker[7]).Thethreemainreasonsfortheirsuccessare:(1)theirabilitytotransformasegmentationmodelingproblemintoapartialdifferen-tialequationframeworkandtheirabilitytoembedandintegratedifferentregularizersintothesemodels;(2)theirabilitytosolvePDE’sinthelevelsetframeworkusingfinitedifferencemethods;and(3)theireasyextensiontoahigherdimensionalspace.
ThispaperisanattempttosummarizePDE’sandtheirsolutionsappliedtoimagediffusion.Thepaperfirstpresentsthefundamentaldiffusionequation.Next,themulti-channelanisotropicdiffusionimagingispresented,followedbytensornon-linearanisotropicdiffusion.WealsopresenttheanisotropicdiffusionbasedonPDEandtheTukey/Huberweightfunctionforimagenoiseremoval.ThepaperalsocoverstherecentgrowthofimagedenoisingusingthecurveevolutionapproachandimagedenoisingusinghistogrammodificationbasedonPDE.Finally,thepaperpresentsthenon-linearimagedenoising.Examplescoveringbothsyntheticandrealworldimagesarepresented.
Keywords:ImageDenoising,Smoothing,Filtering,Diffusion,PartialDifferentialEquations(PDE’s).
toprovidesolutionsinafast,stableandclosedform;andlastly,(8)theirabilitytointeractivelyhandleimageseg-mentationinthePDEframework.
ApplicationofPDEhasrecentlybecomemorepromi-nentinthebiomedicalandnon-biomedicalimagingfields(seeChambolle[20]andMoreletal.[21])forshapere-covery(seetherecently1publishedbookbySapiro[22]).Thisisbecausetheroleofshaperecoveryhasalwaysbeenacriticalcomponentin2-Dand3-Dmedicalandnon-medicalimagery.Thisassistslargelyinmedicaltherapy,andobjectdetection/trackinginindustrialapplications,re-spectively(seetherecentbookbySurietal.[4]andthereferencesthereinandalsoseeWeickertetal.[23],[24]andthereferencestherein).Shaperecoveryofmedicalor-gansinmedicalimagesismoredifficultcomparedtootherimagingfields.Thisisprimarilyduetothelargeshapevari-ability,structurecomplexity,severalkindsofartifactsandrestrictivebodyscanningmethods.WithPDE-basedseg-mentationtechniques,ithasbeenpossibletointegratelowlevelvisiontechniquestomakethesegmentationsystemrobust,reliable,fast,closed-formandaccurate.ThispaperrevisitstheapplicationofPDEinthefieldofcomputervi-sionandimageprocessinganddemonstratestheabilitytomodelsegmentationinPDEandthelevelsetframework.BeforediscussingPDEtechniquesindetail,wewillfirstdiscussthedifferentkindsofPDEapplications.
2.DiffusionImaging:ImageSmoothingand
RestorationViaPDE
Thepresenseofnoiseinimagesisunavoidable.ThiscouldbeintroducedbytheimageformationprocessinMR,CT,X-rayorPETimages,orimagerecordingorevenanim-agetransmissionprocess.Severalmethods2havebeenpresentedinnoiseremovalandsmoothing,butthispa-perfocuseson“noiseremoval”or“noisediffusion”usingPDE.Thishasbeenusedforquitesometimebutrecently,robusttechniquesforimagesmoothinghavebeendevel-oped(seePeronaetal.[8],[9],Gerigetal.[10],Alvarezetal.[11],[12],Kimiaetal.[13],[14],Sapiroetal.[15],Casellesetal.[16],Weickert[17],Blacketal.[18],Ar-1Published2morphological
1.Introduction
Partialdifferentialequations(PDE’s)haverecentlydomi-natedthefieldsofcomputervision,imageprocessingandappliedmathematicsduetothefollowingreasons:(1)theirabilitytotransformasegmentationmodelingproblemintoaPDEframework;(2)theirabilitytoembedandintegrateregularizersintothesemodels;(3)theirabilitytosolvePDE’susingfinitedifferencemethods(FDM);(4)theirabilitytolinkbetweenPDE’sandthelevelsetframeworkforimplementingfinitedifferencemethods;(5)theirabil-itytoextendthePDEframeworkfrom2-Dto3-Dorevenhigherdimensions;(6)theirabilitytocontrolthedegreeofPDEintheimageprocessingdomain;(7)theirability
inJan.2001.
smoothing,linear,non-linear,geometric
ridgeetal.[19],Bajlaetal.[37],Olveretal.[38],Scherzeretal.[39],Romenyetal.[40],[41]andNielsenetal.[42]).Thissectioncoversthesearticlesinthefollowingway:Thefundamentaldiffusionequationisgiveninsub-section2.1.Sub-section2.2presentsmulti-channelanisotropicdiffu-sionimaging.Tensornon-linearanisotropicdiffusionisdiscussedinsub-section2.3.AnisotropicdiffusionbasedonPDEandtheTukey/Huberweightfunctionisdiscussedinsub-section2.4.Imagedenoisingusingthecurveevo-lutionapproachispresentedinsub-section2.5.Finally,imagedenoisingandhistogrammodificationusingPDEispresentedinsub-section2.6.2.1Perona-MalikAnisotropicImageDiffu-sionViaPDE(Perona)
OneofthefirstpapersondiffusionwasfromPeronaandMalik,calledPerona-MalikAnisotropicDiffusion(PMAD)3[8].PMAD’sideawasbasedononeoftheear-lierpapersbyWitkin[29].Peronaetal.[8]gavethefunda-mentalPDE-baseddiffusionequationforimagesmoothingas:
(1)
whereisthedivergence4operator,wastherateofchangeofimage,wasthediffusionconstantat
location
attimeandwasthegradientoftheim-age.Applyingthedivergenceoperator,theaboveequa-tioncanbere-writttenas:
(2)
wherewastheLaplacianoperator,wasthegradientofthediffusionconstantatlocationfortimeandwasthegradientoftheimage.Thediffusionconstantwasthekeyfactorinthesmoothingprocess.Per-onaetal.gavetwoexpressionsforthediffusionconstants:
and
,
wherewastheabsolutevalueofthegradientoftheimageattimeandwasaconstantwhichwaseithermanipulatedmanuallyforsomefixedvalueorcom-putedusinga“noiseestimator”asdescribedbyCanny[34].Usingthefinitedifferencemethod,Eq.2wasdis-cretizedto:
werewherethefinitedifference,diffusion,constantsand
for
timeinfourdifferentdirections(north,south,eastandwest)giventhecentrallocation.Thevaluesoftheseconstantswerechoseneitherexponentiallyorasaratioasdiscussedabove.ToseetheperformanceofthePMAD,wetookthreesetsofexamples:Incaseone,wetookasim-pletwopetalflowerimage,addedtheGaussiannoiseandappliedthePMADoverit.Theresultsoftheinput/ouput
3Alsocallededge-baseddiffusion.
4
operationcanbeseeninFigure1.Wetookamorecompleximageofaflowerimagewitheightpetals,addedthesameGaussiannoiseandappliedthePMADoverit.TheresultscanbeseeninFigure2.Inthesamefigure,wecomparedthePMADwithPollaketal.’sinversediffusionmethod(IDM)5(seePollaketal.[35])andfoundbetterresultsus-ingPollak’setal.’sIDM.Inthethirdexample,weappliedthePMADandPollaketal.’sIDMovernoisyfunctionalMRIdataofthebrain.TheresultscanbeseeninFigure3.
Recently,Perona[9]alsodefinedtheangular6diffu-sionbasedonthemagneticconceptofattractingobjects.Thisresearchisoutofthescopeofthispaper.
ProsandConsofPMADUsingPDE:ThefollowingarethemajoradvantagesofthePMAD:(1)Sincethiswasoneofthefirstpapersintheareaofimagediffusion,itgaveagoodpresentationonscale-spaceandanisotropicdiffusionforimagedenoising.(2)Thismethodtakesrelativelylittleexecutiontimeandwasgoodforgeneralpurposeapplica-tions.Thefollowingarethemajordisadvantagesofthismethod:(1)ThePMADmethodbroughtblurringatsmalldiscontinuitiesandhadapropertyofsharpeningedges(seeGerigetal.[10]).(2)ThePMADmethoddidnotincor-porateconvergencecriteria(seeGerigetal.[10]).(3)Themethodwasnotrobusttohandlelargeamountsofnoise.Themethoddidnotpreservethediscontinuitiesbetweenregions.(4)Themethodneededtoadjusttuningconstantssuchas.(5)Themethoddidnottakeintoaccountinho-mogeneousdatasampling.
Figure1.TwopetalflowerwithGaussiannoiseaddedandPMADappliedtoit.(lefttoright):Noisyim-ageswith20,10,5,3and1db.
(lefttoright):PMAD-PDEdiffusionresultsfromtheabovenoisyimages.Thediffusionconstantusedwas:
,wherewaschosenas10,thetimestepas
1andthetotaliterationswere200.
5Recently,
Pollaketal.[35]proposedadiffusionmethodwhichwas
differentfromPMADin:(1)discontinuingtheinverseflowfunctionand(2)mergingtheregionsduringdiffusion.Thefirstfeaturemadethedif-fusionstable.Everylocalmaximumwasdecreasedandeverylocalmin-imumwasincreased.Thesecondfeaturemadethealgorithmfastandunique.
6orientational
2.2Multi-ChannelAnisotropicImageDiffu-sionViaPDE(Gerig)
Recently,Gerigetal.[10]developedthenon-lineardif-fusionsystemforsmoothingornoisereductioninMRbrainimages.Thismethodwascalledmulti-channel7anisotropicdiffusionorcoupled8anisotropicdiffusion.KeepingPMAD’sdiffusioninmind,themulti-channelanisotropicimagediffusionwasgivenas:
(3)
wherethecoupleddiffusioncoefficient
wascomputedfrommulti-channeldatasetsand.andweretherateofchangeofmulti-channelimages.Thecoupleddiffusionwas
given
as:
and
,
where
wastheabsolutevalueofthe
gradientofthemulti-channelimagesandandwasaconstantasusedbyPMAD.Note,ifdiscontinuitiesweredetectedinbothchannels,thenthecombineddiffusioncoefficientwaslargerthananysinglecomponentandthesignificanceoflocalestimationswasincreased.Ontheotherhand,ifadiscontinuitywasdetectedonlyinoneofthechannels,thecombinedcoefficientrespondedtothediscontinuityandhaltedthediffusion.
ProsandConsofMulti-ChannelAnisotropicDiffusion:Themajoradvantagesofthistechniqueare:(1)Efficientnoisereductioninhomogeneousregions.(2)Edgeen-hancementandpreservationofobjectcontours,boundariesbetweendifferenttissuesandsmallstructuressuchasves-sels.(3)Filteredimagesappearclearerandboundariesarebetterdefined,leadingtoanimproveddifferentiationofad-jacentregionsofsimilarintensitycharacteristics.Thema-jorweaknessesofthistechniqueare:(1)ThepaperdidnotshowhowthePDEflowwouldbehaveandhowtheconver-gencewouldgetaffectedifthe“coupledPDEdiffusion”wascomputed.(2)Thenumberofiterationsinthesmooth-ingprocesswasselectedbyvisualcomparison.Thustheconvergencetosteady-stateandstoppingcriteriawasfuzzy.(3)Therewasnodiscussiononthetimecomputationfor3-Dfilteringfornon-isotropic9volumes.(4)Selectionoftheparameterwasnotautomatic.
7for
example,MRbraindataacquiredfromthreedifferentscantech-niques:-,-and-weighted
8ThisnamecameaboutsincethediffusioncoefficientwascoupledbetweentwodifferentMRdatasets.
9Anon-isotropicvolumeisavolumewhosethreedimensionsarenotsame.
Figure2.Thisflowerimagehasalargersetofpetals
comparedtoFigure1andconvolutions.(lefttoright):Noisyimageswith20,10,5,3,1and1db.
(lefttoright):PMAD-PDEdiffusionre-sultsfromtheabovenoisyimages.Thediffusionconstant
usedwas:
,wherewaschosenas10,thetimestepas1andthetotalnumberofiterationsusedwas200.Note,thelastimage(rightmost)inthebot-tomrowwastheresultduetoPollaketal.’sIDM[35].Thenumberofiterationsusedwas500.
Figure3.NoisyfMRIimageswithPMADandPollaket
al.’sIDMappliedtothem.
:fMRInoisybrainaxialslice.:PMAD-PDEdiffusionresults.Thediffu-sionconstantusedwas:,where
waschosenas10,thetimestepas0.1andthetotalnumber
ofiterationsusedwas200.
:Pollaketal.’sIDMwithtimestep0.1andnumberofiterationsusedwas500.Asseeninthisfigure,Pollaketal.’smethoddidthebestinremovingthenoiseandhighlightingthegraymatter/whitematterinterface(fordetailsontheimportanceofWM/GMinterface,seeSurietal.[5]).
2.3TensorNon-LinearAnisotropicDiffu-sionViaPDE(Weickert)
TocombattheproblemsofPMAD,Weickert[17]pro-posedatrulyanisotropicdiffusion,calledTensorNon-LinearAnisotropicDiffusion(TNAD)andwasmathemat-icallygivenas:
(4)
wherewasthediffusiontensorhavingeigenvectorsand
anddefinedinawaysuchthat:and
theeigenvalues
.Weickertsuggestedchoosingandsuchthat:and.Thesam-pleresultsofthistechniquecanbeseeninFigure4taken
fromthesource(seeWeickertetal.[48]).DetailsonthismethodcanbeseeninthebookbyWeickert[17].Otherauthorswhodidworkinnon-linearanisotropicdiffusionwere:Schn¨orr[25],[26]andCatteetal.[27],[28].
ProsandConsofTensorNon-LinearAnisotropicDiffu-sion:ThefollowingisthemainadvantageoftheTNADmethod:(1)Themethodperformedwellonawidevarietyofimages.ThefollowingistheweaknessoftheTNADmethod:(1)Themethoddidnottakeintoaccountinhomo-geneousdatasampling.
Figure4.Tensorimagediffusionona3-Dultrasoundim-age.:Renderingofa3-Dultrasounddatasetofa
10-weekoldhumanfetus.
:Renderingafterdenois-ingbyTensorNon-LinearAnisotropicDiffusion(TNAD)filtering(seeWeickertetal.[48]).CourtesyofProfessorWeickert,ComputerVision,GraphicsandPatternRecog-nitionGroup,DepartmentofMathematicsandComputerScience,UniversityofMannheim,Mannheim,Germany.
2.4AnisotropicDiffusionUsingthe
Tukey/HuberWeightFunction(Black)
Blacketal.[18]recentlyshowedacomparativestudybe-tweenPerona-MalikAnisotropicDiffusion(PMAD)basedonacombinationofPDEandTukey’sbiweightestima-tor,knownasBlacketal.’sRobustAnisotropicDiffu-sion(BRAD).Inthisimagesmoothingprocess(estimating
piecewiseconstant),thegoalwastominimizethatsatis-fied:
(5)
wherewasthepixellocation,tookoneofthefourneigh-boursandwasthesetoffourneighboursof.wasthe
errornormfunction(calledarobustestimator,seeMeeretal.[43])andwasthescaleparameter.Therelation-shipbetweenandPMADwasexpressedas:IfPMADwasgivenbythefunction,where,
then
,thederivativeoftheerrornormfunc-tion.Blacketal.usedTukey’sBiweightandtheHubermin-maxfunctionforerrornorm.TheTukeyfunctionwasgivenas:,ifand
,if.TheHuber’smin-maxforer-rornormwas.Notethatwas,ifor,if.computedBlacketas:al.used=1.4gradientmedian
de-scentforsolvingtheimagesmoothingminimizationprob-lem.ThusthediscretesolutionofBRADwas:
(6)
wherewasthederivativeoferrornorm,.Thewhole
ideaofbringingtherobuststatisticswastoremoveout-liersandpreserveshapeboundaries.ThusBRAD=PMAD+Huber/Tukey’sRobustestimator.Figure5comparesPerona-Malik’sAnisotropicDiffusion(PMAD)withBlacketal.’sTukeyfunction(BRAD).Thefigureshowsthere-sultsfor100and500iterations.Ascanbeseen,theresultsofTukeyarefarsuperiorcomparedtoPMAD.Also,atin-finity,thePMADwillmaketheimagegoflatandtheTukeywillnot10.ReadersinterestedintheapplicationofHuber’sweightfunctioninmedicalapplicationcanseetheexcel-lentworkbySuri,HaralickandSheehanetal.[44].Here,thegoalwastoremovetheoutlierlongitudinalaxesoftheLeftVentricle(LV)forruledsurfaceestimationtomodelthemovementoftheLVoftheheart.
ProsandConsofAnisotropicDiffusionBasedonRo-bustStatistics:ThefollowingarethemainadvantagesoftheBRADmethod:(1)ThemethodwasmorerobustcomparedtoPMAD.(2)Themethodwasstableforalargenumberofiterationsandtheimagedidnotgetflatwhenwasinfinity.ThefollowingarethemaindisadvantagesoftheBRADmethod:(1)TherewasnodiscussiononhowthewascomputedfordesignoftheHuber’sweightfunction.(2)Therewasnodiscussiononthecomputationaltime.
2.5ImageDenoisingUsingPDEandCurve
Evolution(Sarti)
Recently,Sartietal.[49]presentedanimagedenoisingmethodbasedonthecurveevolutionconcept.Beforewe
10Personal
communicationwithProfessorSapiro.
Figure5.ComparisonofthePerona-MalikfunctionandTukeyfunctionafter100and500iterations.ResultsofPerona-MalikPDEandBlacketal.’sTukeymethods.Leftmost:PMADwithnumberofiterationssetto100.LeftMiddle:Tukeyfunctionwithnumberofiterationssetto100.RightMiddle:PMADwithnumberofiter-ationssetto500iterations.Rightmost:Tukeyfunctionwithnumberofiterationssetto500(CourtesyofProfessorSapiro,DepartmentofElectricalandComputerEngineer-ing,UniversityofMinnesota,Minneapolis,MN,source:Blacketal.[18]).
discusswhatSartietal.did,wewilljustpresentthefunda-mentalequationpresentedbyKichenassamyetal.[50]and
Yezzietal.[51].Theypresentedthecurveevolutionmodelbyintroducinganextrastoppingterm.Thiswasexpressedmathematicallyas:
(7)
Notethatdenotedtheprojectionofanattrac-tiveforcevectoronthenormaltothecurve.Thisforcewasrealizedasthegradientofapotentialfield.Thispotentialfieldforthe2-Dand3-Dcasewasgivenas:
,respectively,whereand
wasthegradientGaussianopera-tor,andweretheconvolutionoperations,respectively.NotethatEquation7issimilartoEquation7givenbyMalladietal.in[52].Mal-ladietal.callstheequationasanadditionalconstraintonthesurfacemotion.RewritingEquation7ofMalladietal.[52]becomes:
(8)
wherewastheedgestrengthconstant,wasacon-stant(1asusedbyMalladietal.),wasthecurvaturedependentspeed,wastheconstanttermcontrollingthecurvaturedependentspeedandwasthesameasdefinedabove.Havingpresentedthelevelsetequationintermsofspeedfunctionsandconstants,Sartietal.’smethodchangedtheaboveequationbyremovingthecon-stantpropagationforce
andsimplysolvedtheremain-ingequationforimagedenoising.Thesmoothingworked
inthefollowingway:If
waslarge,theflowwasslowandtheexactlocationoftheedgewasre-tained.If
wassmall,thentheflowtendedtobefast,therebyincreasingthesmoothingprocess.Thefilteringmodelwasreducedtomeancurvatureflowwhen
wasanequationtounity.Thus,thedataconsistency
term
servedasanedgeindicator.Theconvolutionop-erationsimplyeliminatedtheinfluenceofspuriousnoise.Sartietal.thussolvedEq.7iterativelyandprogressivelysmootheditovertime.Theedgeindicatorfunctionbe-camesmootherandsmootherovertimeanddependedlessandlessonthespuriousnoise.NotethattheminimalsizeofthedetailisrelatedtothesizeoftheGaussiankernel,whichactedlikeascaleparameter.Thevarianceofwastakenaswhichcorrespondedtothedimen-sionofthesmalleststructurethathadtobepreserved.Thesharpeningoftheedgeinformationwasduetothehyper-bolicterm
.ProsandConsofImageDenoisingUsingPDEandCurveEvolution:Thefollowingarethemajoradvan-tagesofsuchamethod:(1)Themethodwasasimpleex-tensionfromthebasiccurveevolutionmethod.(2)Thesharpeningoftheedgeinformationwasduetothehyper-bolicterm
.Thefollowingaretheweaknesses:(1)Therewasnotmuchdiscussiononthescale-spacepa-rameter,thevarianceof
.Thiswasoneofthecriti-calpiecesintheimagedenoising.(2)TherewasnotmuchdiscussiononthestoppingmethodforthePDE.Sincecon-stantforceplaysacriticalroleandwillbeseenaheadasbeingusedasa“regularizerforce”insegmentationmodel-ingandastopperaswell,wefeelthatconstantforcecanbeusedmoreefficientlyratherthanplainremovalinimagedenoisingprocess.
2.6ImageDenoisingandHistogramModifi-cationUsingPDE(Sapiro)
Adescriptionondiffusionwouldbeincompleteifref-erencesfromSapiroweremissed.Recently,Sapiroetal.[30],[31]and[15]developedtheensembleoftwodif-ferentalgorithmsinonePDE.Thismethodusedhistogrammodification(orhistogramequalization)andimagede-noisinginonePDE.Thiswascallededgepreservationanisotropicdiffusion.Theideawastosmooththeimageonlyinthedirectionparalleltotheedges,achievingthisviacurvatureflows.Thiswasdoneusingthefollowingflowequation:
(9)
wherewasthelevelsetfunctionwhichevolvedaccordingtotheaffineheatflowforplanarshapesmoothingwiththevelocitygivenby,wherewastheconvo-lutionoperator.Now,thehistogramequalizationflowwasgivenas:
(10)
whereimage
;representedarea(ornumberofpixels).CombiningEq.9andEq.10
yieldedajointPDEas:
(11)
segmentationsincetheycanhandleanyofthecavities,con-cavities,convolutedness,splittingormerging.(5)FindingtheLocalandGlobalMinima:Thereisnoproblemfind-ingthelocalminimaorglobalminimaissues,unliketheoptimizationtechniquesofparametricsnakes.(6)NormalComputation:Thesetechniquesarelesspronetothenor-Here,isdefinedas:Ifthedensityfunctionofim-age
is,where,then
wastheweighting,likea“cumulated”factor.Theredensityhasbeenfunction.someresearchHere,
whichrelatesanisotropicdiffusion,curveevolutionandsegmentation.InterestedresearcherscanlookatSapiro[33]andShah[36].
ProsandConsofImageDenoisingandHistogramMod-ificationViaPDE:Thefollowingarethemajoradvan-tagesofsuchamethod:(1)Thispapergaveusanex-ampleonhowtousedifferentkindsofdiffusiontogether.(2)Thismethodsmoothedtheimagesonlyinthedirec-tionparalleltotheedges.Thefollowingistheweaknessofsuchamethod:(1)Therewasnodiscussionabouthow
toselecttheparameter.
Havingdiscussedthedif-ferentimagediffusiontechniquesbasedonPDE,andtheirprosandcons,interestedreaderscangointomoredetailonbehavioralanalysisonanisotropicdiffusioninimagepro-cessing(seeYouetal.[32]).Also,seeknowledge-basedtensoranisotropicdiffusionforcardiacMRIbySanchez-Ortizetal.[46].AdetailedreviewonPDE-baseddiffusionandacomparisonbetweendifferentsmoothingtechniquesusingPDE,scale-spacemathematicalmorphologyandin-versediffusiontechniquescanbeseenbySurietal.[45].
3.Advantages,DisadvantagesandConclu-sions3.1AdvantagesofPDEFramework
ThePDEbasedmethodinthelevelsetframeworkoffersalargenumberofadvantagesthatareasfollows:(1)CaptureRange:Thegreatestadvantageofthistechniqueisthatthisalgorithmincreasesthecapturerangeofthe“fieldflow”,whichincreasestherobustnessoftheinitialcontourplace-ment.(2)EffectofLocalNoise:Whentheregionalinfor-mationisintegratedintothesystem,thenthelocalnoiseoredgedoesnotdistractthegrowthprocess.Thistechniqueisnon-localandthusthelocalnoisecannotdistractthefinalplacementofthecontourorthediffusiongrowthprocess.(3)NoNeedofElasticityCoefficients:Thesetechniquesarenotcontrolledbytheelasticitycoefficients,unliketheclas-sicalparametriccontourmethods.Thereisnoneedtofitthetangentstothecurvesandcomputethenormalsateachvertex.Inthissystem,thenormalsareembeddedinthesystemusingthedivergenceofthefieldflow.Thesemeth-odshavetheabilitytomodeltheincrementaldeformationsintheshape.(4)SuitabilityforMedicalImageSegmenta-tion:Thesetechniquesareverysuitableformedicalorgan
malcomputationalerrorwhichisveryeasilyincorporatedinthe“classicalballoonforce”snakesforsegmentation.(7)Automaticity:Itisveryeasytoextendthismodelfromsemi-automatictocompletelyautomaticbecausetheregionisdeterminedonthebasisofpriorinformation.(8)Integra-tionofRegionalStatistics:Thesetechniquesarebasedonthepropagationofcurves(justlikethepropagationofrip-plesinthetankorpropagationofthefireflames)utilizingtheregionstatistics.(9)FlexibleTopology:Thesetech-niquesadjusttothetopologicalchangesofthegivenshapesuchasjoiningandbreakingofthecurves.(10)WideAp-plications:Thistechniquecanbeappliedtounimodal,bi-modalandmulti-modalimagery,whichmeansitcanhavemultiplegrayscalevaluesinit.ThesePDE/levelsetbasedmethodshaveawiderangeofapplicationsin3-Dsurfacemodeling.(11)SpeedoftheSystem:Thesetechniquecanbeimplementedusingthefastmarchingmethodsinthenar-rowbandandthuscanbeeasilyoptimized.(12)Extension:Thistechniqueisaneasyextensionfrom2-Dto3-D.(13)IncorporationofRegularizingTerms:Thiscaneasilyincor-porateotherfeaturesforcontrollingthespeedofthecurve.Thisisdonebyaddinganextratermtotheregion,gradientandcurvaturespeedterms.(14)HandlingCorners:Thesystemtakescareofthecornerseasilyunliketheclassicalparametriccurves,whereitneedsspecialhandlingatthecornersoftheboundary.(15)ResolutionChanges:Thistechniqueisextendabletomulti-scaleresolutions,whichmeansthatatlowerresolutions,onecancomputeregionalsegmentations.Thesesegmentedresultscanthenbeusedforhigherresolutions.(16)Multi-phaseProcessing:Thesetechniquesareextendabletomulti-phase,whichallowsthatiftherearemultiplelevelsetfunctions,thentheyautomat-icallymergeandsplitduringthecourseofthesegmenta-tionprocess.(17)SurfaceTracking:Trackingsurfacesareimplementedusinglevelsetsverysmoothly.(18)Quan-tificationof3-DStructures:Geometricalcomputationscanbedoneinanaturalway,forexample,onecancomputethecurvatureof3-Dsurfacesdirectlywhileperformingnormalcomputations.(19)IntegrationofRegularizationTerms:Allowseasyintegrationofvisionmodelsforshaperecov-erysuchasfuzzyclustering,Gibbs’model,MarkovRan-domFieldsandBayesianmodels(seeParagiosetal.[53]).Thismakesthesystemverypowerful,robustandaccurateformedicalshaperecovery.Onecansegmentanypartofthebraindependinguponthemembershipfunctionofthebrainimage.So,dependinguponthenumberofclassesestimated,onecansegmentanyshapein2-Dor3-D.(20)ConciseDescriptions:Onecangiveconcisedescriptionsofdifferentialstructuresusinglevelsetmethods.Thisisbecauseofthebackgroundmeshresolutioncontrols.(21)HierarchicalRepresentations:Thelevelsetoffersanatu-
ralscalespaceforhierarchicalrepresentations.(22)Repa-rameterization:Thereisnoneedforreparameterizationforcurve/surfaceestimationduringthepropagation,unlikeintheclassicalsnakesmodel.(23)ModelinginaContin-uousDomain:OnecanmodelthesegmentationprocessinacontinuousdomainusingPDE’s.Thustheformalismprocessisgreatlysimplifiedwhichisgridindependentandisotropic.(24)StabilityIssues:Withthehelpofresearchinnumericalanalysis,onecanachievehighlystablesegmen-tationalgorithmsusingPDE’s.(25)ExistenceandUnique-ness:UsingPDE’s,onecanderivenotonlysuccessfulal-gorithmsbutalsousefultheoreticalresults,suchasexis-tenceanduniquenessofsolutions(seeAlvarezetal.[12]).
3.2DisadvantagesofPDEinLevelSets
Eventhoughlevelsetshavedominatedseveralfieldsofimagingscience,thesefrontpropagationalgorithmshavecertaindrawbacks.Theyareasfollows:(1)ConvergenceIssue:Althoughtheedgeswillnotbeblurrywhenoneper-formsthediffusionimaging,theissueofconvergenceal-waysremainsachallenge.Indiffusionimaging,ifthestepsizeissmall,thenittakeslongertoconverge.(2)DesignoftheConstantForce:ThedesignoftheconstantforceinthePDEisanotherchallenge.Thisinvolvescomputationofregionalstatisticsintheregionofthemovingcontour.Thereisatrade-offbetweentherobustnessoftheregionaldesign,computationaltimefortheoperationandtheaccu-racyofthesegmentation.Thedesignofthemodelplaysacriticalroleinsegmentationaccuracyandremainsasachallenge.Anotherchallengeoccursifthedesignforceisinternalorexternal(fordetails,seeSurietal.[6]).(3)Sta-bilityIssues:ThestabilityissuesinPDE’sarealsoimpor-tantduringthefrontpropagation.Theratioof
,calledtheCFLnumber11,isanotherfactorwhichneedstobecare-fullydesigned.(4)InitialPlacementoftheContour:Oneofthemajordrawbacksoftheparametricactivecontoursisitsinitialplacement.Thisdoesnothaveeitherenoughcapturerangeorenoughpowertograbthetopologyoftheshapes.Bothofthesedrawbacksareremovedbylevelsetsprovidedtheinitialcontourisplacedsymmetricallywithrespecttotheboundariesofinterest.Thisensuresthatthelevelsetsreachobjectboundariesalmostatthesametime.Onthecontrary,iftheinitialcontourismuchclosertothefirstportionoftheobjectboundarycomparedtothesecondportion,thentheevolvingcontourcrossesoverthefirstpor-tionoftheobjectboundary.Thisisbecausethestopfunc-tiondoesnotturnouttobezero.Oneofthecontrollingfactorsforthestopfunctionisthegradientoftheimage.Therelationshipofthestopfunctiontothegradientisitsinverseandalsodependsupontheindexpowerinthera-tio.Forstoppingthepropagation,thede-nominatorshouldbelarge,whichmeanstheimageforcesduetothegradientshouldbehigh.Thismeansindexshouldbehigh.Inotherwords,ifishigh,thenthegra-11Courant
number,namedaftertheauthorCourantetal.[54]
dientishigh,whichmeanstheweakboundariesarenotde-tectedwellandwillbeeasilycrossedoverbytheevolving
curve.If
islow(lowthreshold),thenthelevelsetwillstopatnoisyoratisolatededges.(5)EmbeddingoftheOb-ject:Ifsomeobjects(say,theinnerobjects)areembeddedinanotherobject(theouterobject),thenthelevelsetwillnotcapturealltheobjectsofinterest.Thisisespeciallytrueiftheembeddedobjectsareasymmetricallysituated.Undersuchconditions,oneneedsmultipleinitializationsoftheactivecontours.Thismeanstherecanbeonlyoneactivecontourperobject.(6)GapsintheObjectBound-aries:ThisisoneoftheseriousdrawbacksofthelevelsetmethodandhasbeenpointedoutbySiddiqietal.[55].Duetothegapsintheobject,theevolvingcontoursimplyleaksthroughthegaps.Asaresult,theobjectsrepresentedbyincompletecontoursarenotcapturedcorrectlyandfully.Thisisespeciallyprominentinrealisticimages,suchasinultrasoundandinmulti-classMRandCTimages.(7)Prob-lemsDuetoShocks:Shocksareamongthemostcommonproblemsinlevelsets.Kimiaandco-workersin[56],[57],[58]developedsuchaframeworkbyrepresentingshapeasthesetofsingularities(calledshocks)thatariseinarichspaceofshapedeformationsasclassifiedintothefollowingfourtypes:(i)first-ordershocksareorientationdiscontinu-ities(corners)andarisefromprotrusionsandindentations;(ii)second-ordershocksareformedwhenashapebreaksintotwopartsduringadeformation;(iii)third-ordershocksrepresentbends;and(iv)fourth-ordershocksaretheseedsforeachcomponentofashape.Theseshocksariseinlevelsetsandcansometimescauseseriousproblems.(8)Chal-lengeinSegmentation:Althoughthelevelsetsegmentationmethodsucceedsintheobjectandmotionsegmentation,ithasweaknessinsegmentingmanyotherkindsofimages.Theseimagesmostlydonothaveahomogeneousback-ground;instead,theyarecomposedofmanydifferentre-gions,suchasinnaturalsceneryimages(containingstreets,mountains,trees,carsandpeople).Themethodbasedoncurveevolutionwillnotproducethecorrectregionsasde-sired.Suchasegmentationproblemisachallengetobeovercome.
3.3ConclusionsandtheFutureinPDE-basedMethods
Theclassofdifferentialgeometry,alsocalledPDEincon-junctionwithlevelsets,hasbeenshowntodominateim-ageprocessing,inparticulartomedicalimaging,inama-jorway.Westillneedtounderstandhowtheregular-izationtermscanbeintegratedintothelevelsetstoim-provesegmentationschemes.Eventhoughtheapplicationoflevelsetshasgonewellinthefieldsofmedicalimag-ing,biomedicine,fluidmechanics,combustion,solidifica-tion,CAD/CAM,objecttracking/imagesequenceanalysisanddevicefabrication,wearestillfarawayfromachiev-ingstable3-Dvolumesandastandardsegmentationinreal-time.Bystandard,wemeanthatwhichcansegment
the3-Dvolumewithawidevariationinpulsesequenceparameters.Wewillseeinthenearfuturethemodelingoffrontpropagationthattakesintoaccountthephysicalconstraintsoftheproblem,forexample,minimizationofvariationgeodesicdistances,ratherthansimpledistancetransforms.Wewillalsoseemoreincorporationoflikeli-hoodfunctionsandadaptivefuzzymodelstopreventleak-ingofthecurves/surfaces.Agoodexampleofintegra-tionofthelowlevelprocessesintotheevolutionprocess
wouldbegivenas:
,where,whereisthe
lowlevelprocessfromedgedetection,opticalflow,stereodisparity,texture,etc.Thebetterthe,themorero-bustwouldbethelevelsetsegmentationprocess.Wealsohopetoseemorepapersonlevelsetswherethesegmen-tationstepdoesrequireare-initializationstage(seeZhaoetal.[59]andEvansetal.[60]).Itwouldalso,however,behelpfulifwecanincorporateafastertriangulationalgo-rithmforisosurfaceextractionin3-Dsegmentationmeth-ods.
Wealsoseeamassiveeffortbythecomputervisioncommunitytointegrateregularizationtermstoimprovetherobustnessandaccuracyofthe3-Dsegmentationtech-niques.Inthispaper,weshowedtheroleofPDEandlevelsetmethodforimagesmoothingorimagediffusionorim-agedenoising.Alsoshownwashowcurve/surfaceprop-agationhypersurfacesbasedondifferentialgeometryareusedforthesegmentationofobjectsinstillimagery.Wealsoshowedtherelationshipbetweentheparametricde-formablemodelsandcurveevolutionframework;incorpo-rationofclamping/stoppingforcestoimprovetherobust-nessofthesetopologicallyindependentcurves/surfaces.Wespentaconsiderableamountoftimeindiscussingseg-mentationofanobjectinmotionimagerybasedonPDEandthelevelsetframework.Thepaperalsopresentedre-searchintheareaofcoupledPDE’sforedgepreservationandsmoothing.SomecoveragewasalsogivenonPDEinmiscellaneousapplications.Finally,thispaperconcludedwiththeadvantagesandthedisadvantagesofsegmentationmodelingviageometricdeformablemodels(GDM),PDEandlevelsets.
Acknowledgements:ThanksareduetoDr.ElaineKeelerfromMarconiMedicalSystems,Inc.,Cleveland,OH,Dr.GeorgeThoma,NationalInstitutesofHealth,Bethesda,MD,ProfessorLindaShapiro,UniversityofWashington,Seattle,WA,fortheirmotivations.
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