PublishedinNeuralComputation,Vol6,number1,January1994,p127-146
ZhaopingLiandJosephJ.AtickTheRockefellerUniversity
1230YorkAvenueNewYork,NY10021,USA
Abstract
Weexplorethehypothesisthatlinearcorticalneuronsareconcernedwithbuildingaparticulartypeofrepresentationofthevisualworld—onewhichnotonlypreservestheinformationandtheefficiencyachievedbytheretina,butinadditionpreservesspatialrelationshipsintheinput—bothintheplaneofvisionandinthedepthdi-mension.Focusingonthelinearcorticalcells,weclassifyalltransformshavingtheseproperties.Theyaregivenbyrepresentationsofthescalingandtranslationgroup,
’(integers).Anygivenandturnouttobelabeledbyrationalnumbers‘
predictsasetofreceptivefieldswhichcomeatdifferentspatiallocationsandscales(sizes)withabandwidthofoctaves,and,mostinterestingly,withadiversityof‘’cellvarieties.Thebandwidthaffectsthetrade-offbetweenpreserva-tionofplanaranddepthrelations,and,wethink,shouldbeselectedtomatchstruc-turesinnaturalscenes.Forbandwidthsbetweenandoctaves,whicharetheoneswefeelprovidethebestmatching,wefindforeachscaleaminimumoftwodistinctcelltypesthatresidenexttoeachotherandinphasequadrature,i.e.,differbyinthephasesoftheirreceptivefields,asarefoundinthecortex,theyresemblethe“even-symmetric”and“odd-symmetric”simplecellsinspecialcases.Aninterest-ingconsequenceoftherepresentationspresentedhereisthatthepatternofactivationinthecellsinresponsetoatranslationorscalingofanobjectremainsthesamebutmerelyshiftsitslocusfromonegroupofcellstoanother.Thisworkalsoprovidesanewunderstandingofcolorcodingchangesfromtheretinatothecortex.
1.Introduction
Whatisthepurposeofthesignalprocessingperformedbyneuronsinthevisualpath-way?Aretherefirstprinciplesthatpredictthecomputationsoftheseneurons?Recentlytherehasbeensomeprogressinansweringthesequestionsforneuronsintheearlystagesofthevisualpathway.InAtickandRedlich(1990,1992)aquantitativetheory,basedontheprincipleofredundancyreduction,wasproposed.Ithypothesizesthatthemaingoalofretinaltransformationsistoeliminateredundancyininputsignals,particularlythatduetopairwisecorrelationsamongpixels—second-orderstatistics.2Thepredictionsofthetheoryagreewellwithexperimentaldataonprocessingofretinalganglioncells(AtickandRedlich1992,Aticketal1992).
Giventhesuccessesofthistheory,itisnaturaltoaskwhetherredundancyreductionisacomputationalstrategycontinuedintothestriatecortex.Onepossibilityisthatcor-ticalneuronsareconcernedwitheliminatinghigher-orderredundancy,whichisduetohigher-orderstatistics.Wethinkthisisunlikely.Toseewhy,werecallthefactsthatmakeredundancyreductioncompellingwhenappliedtotheretina,andseethatthesefactsarenotasrelevantforthecortex.
First,theretinahasaclearbottleneckproblem:theamountofvisualdatafallingontheretinapersecondisenormous,oftheorderoftensofmegabytes,whiletheretinaloutputhastofitintoanopticnerveofadynamicrangesignificantlysmallerthanthatoftheinput.Thus,theretinamustcompressthesignal,anditcandosowithoutsignificantlossofinformationbyreducingredundancy.Incontrast,afterthesignalispasttheopticnerve,thereisnoidentifiablebottleneckthatrequirescontinuedredundancyreductionbeyondtheretina.
Second,eveniftherewerepressuretoreducedata3,eliminatinghigher-orderstatisticsdoesnothelp.Thereasonisthathigher-orderstatisticsdonotcontributesignificantlytotheentropyofimages,andhencenosignificantcompressioncanbeachievedbyeliminat-ingthem(forreviewsofinformationtheoryseeShannonandWeaver1949;Atick1992).Thedominantredundancycomesfrompairwisecorrelations.4
Thereisanotherintrinsicdifferencebetweenhigherandsecond-orderstatisticsthatsuggeststheirdifferenttreatmentbythevisualpathway.Fig.showsimageandan-otherimagewhichwasobtainedbyrandomizingthephasesofthefouriercoefficientsof.thushasthesamesecond-orderstatisticsasbutnohigherorderones.Contraryto,hasnoclearformsorstructures(cf.Field1989).Thissuggeststhatsecond-orderstatisticsareuseless,whilehigher-orderonesareessential,fordefiningformsandfordis-criminatingbetweenimages.Actually,eliminatingtheformerhighlightsthehigher-orderstatisticswhichshouldbeusedtoextractformsignalsfrom“noise.”5
Sowhatisthecortexthentryingtodo?Ultimately,ofcourse,thecortexisconcernedwithobjectandpatternrecognition.Onepromisingdirectioncouldbetousestatisticalregularitiesofimagestodiscovermatchedfilterswhichleadtobetterrepresentationsforpatternrecognition.Researchinthisdirectioniscurrentlyunderway.However,thereisanotherimportantproblemthataperceptualsystemhastofacebeforetherecognitiontask.Thisistheproblemofsegmentation,orequivalently,theproblemofgroupingfeaturesaccordingtoahypothesisofwhichobjectstheybelongto.Itisacomplexproblem,whichmayturnoutnottobesolvableindependentlyfromtherecognitionproblem.However,sinceobjectsareusuallylocalizedinspace,wethinkanessentialingredientforitssuc-cessfulsolutionisarepresentationofthevisualworldwherespatialrelationships,bothintheplaneofvisionandinthedepthdimension,arepreservedasmuchaspossible.
Inthispaperwehypothesizethatthepurposeofearlycorticalprocessingistopro-ducearepresentationthat1.preservesinformation,2.isfreeofsecond-orderstatistics,and3.preservesspatialrelationships.Thefirsttwoobjectivesarefullyachievedbytheretinasowemerelyrequirethattheybemaintainedbycorticalneurons.Wethinkthe
retinotopicandscaleinvariantsampling);thirdobjectiveisattemptedintheretina(
however,itisonlycompletedinthecortexwheremorecomputationalandorganizationalresourcesareavailable.
Here,wefocusonthecorticaltransformsperformedbytherelativelylinearcells,thefirsttworequirementsimmediatelylimittheclassoftransformsthatlinearcellscanper-formontheretinalsignalstotheclassofunitarymatrices6,with.Sotheprincipleforderivingcorticalcellkernelsreducestofindingthethatbestpreservesspatialrelationships.Actually,preservingplanaranddepthrelationshipssimultaneouslyrequiresatradeoffbetweenthetwo(sectiontwo).Thisimpliesthatthereisafamilyof’s,oneforeverypossibletradeoff.Eachislabelledbythebandwidthoftheresultingcellfiltersandformsarepresentationofthescalingandtranslationgroup(sectionthree).Weshowthattherequirementofunitaritylimitstheallowedchoicesofbandwidths,andforeachchoicepredictstheneededcelldiversity.Thebandwidththatshouldultimatelybeselectedistheonethatbestmatchesstructuresinnaturalscenes.Forbandwidthsaroundoctaves,whicharetheoneswefeelaremostrelevantfornaturalscenes,thepredictedcellkernelsandcelldiversityresemblethoseobservedinthecortex.
Theresultingcellkernelsalsopossessaninterestingobjectconstancyproperty:whenanobjectinthevisualfieldistranslatedintheplaneorperpendiculartotheplaneofvision,thepatternofactivationitevokesinthecellsremainsintrinsicallythesamebutshiftsitslocusfromonegroupofcellstoanother,leavingthesametotalnumberofcellsactivated.Theimportanceofsuchrepresentationsforpatternrecognitionhasbeenstressedrepeat-edlybymanypeoplebeforeandrecentlybyOlshausenetal(1992).Furthermore,thisworkprovidesanewunderstandingofcolorcodingchangefromthesingleopponencyintheretinatothedoubleopponencyinthecortex.2.ManifestingSpatialRelationships
Inthissectionweexaminethefamilyofdecorrelatingmapsandseehowtheydifferinthedegreewithwhichtheypreservespatialrelationships.Westartwiththeinput,repre-sentedbytheactivitiesofphotoreceptorsintheretina,xwherexlabelsthespatial
¯¯
locationofthe’thphotoreceptorinatwo-dimensional()grid.Forsimplicitywetakethegridtobeuniform.Tofocusontherelevantissueswithoutthenotationalcomplex-ityof,wefirstexaminetheone-dimensional()problemandthengeneralizetheanalysistoinsectionfour.Theautocorrelatorofthesignalsis
(1)
wherebracketsdenoteensembleaverage.Toeliminatethisparticularredundancyonehastodecorrelatetheoutputandthenapplytheappropriategaincontroltofitthesignalsintoalimiteddynamicrange.Thiscanbeachievedbyalineartransformation
(2)
where
denotematrices:
andthekernel
istheproductoftwomatrices.Usingbold-faceto
(3)
istherotationtotheprincipalcomponentsof:,wherearetheeigenvaluesof.Whileisthegaincontrolwhichisadiagonalmatrixwithelements.Thustheoutputhastheproperty
(4)
Animportantfacttonoteisthatredefiningbywhereisaunitaryma-trix()doesnotalterthedecorrelationproperty(4).(Actuallyshouldbeanorthogonalmatrixforreal,butsincewewillforconvenienceusecomplexvariables,unitaryisappropriate.)Therefore,thereisawholefamilyofequallyefficientrepresen-tationsparametrizedby.Anymemberisdenotedby
(5)
whereisthetransformationtotheprincipalcomponents.Withoutcompro-misingefficiency,thisnon-uniquenessallowsonetolookforaspecificthatleadstowithotherdesirablepropertiessuchasmanifestspatialrelationships.7
Toseethisletusexhibitthetransformationmoreexplicitly.Fornaturalsignals,theautocorrelatoristranslationallyinvariant,inthesensethat.Onecan
thendefinetheautocorrelatorbyitsfouriertransformoritspowerspectrum,whichin
isff,wherefisthespatialfrequency(Field1987,Ruderman1992).For
¯¯¯
illustrationpurposeswetakeinthissectiontheanalogous“scaleinvariant”spectrum,namely.Intheanalysisofsectionfourweusethemeasuredspectrum
f.¯
Foratranslationallyinvariantautocorrelator,thetransformationtoprincipalcompo-nentsisafouriertransform.Thismeans,theprinciplecomponentsofnaturalscenesortherowvectorsofthematrixaresinewavesofdifferentfrequencies
(6)
where
ifisodd
ifiseven
Whilethegaincontrolmatrixbecomes
is
.Thetotaltransformthen
(7)
Thisperformsafouriertransformandatthesametimenormalizestheoutputsuchthatthepowerisequalizedamongfrequencycomponentsconst.,outputiswhitened.Oneundesirablefeatureofthetransformationisthatitdoesnotpreservespatialrelationshipsintheplane.Asanobjectistranslatedinthefieldofviewthelocusofresponsewillnotsimplytranslate.Alsotwoobjectsseparatedintheinputdonotactivatetwoseparategroupsofcellsintheoutput.Typicallyallcellsrespondtoamixtureoffeaturesofallobjectsinthevisualfield.Segmentationisthusnoteasilyachievableinthisrepresentation.
Mathematically,wesaythattheoutputpreservesplanarspatialrelationshipsintheinputif
when(8)
where.Inotherwords,atranslationintheinputmerelyshiftsthe
outputfromonegroupofcellstoanother.Implicitly,preservingplanarspatialrelation-shipalsorequires,andwewillthereforeenforce,thatthecellreceptivefieldsbelocal,soa
spatiallylocalizedobjectevokesactivitiesonlyinalocalcellgroup,whichshiftsitsloca-tionwhentheobjectmovesandisseperatedfromanothercellgroupevokedbyanotherspatiallydisjointobjectintheimageplane.Technicallyspeakinganthatsatisfies(8)issaidtoformarepresentationofthediscrete“translationgroup”.
Insistingon(8)picksupauniquechoiceof.Infactinthiscaseisgivenby
(9)
whichisjusttheinversefouriertransform.Theresultingtransformation
givestranslationallyinvariantcenter-surroundcellkernels
U=
..
.
Eachsubmatrixhasdimensionandgivesrisetocellswithoutputslo-catedatlatticepointsfor.Sincetheblockmatricesact
whicharethefouriermodesoftheinputs,theresultingcellsinanygivenblockon
filtertheinputsthroughalimitedandexclusivefrequencybandwithfrequenciesfor
.Sincethesecellssamplemoresparselyontheoriginal
visualfield.Notice,thecellsfromdifferentblocksarespatiallymingledwitheachotherandtheirtotalnumberaddupto.Thehopeistohavetranslationinvariancewithineachblockandscaleinvariancebetweenblocks,i.e.,
forfor
and
(11)(12)
Eachblock‘’thusrepresentsaparticularscale,thetranslationinvariancewithinthatscalecanbeachievedwitharesolution,inverselyproportionalto.Largerblocksorlargerthusgivebettertranslationinvariance,andthesingleblockmatrix
achievesthissymmetrytothehighestpossibleresolution.Ontheother
hand,ahigherresolutioninscalinginvariancecallsforasmaller.Aswewillseebelow,,whereisthesmallestfrequencysampledbytheblock.Henceabetterscalinginvariancerequiressmallerblocksizes.Atrade-offbetweenbettertranslationandscalinginvariancereducestochoosingthescalingfactor,orthebandwidthdependingonit.Thiswillbecomecleareraswenowfollowthedetailedconstructionof.Theunitarityconditionnowrequireshavingforeach,resultinginoutputcellsuncorrelatedwithineachscaleandbetweenscales.Toconstruct,onenoticesthattherequirementoftranslationinvarianceisequiva-lenttohavingidenticalreceptivefields,exceptforaspatialshiftofthecenters,withineachscale.Itforces.Forageneral,itturnsoutthattheconstraintforcannotbesatisfiedifoneinsistsononlyonecellorreceptivefieldtypewithinthescale.Howeverifoneallowstheexistenceofseveral,say‘’,celltypeswithinthescale,isagainpossible.Inthiscase,eachcellisidenticalto(oristheoff-celltypeof)theonethatislatticespacesawayinthesamescalelattice(i.e.).Themostgeneralchoiceforrealreceptivefieldsisthen
if
if
(13)
6
whereisanarbitraryphasewhichcanbethoughtofaszeroforsimplicityatthemoment,and
for.Includingboththepositiveandthenegativefrequencies,thetotalnumberoffrequenciessampled,and,sinceisasquarematrix,thetotalnumberofcellsinthisscale,is.Theconstraintofunitarityforleadstotheequation
thenleadstothenon-trivialconsequence
(17)
Inadiscretesystem,theonlyacceptablesolutionsarethosewhereisaninteger.Forexamplethechoiceofandleadstothescaling.Thisisthemostinterestingsolutionasdiscussedbelow.Mathematicallyspeaking,inthecontinuumlimitalargeclassofsolutionsexists,sinceinthatlimitonetakesandsuchthatremainsfinite,thenwearesimplyleadtoforanyand.Thusrepresentationsofthescalingandtranslationgrouparepossibleforallrationalscalingfactors.Thebandwidth,,ofthecorrespondingcellsis
.
Interestingconsequencesfollowfromtherelationshipbetweencellbandwidthanddiversity:
Suchconstructionsneedonlyonefiltertypeineachscaleandgivescalefactorsnolargerthan(equivalentlythelargestbandwidthisoctave—e.g.,thewell-knownHaarbasiswavelets(Daubechies1988)).Thisagreeswithwhatwederivedaboveforthespecialcaseofwhere.However,allowinggivesmorebandwidthchoicesinourconstruction.Forexample,gives,howevernolargerthanoctaves,andgives,nolargerthanoctaves,etc.TheseresultsalsoagreewiththerecenttheoremofAuscher(1992)whoprovedthatmultiscalerepresentationscanexistforscalingsbyanyrationalnumber,providedfiltertypesareallowedineachscale.Ourconclusionaboveyieldsexactlythesameresultbyredefiningand.Wearrivedatourconclusionindependentlythroughtheexplicitconstructionpresentedabove.8
Theconnectionbetweenthenumberofcelltypesandthebandwidththatispossibletoachieveissignificant.Webelievethebandwidthneededbycorticalcellsisdeterminedbypropertiesofnaturalimages.Itsvalueshouldbethebestcompromisebetweenplanaranddepthresolutionpreservationforthedistributionofstructuresinnaturalscenes.Actually,Field(1987,1989)examinedtheissueofbestbandwidthforfiltersthatmodelledcorticalcellsandfoundthatbandwidthsbetweenandoctavesbestmatchednaturalscenestructures.Ourresultshereshowthatcorticalcellscannotachievebandwidthsmorethanoneoctavewithouthavingmorethanonecelltype.
Nextweshowwhatthepredictedcellkernelslooklike.Forgenerality,wegivetheex-pressionforthekernelsinthecontinuumlimitforanyscalefactor
(19)
comeinvarieties.EvenandoddvarietiesForanygivenandthekernelsfor
areimmediatelyapparentwhenonesets,,and(areevenor
forevenorodd).InFig.weexhibittheevenandoddkernelsoddfunctionsof
intwoadjacentscalesandtheirspectra.Thekernels,whereischosen,aresimilartothecenter-surroundretinalganglioncells(howevertheyarelargerinsize),andhenceweneednottoexhibitthemhere.Ingeneral,though,cantakeanyvalue,and
theneighboringcellswillsimplydifferbyaphaseshift,orinquadrature,withoutnecessarilyhavingevenoroddsymmetryintheirreceptivefieldshapes.From(18),itiseasytoshowthatthekernelsforsatisfythefollowingrecursiverelations
wasnotthereineqn.(18).Theseresultscanbeextendedto
thewhiteningfactoris
where
Onenoticesthatthisextensionto2Drequiresachoiceoforientationssuchasthex-yaxes,breakingtherotationalsymmetry.Furthermore,itisnaturaltoaskiftheobjectconstancybytranslationsandscalingsshouldbeextendedtotheobjectrotationsintheimageplane—requiringthecellsberepresentationsoftherotationgroup.Atthispoint,itisnotclearwhethertherotationalinvarianceisnecessary(notingthatweusuallytiltourheadstoreadatiltedbookorfailtorecognizeafaceupsidedown),andwhethertherotationalinvariancecanbeincorporatedsimultanouslywiththetranslationandscalingoneswithoutincreasingthenumberofcells.Wewillleavethisoutsidethepaper.
9
9
Figs.,and,showthefivecelltypesoneencountersforthebreakinginandtheninecelltypesforthebreakinginFig.,respectively,forachoiceofscalingfactor.Finally,theobjectconstancyeqns.(11),(12)stillholdsince(20)and(21)extendtoas
xn¯¯
x¯
ff.¯¯
From(18)and(19),itisclearthatthecorticalkernels
onlybytherange
ofthefrequencyintegrationorselectivity.Thecorticalreceptivefieldsarelowpassorbandpassversionsoftheretinalones.Oneimmediateconsequenceofthisisthatmostcorticalcells,especiallythelowpassoneslikethoseintheCytochromeOxidaseBlobcells,havelargerreceptivefieldsthantheretinalones.Second,whenconsideringcolorvision,
andfortheluminanceandchrominancechannelsre-thepowerspectrums
spectively,differintheirmagnitudes.Inrealitywhennoisesareconsidered,thereceptivefieldfiltersarenotsimply
cortex.Theanalysisalsopredictsaninterestingrelationshipbetweenbandwidthsofcellsandtheirdiversityaswasdiscussedinsectionsthreeandfour.Oneconsequenceofthatrelationshipisthatforcellstoachievearepresentationoftheworldwithsamplingband-widthbetweenandoctavestheremustbeatleasttwocelltypesadjacenttoeachotheranddifferbyintheirreceptivefieldphases(Fig.).Thisbandwidthrangeistherangeofmeasuredbandwidthsofsimplecells(KulikowskiandBishop1981;AndrewsandPollen1979)andalso,wethink,isbestsuitedformatchingstructuresinnaturalscenes(cf.Field1987,1989).Thisanalysisthusexplainsthepresenceofphasequadrature(e.g.,pairedeven-oddsimplecells)observedinthecortex,(PollenandRonner1981):suchcelldiversityareneededtobuildafaithfulmultiscalerepresentationofthevisualworld.Theanalysisalsorequiresbreakingorientationsymmetry.Herewedonotwishtoadvocatescalingsymmetryasanexplanationfortheexistenceoforientedcellsinthecortex.Itmaybethatorientationsymmetryisbrokenforamorefundamentalreasonandthatscalingsymmetrytakesadvantageofthat.Eitherway,orientationsymmetrybreakingisanimportantingredientinbuildingthesemultiscalerepresentations.
Inthepast,therehasbeenasizeablebodyofworkontryingtomodelsimplecellsintermsof“Gabor”and“logGabor”filters(Kulikowskietal1982;Daugman1985,Field1987,1989).Suchfiltersarequalitativelyclosetothosederivedhere,andtheydescribesomeofthepropertiesofsimplecellswell.Ourworkdiffersfrompreviousworkinmanyways.Thetwomostimportantdifferencesarethefollowing.First,thefiltersherearederivedbyunitarytransformsonretinalfilterswhichreduceredundancyininputsbywhitening.Byselectingtheunitarytransformationthatmanifestsspatial-scalere-lationshipsinsignals,onearrivesatarepresentationthatexhibitsobjectconstancy—theoutputresponsetoaninputanditsplanaranddepthtranslatedversion(i.e.,
)arerelatedby
(22)
Henceavisualobjectmovedinspacesimplyshiftstheoutputsfromonegroupofcellstoanother.Second,wefindadirectlinkagebetweencellbandwidthanddiversity.Suchlinkagedoesnotappearinpreviousworkswhereorthonormalityorunitaritywasnotrequired.
Morerecentlytherehasalsobeenalotofworkonorthonormalmultiscalerepresen-tationsofthescalingandthetranslationgroup,alternativelyknownaswavelets(Meyer1985,Daubechies1988,Mallat1989).Therelationshipofourworktowaveletswasdis-cussedinsectionthree.Hereweshouldaddthatinthispaperweprovideexplicitcon-structionoftheserepresentationsforanyrationalscalingfactor.Furthermore,ourfilterssatisfy
.Thisdifferencestems
fromthefactthatourfiltersaretheconvolutionofthewhiteningfilterandthestandard-typewavelet.Thewhiteningfilter—givenby
constructionwithclassesofcellsinthecortex.First,thereistheclassoflowpasscells,whichhavelargereceptivefields,andnoorientationtuning(actuallysincetheirkernelshaveawhiteningfactor,theyarenotcompletelylowpassbutanincompletebandpass–weaksurround).WethinkagoodcandidateforthesecellsarethecellsintheCytochromeOxidaseBlobareasinthecortex.Whenweaddcolortoouranalysis,thisclasswillcomeouttobecoloropponent10.Thesecells,alowpassversionofthesingleopponentretinalcells,turnouttobedoubleopponentorcolor-opponent-center-only(seeFig.5)fromthismathematicalconstruction,inagreementwithobservations.Second,therepresentationrequiresseveralorientationclassesineverychoiceofhigherscale,theyarenotaslikelytobecolorselectiveand,withineachorientationandscale,therearetwotypesofcells—inphasequadrature(e.g.,evenandoddsymmetric)—ifthebandwidthofthecellsisgreaterthanoneoctave.Thesehavekernelssimilartosimplecells’.Also,insomechoicesofdivisionofthetwodimensionalfrequencyspaceintobands(seeFig.)oneencounterscellsthatareverydifferentfromsimplecells.Thesecellscomefromthebandpassregion
)andassuchpossessrelativelyinboththeanddirections(the‘bb’regioninFig.
smallreceptivefieldsinspace.ItisamusingtonotetheirresemblancetothetypeofcellsthatVanEssendiscoveredin(privatecommunication).
Itisimportantatthisstagetolookindetailforevidencethatcorticalneuronsarebuildingamultiscale,translationallyinvariantrepresentationoftheinputalongthelinesdescribedinthispaper.However,inlookingforthosewemustallowforthepossibilitythattheserepresentationsareformedinanactiveprocessstartingasearlyasthestriatecortex,aswasproposedrecentlyby(Olshausenetal1992).Wealsomustkeepinmindthattoperformdetailedcomparisonwithrealcorticalfilters,ourfiltershavetomodifiedtotakenoiseintoaccount.
AcknowledgmentsWewouldliketothankD.Field,C.GilbertandN.Redlichforusefuldiscussions,andtheSeaverInstituteforitssupport.Appendix
Inthisappendixweexaminetheconditionofunitarityonthematrix.Thematrixelementsofinthescalearegeneralizedfromthecasesimplyas
nj
¯¯
.
ApriorithecellsinsamplefromthefrequencyregioninsidethebigsolidboxbutoutsidethedashedboxinFig..Thecriticalfactthatmakesthecasedifferentfrom
isthattherearecellsinthe’thclass,whilethetotalnumberofcellsis,then.
Theunitarityrequirement
(
)canbeshowntobeequivalentto
(24)
isanyinteger.Asimilarconditioninthedirectionshouldalsohold.Towhere
satisfy(24)onecanonlyhopethatthecosinefactoriszeroforoddandthesinefactoriszerofortherest.Thisisimpossibleinalthoughpossiblein.Toseethisdifference,notethatin,andtheargumentofthesineiswhichleadstovanishingsineforeven.Onethenmakescosinetermzeroforoddbychoosingsuchthat.Thisisexactlyhowequation(16)isreached.In,
foreven.AlthoughwecannotprovethatthenegativeresultinisnotcausedbythefactthatwehaveaEuclideangrid,wethinkitnotpossibletoconstructtherepresentationevenwhenusingaradiallysymmetriclattice.Toensureunitarityof,weneedtoallowforcelldiversityofadifferentkind—cellsinscaleneedtobefurtherbrokendownintodifferenttypesororientations,eachtypesamplingfrom,alimitedregionofthefrequencyspaceasshownforexampleinFig..
13
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15
Figure1::Demonstrationoftheuselessnessofsecond-orderstatisticsforformdefinitionanddiscrimination.Fol-lowing(Field1989),imageisconstructedbyfirstfouriertransforming,randomizingthephasesofthecoefficientsandthentakingtheinversefouriertransform.Thetwoimagesthushavethesamesecond-orderstatisticsbuthasnohigher-orderones.Allrelevantobjectfeaturesdisappearedfrom.
AB16
ReceptivefieldsSensitivity
AB
CD
SpatialDistanceSpatialfrequency
Figure2:Even-symmetric”,,,and“odd-symmetric”,,,kernelspredictedforthescalefactor(equivalently
octaves)fortwoneighboringscales(topandbottomrows,respectively),togetherwiththeirspectrafor
(frequencysensitivitiesorselectivities).
17
Figure3::Proliferationofmorecelltypesbythebreak-downofthefrequencysamplingregioninwithinagivenscale.Ignoringthenegativefrequencies,thefrequencieswithinthescaleareinsidethelargesolidboxbutoutsidethesmalldashedbox.Thesolidlineswithinthelargesolidboxfurtherpartitionthesamplingintosubregionsdenotedby‘bl’,‘lb’,and‘bb’,whichindicatebandpass-lowpass,lowpass-bandpass,andbandpass-bandpass,respec-andgiveasymmetricbreakdownbetweenanddirections,the‘lb’cellsarenottively,in-directions.
rotationofthe‘bl’cells.givessymmetricbreak-downbetweenanddirections.The‘bb’cellsequivalenttoa
aresignificantlydifferentfromtheothers,seeFig..
18
Figure4:Fig.:Thepredictedvarietyofcellreceptivefieldsin.ThefivecelltypesinandtheninecelltypesinarisefromthefrequencypartitioningschemesinFig.andFig.,respectively.Thekernelsinthelower-leftcornerofbothimagesdemonstratethelowpass-lowpassfilterinandtheyarenon-oriented.Allothersarebandpassinatleastonedirection.Thoseareactuallysignificantlysmallerbutareexpandedinsizeinthisfigurefordemonstration.The‘bb’cellsintheupper-rightpartofcomeinfourvarieties(even-even,odd-odd,even-oddand
istakenforbothxandydirections)andshouldexistinthecortexiftheschemeinFig.isodd-evenwhen
favored.AllkernelsareconstructedtakingintoaccounttheopticalMTFoftheeye.
AB19
Luminance (Solid) & Chrominance (Dashed)
100.
Sensitivity10.
1
0.11
SpatialFrequency(c/deg)
10.100.
Ganglion redGanglion green
StrengthBlob redBlob green
SpatialDistance
Figure5:Changeofcolorcodingfromretinatocortex.Thetopplotshowsthevisualcontrastsensitivitiestotheluminanceandchrominancesignals.Thebottomplotdemonstratesthereceptivefieldprofiles(sensitivitytoredorgreenconeinputs)ofthecolorselectivecellsintheretina(organglion)andthecortex.TheparametersusedfortheganglioncellsarethesameasthoseinAticketal1992.Theblobcellsareconstructedbylowpassfilteringtheganglion
wherec/deg.Thestrengthsofthecellprofilescelloutputswithafilterfrequencysensitivityof
areindividuallynormalizedforboththeganglionandtheblobcells.Therangeofthespatialdistanceaxes,orthesize,oftheblobcellsis3.7timeslargerthanthatofganglioncells.Thismeansthateachblobcellsumstheoutputsfrom(ontheorderof)atleastaboutlocalganglioncells.
20
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