scale space.docx

scale space.docx

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scalespace

Scalespace

Scale-spacetheoryisaframeworkformulti-scalesignalrepresentationdevelopedbythecomputervision,imageprocessingandsignalprocessingcommunitieswithcomplementarymotivationsfromphysicsandbiologicalvision.Itisaformaltheoryforhandlingimagestructuresatdifferentscales,byrepresentinganimageasaone-parameterfamilyofsmoothedimages,thescale-spacerepresentation,parametrizedbythesizeofthesmoothingkernelusedforsuppressingfine-scalestructures.[1][2][3][4][5][6][7]Theparametertinthisfamilyisreferredtoasthescaleparameter,withtheinterpretationthatimagestructuresofspatialsizesmallerthanabout

havelargelybeensmoothedawayinthescale-spacelevelatscalet.

Themaintypeofscale-spaceisthelinear（Gaussian）scale-space,whichhaswideapplicabilityaswellastheattractivepropertyofbeingpossibletoderivefromasmallsetofscale-spaceaxioms.Thecorrespondingscale-spaceframeworkencompassesatheoryforGaussianderivativeoperators,whichcanbeusedasabasisforexpressingalargeclassofvisualoperationsforcomputerizedsystemsthatprocessvisualinformation.Thisframeworkalsoallowsvisualoperationstobemadescaleinvariant,whichisnecessaryfordealingwiththesizevariationsthatmayoccurinimagedata,becausereal-worldobjectsmaybeofdifferentsizesandinadditionthedistancebetweentheobjectandthecameramaybeunknownandmayvarydependingonthecircumstances.[8]

Definition

Thenotionofscale-spaceappliestosignalsofarbitrarynumbersofvariables.Themostcommoncaseintheliteratureappliestotwo-dimensionalimages,whichiswhatispresentedhere.Foragivenimagef（x,y）,itslinear（Gaussian）scale-spacerepresentationisafamilyofderivedsignalsL（x,y;t）definedbytheconvolutionoff（x,y）withtheGaussiankernel

suchthat

wherethesemicolonintheargumentofLimpliesthattheconvolutionisperformedonlyoverthevariablesx,y,whilethescaleparametertafterthesemicolonjustindicateswhichscalelevelisbeingdefined.ThisdefinitionofLworksforacontinuumofscales

buttypicallyonlyafinitediscretesetoflevelsinthescale-spacerepresentationwouldbeactuallyconsidered.

tisthevarianceoftheGaussianfilterandasalimitfort=0thefiltergbecomesanimpulsefunctionsuchthatL（x,y;0）=f（x,y）,thatis,thescale-spacerepresentationatscalelevelt=0istheimagefitself.Astincreases,Listheresultofsmoothingfwithalargerandlargerfilter,therebyremovingmoreandmoreofthedetailswhichtheimagecontains.Sincethestandarddeviationofthefilteris

detailswhicharesignificantlysmallerthanthisvaluearetoalargeextentremovedfromtheimageatscaleparametert,seethefollowingfigureand[9]forgraphicalillustrations.

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Scale-spacerepresentationL（x,y;t）atscalet=0,correspondingtotheoriginalimagef

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Scale-spacerepresentationL（x,y;t）atscalet=1

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Scale-spacerepresentationL（x,y;t）atscalet=4

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Scale-spacerepresentationL（x,y;t）atscalet=16

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Scale-spacerepresentationL（x,y;t）atscalet=64

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Scale-spacerepresentationL（x,y;t）atscalet=256

[edit]WhyaGaussianfilter?

Whenfacedwiththetaskofgeneratingamulti-scalerepresentationonemayask:

Couldanyfiltergoflow-passtypeandwithaparametertwhichdeterminesitswidthbeusedtogenerateascale-space?

Theanswerisno,asitisofcrucialimportancethatthesmoothingfilterdoesnotintroducenewspuriousstructuresatcoarsescalesthatdonotcorrespondtosimplificationsofcorrespondingstructuresatfinerscales.Inthescale-spaceliterature,anumberofdifferentwayshavebeenexpressedtoformulatethiscriterioninprecisemathematicalterms.

TheconclusionfromseveraldifferentaxiomaticderivationsthathavebeenpresentedisthattheGaussianscale-spaceconstitutesthecanonicalwaytogeneratealinearscale-space,basedontheessentialrequirementthatnewstructuresmustnotbecreatedwhengoingfromafinescaletoanycoarserscale.[2][3][5][8][10][11][12][13][14][15]Conditions,referredtoasscale-spaceaxioms,thathavebeenusedforderivingtheuniquenessoftheGaussiankernelincludelinearity,shiftinvariance,semi-groupstructure,non-enhancementoflocalextrema,scaleinvarianceandrotationalinvariance.

Equivalently,thescale-spacefamilycanbedefinedasthesolutionofthediffusionequation（forexampleintermsoftheheatequation）,

withinitialconditionL（x,y;0）=f（x,y）.Thisformulationofthescale-spacerepresentationLmeansthatitispossibletointerprettheintensityvaluesoftheimagefasa"temperaturedistribution"intheimageplaneandthattheprocesswhichgeneratesthescale-spacerepresentationasafunctionoftcorrespondstoheatdiffusionintheimageplaneovertimet（assumingthethermalconductivityofthematerialequaltothearbitrarilychosenconstant½）.Althoughthisconnectionmayappearsuperficialforareadernotfamiliarwithdifferentialequations,itisindeedthecasethatthemainscale-spaceformulationintermsofnon-enhancementoflocalextremaisexpressedintermsofasignconditiononpartialderivativesinthe2+1-Dvolumegeneratedbythescale-space,thuswithintheframeworkofpartialdifferentialequations.Furthermore,adetailedanalysisofthediscretecaseshowsthatthediffusionequationprovidesaunifyinglinkbetweencontinuousanddiscretescale-spaces,whichalsogeneralizestonon-linearscale-spaces,forexample,usinganisotropicdiffusion.Hence,onemaysaythattheprimarywaytogenerateascale-spaceisbythediffusionequation,andthattheGaussiankernelarisesastheGreen'sfunctionofthisspecificpartialdifferentialequation.

Motivations

Themotivationforgeneratingascale-spacerepresentationofagivendatasetoriginatesfromthebasicobservationthatreal-worldobjectsarecomposedofdifferentstructuresatdifferentscales.Thisimpliesthatreal-worldobjects,incontrasttoidealizedmathematicalentitiessuchaspointsorlines,mayappearindifferentwaysdependingonthescaleofobservation.Forexample,theconceptofa"tree"isappropriateatthescaleofmeters,whileconceptssuchasleavesandmoleculesaremoreappropriateatfinerscales.Foracomputervisionsystemanalysinganunknownscene,thereisnowaytoknowaprioriwhatscalesareappropriatefordescribingtheinterestingstructuresintheimagedata.Hence,theonlyreasonableapproachistoconsiderdescriptionsatmultiplescalesinordertobeabletocapturetheunknownscalevariationsthatmayoccur.Takentothelimit,ascale-spacerepresentationconsidersrepresentationsatallscales.[8]

Anothermotivationtothescale-spaceconceptoriginatesfromtheprocessofperformingaphysicalmeasurementonreal-worlddata.Inordertoextractanyinformationfromameasurementprocess,onehastoapplyoperatorsofnon-infinitesimalsizetothedata.Inmanybranchesofcomputerscienceandappliedmathematics,thesizeofthemeasurementoperatorisdisregardedinthetheoreticalmodellingofaproblem.Thescale-spacetheoryontheotherhandexplicitlyincorporatestheneedforanon-infinitesimalsizeoftheimageoperatorsasanintegralpartofanymeasurementaswellasanyotheroperationthatdependsonareal-worldmeasurement.[4]

Thereisacloselinkbetweenscale-spacetheoryandbiologicalvision.Manyscale-spaceoperationsshowahighdegreeofsimilaritywithreceptivefieldprofilesrecordedfromthemammalianretinaandthefirststagesinthevisualcortex.Intheserespects,thescale-spaceframeworkcanbeseenasatheoreticallywell-foundedparadigmforearlyvision,whichinadditionhasbeenthoroughlytestedbyalgorithmsandexperiments.[3][8]

[edit]Gaussianderivativesandthenotionofavisualfront-end

Atanyscaleinscale-space,wecanapplylocalderivativeoperatorstothescale-spacerepresentation:

DuetothecommutativepropertybetweenthederivativeoperatorandtheGaussiansmoothingoperator,suchscale-spacederivativescanequivalentlybecomputedbyconvolvingtheoriginalimagewithGaussianderivativeoperators.ForthisreasontheyareoftenalsoreferredtoasGaussianderivatives:

Interestingly,theuniquenessoftheGaussianderivativeoperatorsaslocaloperationsderivedfromascale-spacerepresentationcanbeobtainedbysimilaraxiomaticderivationsasareusedforderivingtheuniquenessoftheGaussiankernelforscale-spacesmoothing.[3][16]

TheseGaussianderivativeoperatorscaninturnbecombinedbylinearornon-linearoperatorsintoalargervarietyofdifferenttypesoffeaturedetectors,whichinmanycasescanbewellmodelledbydifferentialgeometry.Specifically,invariance（ormoreappropriatelycovariance）tolocalgeometrictransformations,suchasrotationsorlocalaffinetransformations,canbeobtainedbyconsideringdifferentialinvariantsundertheappropriateclassoftransformationsoralternativelybynormalizingtheGaussianderivativeoperatorstoalocallydeterminedcoordinateframedeterminedfrome.g.apreferredorientationintheimagedomainorbyapplyingapreferredlocalaffinetransformationtoalocalimagepatch（seethearticleonaffineshapeadaptationforfurtherdetails）.

WhenGaussianderivativeoperatorsanddifferentialinvariantsareusedinthiswayasbasicfeaturedetectorsatmultiplescales,theuncommittedfirststagesofvisualprocessingareoftenreferredtoasavisualfront-end.Thisoverallframeworkhasbeenappliedtoalargevarietyofproblemsincomputervision,includingfeaturedetection,featureclassification,imagesegmentation,imagematching,motionestimation,computationof