Fourier analysis and cortical architectures the ex

Fourieranalysisandcorticalarchitectures:theexponentialchirptransformGiorgioBonmassarEricL.SchwartzDept.BiomedicalDept.CognitiveandEngineeringNeuralSystemsBostonUniversityBostonUniversityBoston,MA02146Boston,MA02146e-mail:[e-mail]-mail:[e-mail]lrepresentationsinwhichpixel-sizeandlocalneighbor-hoodtopologyarenotconstantistermedspace-variantvision.Thisisthedominantvisualarchitectureinallhighervertebratevisualsystems,andiscomingtoplayanimportantroleinreal-timeactivevisionapplicationsintheformoflog-polar,foveatingpyramid,andrelatedapproachestomachinevision.Thebreakingoftranslationsymmetrythatisunavoidablyassociatedwithspace-variantvisionpresentsamajoralgorithmiccomplicationforim-ageprocessing.InthispaperweuseaLiegroupapproachtoderiveakernelwhichprovidesageneralizationoftheFourierTransformthatprovidesaquasi-shiftinvarianttemplatematchingcapabilityinthedistorted(range)coordinatesofthespace-variantmapping.Weworkoutthespecialcaseofthelog-polarmapping,whichistheprinciplespace-variantmappinginuse;inthiscase,wecalltheassociatedintegraltransformthe“exponentialchirpWorksupportedbyARPAANNT-ONRN00014-92-C-0119andONRMURIN60014-95-I-0409Weusethetermquasi-shiftinvarianttorefertothenonuniformsamplingnatureofspace-variantmapslikethelog-polarmapping.Thus,forexample,ifanobjectisshiftedacrosstheapertureofaspace-variantsystem,ourmethodswillproduceinvarianceupto,butnotincluding,theapplicationofaband-passlterthatispositiondependent.Thiswillbemadeclearinthetext.1transform”(ECT).Themethodis,however,generalforotherformsofmap-ping,orwarp,function.Examplesfromthetwo-dimensional(imageprocessing)log-polartrans-formationarepresentedalongwiththedemonstrationthattheECTpre-servesthefoveatingaspectofthespacedomainmapping,andthereforepro-videsaquasi-shift-invariantrealizationfortheapplicationsofmatchedlterandphase-onlylter.Thisworkprovides,forthersttime,aconceptualba-sisforcombiningglobalspatialfrequencymethodswithspace-variantmap-pingsinawaywhichisconsistentwiththeanatomicalfactthathumanvision,atthecorticallevel,takesplaceinlog-polarcoordinates.Categories:Biologicalvision,real-timeactivevision,Low-LevelProcess-ing,ShapeandObjectRepresentation.1IntroductionThispaperaddressesafundamentaldifcultyinperformingfrequencydomainimageprocessingonimagearchitectureswhicharestronglyspace-variant,and,inparticular,aredescribedbythelog-polarmap,oroneofitsvariants.Thelog-polarmapisofinterestincomputervisionforthreemajorreasons:1.Ithasbeenshowntobeagoodapproximationtotheimageformatusedinprimateandhumanvisualcortex(see(Schwartz,1994)forareviewofthemathematicalcharacterizationoftheanatomicalstruc-tureofV-1),andwouldseemtoprovideadvantagestomachinevi-sionwhicharesimilartothosethatarealreadyunderstoodtoapplytohumanvision.2.Itprovidesacontinuummodelforvariableresolution,orfoveatingpyramidarchitecturesinmachinevision.3.Itprovidesawide-angleyethighresolution(i.e.foveal)imagefor-matwithattractivespace-complexity.Thecompressionrelativetoacomparablespace-invariantformatisuptofourordersofmagnitudeforhumanvision(RojerandSchwartz,1990),anduptotwoorthreeordersofmagnitudeforcurrentlyrealizablemachinevisionsystems(Schwartzetal.,1995).2Real-timevisionisanobviousareaofapplicationforthisarchitecture,asthevertebratevisualsystemseemstosuggest.Recently,anumberofre-searchgroupshaveconstructedmachinevisionsystemsbasedonthisar-chitecture(Weiman,1989;BalochandWaxman,1991;Bedersonetal.,1992;Engeletal.,1994;SandiniandDario,1989;Baronetal.,1995).However,applicationinmachinevisionhasbeenimpededbythegenerallackofsuitableimageprocessingalgorithmsforthedifcultapplicationdomainpresentedbythelog-polarmapping.Thepresentpaperprovidesapossi-blesolutiontothisquandary.Thespecicproblemaddressedinthispaperistodevelopageneral-izedformofFourierTransformwhichmaybeappliedinthespace-variantrangecoordinatesofanimagewarp.ThemethodusedhereisbasedontheuseofLieGroupmethodstoformulateandthensolveaspecicpar-tialdifferentialequation(PDE),followingtheworkof(FerraroandCaelli,1988)and(Rubinsteinetal.,1991),whousedthesemethodsinthecontextofrecognitionofdistortedpatterns.Inthepresentcase,thesolutionofthePDEdeterminedbytheLieGroupapproachproducesaspace-variantkernelwhichprovidesashift-invariantpropertyanalogoustothatoftheconventionalFouriertransform.Usingthekernelthatwehavederived,whichwecalltheexponentialchirpkernel,itispossibletocomputeanin-tegraltransformwithanimagein,forexample,log-polarcoordinateform,andyetretaintheadvantageousshift(andalsosizeandrotation)invari-ancesassociatedwithconventionalFourieranalysis.Thelog-polar(ormoreaccurately,complexlogarithmic)mapisde-nedasfollows:(1)Inthisequation,isanexperimentalconstantwhichisonlyofrelevancetothebiologicalscalefactorofaparticularmap,andwillbedroppedinthefollowingdiscussion(see(Schwartz,1994)forareviewofestimatesofthisparameter),“”arealconstant(RojerandSchwartz,1990)thatdealswiththesingularityontheorigin,“”representsvisualpixelcoordinates,and“”representslog-polarcoordinates.Inthecontextof