Facts and Myths About image Patterns

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1 CAR-TR-968 CS-TR-4251 CorneliaFermuller1,HenrikMalm2andYiannisAloimonos1 StatisticsExplainsGeometricalOpticalIllusions 1CenterforAutomationResearchIIS CollegePark,MD UniversityofMaryland May2001 2MathematicalImagingGroup(MIG) DepartmentofMathematics(LTH) LundInstituteofTechnology/ S-22100Lund,Sweden LundUniversity P.O.Box118 estimatedfromtheinputdata.however,noiseintheimageintensityanditsderivativescauses Itisproposedinthispaperthatmanygeometricalopticalillusions,aswellasillusionary problemsintheestimationofthefeatures;inparticular,itcausesbias.asaresult,thelocations intersectionsoflines,orlocalimagemovementmustbederived.thatis,thesefeaturesmustbe patternsduetomotionsignalsinlinedrawings,areduetothestatisticsofvisualcomputations. Theinterpretationofimagepatternsisprecededbyastepwhereimagefeaturessuchaslines, Abstract Thisworkrevealsageneraluncertaintyprinciplegoverningtheworkingsofvisionsystems.A offeaturesareperceivederroneouslyandtheappearanceofthepatternsisaltered.thebias occurswithanyvisualprocessingoflinefeatures;underaverageconditionsitisnotlargeenough tobenoticeable,butillusionarypatternsaresuchthatthebiasishighlypronounced.ingeneral, thisbiascannotbeavoided,andanyvisionsystem,biologicalorarticial,mustcopewithit. consequenceofthisprincipleisthepredictionofalargeclassofgeometricopticalillusions. acknowledged,asisthehelpofsaralarsoninpreparingthispaper. ThesupportofthisresearchbytheNationalScienceFoundationundergrantIIS isgratefully

2 thestreets.everybookstorehasasectiononthetopic,andmanycuriosityandsouvenirstores carryparaphernaliawithdecorativeillusionarypatterns,suchasthedrawingsofm.c.escher Opticalillusionsareasourceoffascinationtomanypeople.Almosteveryoneisfamiliarwith 1Introduction someillusions,asinrecentyearstheyhaveinltratedthepopularsciencesaswellastheartof ancientgreecedevelopedaverygoodunderstandingofthedistortioneectsinstructuresof numberofdierentexplanationsoeredisevenlarger. probablygettheimpressionthattherearehundredsofseeminglyunrelatedillusions,andthe ationofillustrativeartandexplanatorymaterial.butifonetriestondexplanationshewill orgureswithdoubleinterpretations. largeextentandtheyappliedcompensatorymeasures.asverticalcolumnswouldappearto projects,interactivedemonstrations,lotsofartwork,andevencompaniesdevotedtothecre- Someopticalillusionshavebeenknownsinceantiquity.Forexample,thearchitectsof IfonesearchestheInternet,onewillndmanycollectionsofopticalillusions,manyschool beconvextowardtheground,theyconstructedthemtobeconcave;astallcolumnsarelikely leanoutwardatthetop,theyinclinedtheminward;aslonghorizontalbeamswouldappearto toappearsomewhatshrunkeninthemiddle,theymadethemslightlyswollen.anothereect, appearssmalleratthezeniththanatthehorizon.ptolemyprovidedanexplanationforthis documentedveryearlyinhistory,isthemoonillusion,thatis,thephenomenonthatthemoon canbefoundinhelmholtz'streatiseonphysiologicaloptics[34].sincethen,illusionshave eectthatisstillconsideredtoday. sionsbecamethesubjectofsystematicstudy.alargepartofthendingsofthelastcentury beenstudiedmostlyintheeldofpsychology,andovertheyearstheyhavereceivedvarying amountsofattention.inthelastdecades,withgrowinginterestinthestudyofthemindand eldssuchasneurology,physiology,philosophy,andthecomputationalsciences,inparticular perception,illusionshavealsobeenlookedatoccasionallybyscientistsinothermind-related bythoseinterestedinthemodelingofneuralnetworks. Inthenineteenthcenturyscientistsstartedtobeinterestedinperception,andopticalillu- revealsomethingabouthumanlimitationsandbytheirnatureareobscureandthusfascinating. Butthishasnotbeenthesolereasonforscientists'interest.Fortheoristsofperceptiontheyhave beenusedastestinstrumentsfortheory;thiseortoriginatedfromthefoundersofthegestalt school.astrategyinndingouthowcorrectperceptionoperatesistoobservesituationsin whichmisperceptionoccurs.bycarefullyalteringthestimuliandtestingthechangesinvisual performance,psychologistshavetriedtogaininsightintotheprinciplesofperception. Whatisitthathascausedthislong-standinginterestinopticalillusions?Forone,they inuencesonillusionarypatterns thatis,theattitudeoftheobserver,orientation,thickness, numberoflines,etc.,whichcausethemaximumorminimumillusionaryeectinapattern. animals.usingthisknowledgeitispossibletoinventvariationsonillusionarygures.todo oftheobserver'sagehasbeeninvestigated,andafewstudieshaveevenbeenconductedwith thisthereisnoneedtounderstandinanydeepwayhowanillusionworks. Forsomeofthebestknownillusionscross-culturalstudieshavebeenperformed,theinuence Fromalargenumberofexperimentalstudiesalothasbeenlearnedabouttheparametric areanindicatoroftheforlornnessofthehopeofndingageneraltheorywhichwouldaccount edgethatanyexistingtheorysatisfactorilyexplainsopticalillusions,orthattherearemajor principlesunderlyingclassesofillusions.mostwouldagreewithrobinson[29],whowritesin hisintroductiontogeometricalopticalillusionsthatthelargenumberofillusionsinthisclass Atthistime,however,hardlyanyoneinvolvedinthestudyofperceptionwouldacknowl- 1

3 wewillreducetheobservationstoafewprinciples,expressedinonegeneraltheorywhichis capableofpredictingthefailuresaswellasthesuccessesoftheperceptualsystem. forallofthem.butthestudyofperceptionisstillataveryearlystage.forcomparison,the universallaw.thuswehavegoodreasontobelievethatasmoreknowledgebecomesavailable, manyphysicallawsexplainingtheseobservations,andmodernphysicsisnowinsearchofthe muchmorematureeldofphysicsstartedoutinasimilarway,withmanyobservationsand termisatranslationofthegermangeometrisch-optischetauschungenandhasbeenused foranyillusionseeninlinedrawings.itwascoinedbyoppel[26]inapaperaboutthe overestimationofaninterruptedascomparedwithanuninterruptedextent,latercalledthe largeextentnexttoasmalloneandtheunderestimationofasmallextentnexttoalargeone, Oppel-Kundtillusion[19]. andzollnerillusions,thefraserspiral,andthecontrasteect(thatis,theoverestimationofa Thebestknownandmoststudiedofallillusionsarethegeometricalopticalillusions.The theoverestimationinbrightnessofabrightentitynexttoadarkone,etc.).thenumberof bequitediverse. illusionarypatternsthatfallinthisclassisverylarge,andtheunderlyingphenomenaseemto SomeotherveryfamousillusionsinthisclassincludetheMuller-Lyer[23],Poggendor, oflines,orlocalimagemotionmustbederived,thatis,theymustbeestimatedfromtheinput sionillusionarypatternsduetomotionsignalsinlinedrawings,areduetothestatisticsofvisual computations.wheninterpretingapattern,featuresintheimagesuchaslines,intersections data.noiseintheinputcausestheestimationofthefeaturestobebiased.asaresult,the enoughtobenoticeable,butillusionarypatternsaresuchthatthebiasisstronglypronounced. locationsoffeaturesareperceivederroneouslyandtheappearanceofthepatternisaltered.the biasoccurswithanyvisualprocessingoflinefeatures;underaverageconditionsitisnotlarge Theclaimofthispaperisthatmanyofthegeometricalopticalillusions,andalsobyexten- astaticpatternweperformeyemovementsandgatheraseriesofimages(eitherbymoving oneyemovements[4]thedatafromanumberofretinalimagesiscombinedintooneimage theeyefreelyoverthepatternorbyxatingatsomepointonit).accordingtoexperts representation.helmholtzreferstotherepresentationoftheretinalimageastheeldofxation (Blickfeld)andtotherepresentationthatmoveswiththeeyeasthevisualglobe(Sehfeld).More recentlythetermsretinotopicframeandstablefeatureframehavebeenused[7]. Insomewhatmoredetail,oureyesreceiveasinputasequenceofimages.Evenifweview sequencewhichserveasinputtofurtherestimationprocesses.forthemodelingconducted hereweconsiderthreekindsofimagemeasurements:thegrayvaluesofimagepoints;small imagemeasurementscanonlybederivedwithinarangeofaccuracy.inotherwords,there edgeelements(edgels);andimagemotionperpendiculartolocaledges(normalow).these rstinterpretationprocessesconsistofestimatinglocaledgesusingasinputalltheavailable orientationsofedgeelements,andinthedirectionsandlengthsofnormalowvectors.the isnoiseinthepositionsandgrayvaluemagnitudesofimagepoints,inthepositionsand Theearlyvisualprocessingapparatusextractslocalimagemeasurementsfromtheimage imagepoints,orestimatingtheintersectionsoflinesusingalltheedgels,orcomputinglocal 2Dimagemotionusingallthenormalowmeasurementsinasmallregion.Theseestimation processesarebiased.thustheperceivedpositionsofedgelsareshifted,theirdirectionsare tilted,andtheirmovementsareestimatedwrongly. risetotiltedanddisplacedstraightlinesanddistortedcurvesasperceivedinmanyillusionary patterns.inthecaseofmotion,thelocalimagemeasurementsarecombinedinsegmentation pretationprocesses.longstraightlinesorgeneralcurvesarettedtotheedgelsandthisgives Thelocaledgelandimagemotionestimatesserveasinputtothenexthigherlevelinter- 2

4 and3dmotionestimationprocessestointerpretthespatiotemporalcontentofthescene. visualsystem,butrathergeneralcomputationsthatinsomeformmustbeconductedinvisual interpretationprocesses.wechoosetoemploythreedierentmodelsforthreekindsofdata wewillalsoshowthatthebiasexistsforother,moreelaborateestimationprocesses,andthat ingeneralitcannotbeavoided.wedonotattempttomodelthespecicsofthehuman temporalderivatives) forthepurposeofclearandsimpleexplanation.foreachmodelwe discussanumberofillusionsthatarebestexplainedbyit.itshouldbeemphasizedthatthe grayvalues,edgels(spatialderivativesataspatiallocation),andnormalow(spatialand Inthispaperwewilldemonstrateandanalyzethebiasinlinearestimationprocesses,but thedierentiationprocess[25],buttherearemanysourcesofnoisethataectthevisualdata, Lettheirradiancesignalcomingfromthesceneparameterizedbyimageposition(x;y)and dataandnoisyspatiotemporaldatacanbederivedfromnoisygraylevelvaluesbymodeling andthedierentiationprocesscertainlyisoneofthem. 2Erroringrayvalues visionsystem,inalmostanyinterpretationprocess,usesallthreekindsofdata.thenoisyedgel graylevelvalue,andthereisnoiseinspatiotemporallocation.wemodelthesenoisesources asfollows: theidealsignal.ingeneral,twokindsofnoisesourcesmustbeconsidered:thereisnoisein tributednoisetermi(x;y;t)withzeromeanandcovariancematrixg.thisrepresentsthe time(t)bei(x;y;t).theimagereceivedontheretinacanbethoughtofasanoisyversionof bution.theexpectedvalueoftheintensitythusamountstotheconvolutionofthesignalwith erroringraylevelvalue.wemodelitasagaussian,i.e.i(x;y;t)n(0;g).asamodelfor thepositionalerror,weconsiderthevalueatapointashavingagaussianprobabilitydistri- aspatiotemporalgaussiankernel,thatis, Ateveryimagepoint(x;y;t)weconsideranadditive,independentlyandidenticallydis- withx=(x;y;t)and g(x;p)= E(I(x)+N(0;g)g(x;p)) standarddeviationinthegraylevelnoise.theexpectedvalueoftheintensityisthen y-directionstobethesame,withpthestandarddeviationinthepositionalnoiseandgthe forsimplicity,throughoutthepaperweassumethespatialnoisecomponentsinthex-and Inthemodelingofstaticsignalsonlythespatialvariablesneedtobeconsidered.Furthermore, (2)3=2jpj1=2e?12xt?1 1 E(I(x;y)+N(0;g)g(x;y;p))=E(I(x;y)g(x;y;p))+E(N(0;g)g(x;y;p))(1) px oftheintensitycanbewrittenasthesumoftwoterms.thesecondterm,theexpectedvalue second-orderderivatives(thelaplacian,r2)[22].ascanbeseenfrom(1),theexpectedvalue ofn(0;g)g(x;y;p)anditsderivativesareindependentofposition(theyarezero).this meansthatnoiseinthegrayleveldoesnotaecttheexpectedlocationsofedgesandforthe Edgesareobtainedastheextremaoftherst-orderderivatives[3]orasthezerocrossingsof 3

5 purposeofthisanalysisitcanbeignored.1thusweareinterestedinthelocations(x;y)where Beforeanalyzingedgesundersmoothing,letuslookatthesourcesofthenoise.Oneinterpretationisthatthereisuncertaintyinthepositionsinceasequenceofimagesiscombinedinto hasalocalmaximum,orwherer2g(x;y;p)i(x;y)=0 @yi(x;y))2]1=2 onecoordinatesystemanderrorsoccurinthegeometriccompensationforlocation.another withagaussianwithastandarddeviationthatisthesumofthestandarddeviationsofthe obtainedisaprocessedversionoftheidealsignal,whichevenforonlyoneimagetakenatone [22,30].SincerepeatedlteringwithdierentGaussiankernelsisequivalenttoconvolution predictsthatedgesintheearlyvisualsystemarecomputedusingadierenceofgaussian lter,thatis,aband-passlterwhichisapproximatedwellbythelaplacianofagaussian instantintimecanbemodeledbygaussiansmoothing.furthermore,tocomputethederivativessomelow-passltering(thatis,smoothing)isnecessary.forexample,acurrenttheory interpretationisthattheeyecanbeviewedasade-focusedimagingsystemandtheimage thatwecanemploystraightforwardlyinourstudy. scalespacetheory,andalargenumberoftheoreticalresultsonthistopichavebeenderived ofedgepointsinthedirectionnormaltoedges,theso-calleddriftvelocity. kernels,anycombinationoftheabovenoisesourcesisalsowellapproximatedbyourmodel. paralleltothespatialgradientdirectionatp0andtheu-axisperpendiculartoit.ifedgesare ofthesmoothingparameter.lindebergin[20]derivesformulaefortheinstantaneousvelocity Weareinterestedinthepositionsofedges,orthechangeinpositionsofedgeswithvariation ConsiderateveryedgepointP0alocalorthonormalcoordinatesystem(u;v)withthev-axis Gaussiansmoothingofstaticimageshasbeenintensivelystudiedintheliteratureonlinear scaleparameter)amountsto 2(r2Iu)2+(r2Iv)2r2Iu;r2Iv r2?r2i Thedriftvelocityrepresentsthetendencyofthemovementofedgesinscalespace.Toobtain esto Forastraightedge,whereallthedirectionalderivativesintheu-directionarezero,itsimpli- Ivvv(0;1) (3) (2) thedriftvelocity.ifthescaleintervalissmallthedriftvelocityinthesmoothedimageprovides, smoothedimage,wouldrequiretracingtheedgepointoverthescaleintervalandintegrating inmanycases,asucientapproximationtothetotaldrift,andthisiswhatwewillshowin laterillustrations. tobeconsidered:edgesbetweenadarkandabrightregiondonotchangelocationunderscale thetotaldriftofanedgepointp0intheoriginalimagetothecorrespondingedgepointinthe intensitypropagatesintolargeruncertaintyintheestimationofthederivativesoftheintensity.thismighthave someeectonhumanperception,anissuethatcouldbeinvestigated. 1Noiseinthegraylevelaectsthesecondmoment,thatis,thevariance.Uncertaintyintheestimationof ThescalespacebehaviorofstraightedgesisillustratedinFigure1.Therearethreecases 4

6 twoedgestowardeachother.theseobservationssucetoexplainanumberofillusions.the spacesmoothing(figure1a).thetwoedgesattheboundariesofabrightline,orbar,ina underlyingmathematicsisexplainedinfigure2. ofsmoothingonalineofmediumbrightnessnexttoabrightandadarkregionistomovethe parameterislargeenoughthatthewholebaraectstheedges(figure1b).finally,theeect darkregion(or,equivalently,adarklineinabrightregion)driftapart,assumingthesmoothing Figure1:Aschematicdescriptionofthebehaviorofedgemovementinscalespace:(a)no movement,(b)driftingapart,(c)gettingcloser. illusionarypatternsareshownwhichconsistofregulargridsorcheckerboardpatternswithlittle squaressuperimposedonthem.theeectofthesquaresistochangetheappearanceofthe AttheWebsiteoftheAkitaUniversityDepartmentofPsychology[1],anumberofnew (b) (c) blacksquaressuperimposed,andmanyofthesesquaresalignedwiththegrid.figure3bshows asmallpartoftheguremagnied.theblacksquaresinthecenterofthegridallhave straightlinestoconcaveandconvexcurves. beenremovedforclarity,astheydonotnotablyaecttheillusionaryperception.figure3c Figure3dshowsthesmoothedimagewhichresultsfromlteringwithaGaussianwithstandard showstheresultsofedgedetectionontherawimageusingthelaplacianofagaussian(log). deviation5/4timesthewidthofthebarsandfigure3eshowstheresultofedgedetectionon thesmoothedimageusingalog.(thisisclearlythesameasperformingedgedetectiononthe Figure3ashowsoneofthegures,ablacksquaregridonawhitebackgroundwithsmall originallocationswherethegridboundslittlesquaresandareshiftedinsidethewhiteblocks rawimagewithalogoflargerstandarddeviation.) atallotherlocations,andasaresulttheedgesappeartobecurved(figure3e).ascan todriftthetwoedgesapart.atthelocations,however,whereasquareisalignedwiththe grid,thereisonlyoneedge,andthisedgedoesn'tchangelocationunderscale-spacesmoothing. Theneteectofsmoothingisthatedgesofgridlinesarenolongerstraight;theystayatthe beveried,thecurvatureobtainedfromscale-spacesmoothingisqualitativelysimilartothe curvatureperceivedinthegure. Thegridconsistsoflines(orbars),andtheeectofscalespacesmoothingonthebarsis 5

7 Figure2:Therstrowshowsgraylevelguresof(a)adarkregionnexttoabrightregion,(b)a brightlineinadarkregion,and(c)agraylinenexttoadarkandabrightregion.superimposed (b) =7).Thethirdrowshowstherstderivatives,andthefourthrowthesecondderivatives onthegraylevelguresaretheedgeswhichhavebeencomputedasthezerocrossingsofthe versionsofthegraylevelgures(thesolidlinecorrespondsto=3andthedashedlineto toastandarddeviation()ofthreeandthedarkerlinestoaofsevenpixels).thesecond LaplacianofaGaussianfortwodierentwidthsoftheGaussian(thebrighterlinescorrespond rowshowsthefunctionsresultingfromcrosssections(onerow)ofthetwodierentlysmoothed 0.01 ofthesefunctions.themaximaoftherstderivativesandthezerocrossingsofthesecond derivativesaremarkedbyverticallines.fromtheirchangeinpositiontheobservationsabout themovementofedgesinscalespaceasshowninfigure1canbededuced

8 (a) (b) (c) Figure3:(a)Illusionarypattern:\spring."(b)Smallpartoftheguretowhich(c)edge detection,(d)gaussiansmoothing,and(e)smoothingandedgedetectionhavebeenapplied. 7 (e)

9 checkerboardtile).thisexplainstheeect:theedgeswhicharetheboundariesofthecreated barsdriftapartunderscale-spacesmoothingandtheotheredges betweentheblackandwhite tilesofthecheckerboard stayinplace.theresultisthattheedgesnearthelocationsofthe boardwithlittlewhitesquaressuperimposedincornersoftheblacktilesclosetotheedges. square)nexttoablackbar(fromablackcheckerboardtile)nexttoawhitearea(fromawhite whitesquaresarebumpedoutwardtowardthewhitecheckerboardtiles.thisisillustrated Inthispattern,nexttothewhitesquaresshortbarsarecreated awhitearea(fromalittle Figure4ashowsanotheroneofthepatternsfromtheWebsite:ablackandwhitechecker- from(2).againtheeectobtainedfromsmoothingisthesameastheoneperceivedwiththis infigure4bwhichshowsthecombinedeectofsmoothingandedgedetectionforapartof linesseparatingtherows.themortarlinesshouldbemid-wayinluminancebetweentheblack illusionarypattern. Aninteractivedemonstrationofthisillusioncanbefoundat[17].Itconsistsofablackand andwhitesquares;thentheillusionaryeectofatiltinthemortarlines,withalternatemortar whitecheckerboardpatternwithalternaterowsshiftedonehalf-cycleandwiththinmortar thepattern.figures4canddzoominonthedriftvelocityinthesmoothedimageasderived linestiltedinoppositedirections,isstrongest.thecafewallgureiscloselyrelatedtothe Munsterberggure[24],whoseonlydierenceisthatthemortarlinesarethesameluminance aseitherthelightordarktiles;inthisgurethetiltingisperceivedtobemuchweaker. Anotherillusionarypatterninthiscategoryisthe\cafewall"illusionshowninFigure5a. brighttilethetwoedgesmovetowardeachotherunderscalespacesmoothing,andforthinlines ittakesarelativelysmallamountofsmoothingforthetwoedgestomergeintoone.where themortarlineisbetweentwobrightregionsorwhereitisbetweentwodarkregionstheedges moveawayfromeachother.theresultsofsmoothingandedgedetectionareshowninfigure 5cforasmallpartofthepatternshowninFigure5b.Fromthisgureitcanbeseenthatthe edgeelementswhichformboundariesofthetiles(thatis,edgeelementsthatformthelower boundaryofawhitetileandtheupperboundaryofablacktile,andedgeelementsthatform Inthecafewallgure,atthelocationswhereamortarlinebordersbothadarktileanda samesignofslopeasperceived.figures5dandeshowthedriftvelocityinlargemagnication thelowerboundaryofablacktileandtheupperboundaryofawhitetile)aretiltedwiththe tile,therewillbeanedgecloserinpositiontotheblacktilethanwiththegraymortarline. themunsterbergillusioniseasilyunderstood.if,forexample,themortarlineischangedfrom graytowhitetherewillbenoedgesbetweenwhitetiles,andattheborderofawhiteandblack whichillustratesthetendencyofthemovementoftheedges.theweakerillusionaryeectin showsasmallpartofthepatternandfigure6cshowstheedgesdetected.asthelattergure shows,theinsertedsquarespartlycompensateforthedriftinginoppositedirectionsofedges whiteandblacksquaresputinthecornersofthetilesremovetheillusionaryeect.figure6b Theresultisfeweredgeelementswithlesstiltandthusaweakerillusion. andasaresultastraightlinewithouttiltisseen. alongthemortarlineseparatingtilesofthesamegraylevel.asaresultslightlywavyedgels areobtained;butthe\waviness"istooweaktobeperceived(lowamplitude,highfrequency) eectbyintroducingadditionalelements.thishasbeenpursuedinfigure6a;theadditional Ifbiasisindeedthemaincauseoftheillusionthenweshouldbeabletocounteractthe explainsthetiltingoftheseelements.thesecondstageconsistsoftheintegrationofthese localelementsintolongerlines.ourhypothesisisthatthisintegrationiscomputationallyan isduetotwoprocessingstages.intherststagelocaledgeelementsarecomputedandbias approximationofthelongerlinesusingasinputthepositionsandorientationsoftheshortline Afullaccountoftheperceptionoftiltedlinesrequiresadditionalexplanation.Theillusion 8

10 (a) (b) Figure4:(a)Illusionarypattern:\waves."(b)Theresultofsmoothingandedgedetection onapartofthepattern.(c)and(d)thedriftvelocityatedgesinthesmoothedimage logarithmicallyscaledforpartsofthepattern. (d) 9

11 (a) (b) (c) Figure5:(a)Cafewallillusion.(b)Smallpartofthegure.(c)Resultofsmoothingandedge detection.(d)and(e)zoom-insonthedriftvelocity. 10 (e)

12 (a) inthelteredimage,whichcounteractstheillusionaryeect. Figure6:Modiedcafewallpattern.Theadditionalblackandwhitesquareschangetheedges (b) (c) 11

13 spaceandcouldberealizedinamultiresolutionarchitecture.ateveryresolutiontheaverage ofthedirectionsofneighboringelementsiscomputed,andallcomputationsarelocal.ifthe elements.thiscouldbeeasilyimplemented.itbasicallyamountstoasmoothingindirection becomputedashavingatilt. approximationiscarriedoutonthebasisoftheorientationsoflocaledgeelementsthelinewill informationabouttheedgeelementswhichwasderivedearlierandgobacktoimagepoint asinputonlythepositionsofimagepointsontheedgels;suchanapproximationwouldnot information.wedoubtthatevolutionwouldproduceawastefulapproachofthiskind. leadtoatiltintheestimatedlines.however,insuchacasethesystemwouldthrowoutthe Theonlyotherwayofcarryingouttheintegrationwouldbetoapproximatethelines,using twistedcordgivestheperceptionofbeingspiral-shaped,ratherthanasetofcircles.the appearsinbothblackandwhite.inthefraserillusioninfigure8theshortlinesarealltilted principleofthepatterniseasilyexplainedusingthefraserillusioninfigure8.figure8c showsthebasicunitofallfraser'sgures,ashortlinewithatriangleateitherend.this whichtogetherformsomethingratherlikeatwistedcord,onacheckerboardbackground.the Fraser'stwistedcordillusion,ortheFraserspiral,andrelatedFrasergures. TheFraserspiralpattern(Figure7)consistsofcirclesmadeofblackandwhiteelements Ouraccountoftheintegrationprocessalsoexplainsoneofthemostforcefulofallillusions, lines(figure8)orspirals(figure7).wecanalsoaccountfortheincreaseintheperceptionof lines.anapproximationprocessthatusestheseedgelstotextendedcurveswillderivetilted withthesameslopeandatiltedlineisperceived;inthefraserspiralthelineelementsare lineelementsandtheintersectionoftheselineelementswiththetiltedlineelementsofthe thetiltwithinclusionofthebackground.inthiscasetheadditionaltrianglescreateadditional cordonagraybackground(figure8b)butitisslightlyweakerwithoutthepatternofthe backgroundincluded. sectionsofspiralsandspiralsareperceived.theillusionaryeectisalsoobtainedforatwisted gureisbiased,andthisincreasesthetiltofthelineelements. themodelintroducedinthissectionthattheintersectionpointoftwolineswhichintersectat TheperceptualeectatintersectinglinesisillustratedinFigure9.Itcanbeshownwith TheedgeswhicharecomputedinFraser'sguresareatthebordersoftheblackandwhite 3Errorsinlineelements anacuteangleisdisplaced.theeectisobtainedbysmoothingtheimageandthendetecting ofintersectinglinesisthetopicofthenextsection. edgesusingnon-maximumsuppression(seefigure10).amoredetailedanalysisofthebehavior slightlydierentnoisemodel,withthenoisebeingdeneddirectlyontheedgeelements.we areadecisivefactorintheillusion.toanalyzetheselinedrawings,weadoptinthissectiona Thereisalargegroupofillusionsinwhichlinesintersectingatangles,particularlyacuteangles, thecellsanalyzingorientation[15]. edgeelements,parameterizedbytheimagegradient(avectorinthedirectionnormaltothe Anotherphysiologicallyplausibleexplanationisthatsuchnoiseisduetoquantizationerrorsof thatonlythenoiseinthedirectionhasaninuenceonthepositionsoftheintersectionpoints givethesameeects.noiseingraylevelvaluesresultsinnoiseintheestimatededgeelements. assumeadditivenoiseinthepositionsanddirectionsofedgeelements.letusnotebeforehand edge)(ix;iy)andthepositionofthecenteroftheedgeelement(x0;y0). oflines,andthusamodelthatdoesn'tconsideranynoiseinthepositionsofedgeelementswill Wenextanalyzetheestimatedpositionofanintersectionofstraightlines.Theinputsare 12

14 Figure7:Fraser'sspiral[9]. ateitherend. Figure8:From[29].(a)Frasergurewithstraighttwistedcords.(b)Theillusionisweaker withagraybackground.(c)thebasicelementsofthegurearetiltedelementswithatriangle (b) 13

15 Figure9:[34]ThenelineasshowninAappearstobebentinthevicinityofthebroader blackline,asindicatedinexaggerationinb. Figure10:(a)Alineintersectingabaratanangleoffteendegrees.(b)Theimagehasbeen smoothedandthemaximaofthegraylevelfunctionhavebeendetectedandmarkedwithstars. (c)magnicationofintersectionarea. (b) (c) 14

16 rameters.inthesequelunprimedlettersareusedtodenoteestimates,primedletterstodenote actualvalues,and'stodenoteerrors,whereix=i0x+ix,iy=i0y+iy,x0=x0+x0and y0=y0+y0. Consideradditive,independentlyidenticallydistributed(i.i.d.)zero-meannoiseinthepa- Thisequationisapproximatedbythemeasurements.Letnbethenumberofmeasurements. Eachmeasurementiprovidesoneequation Foreverypoint(x;y)onthelinesthefollowingequationholds: andweobtainasystemofequationswhicharerepresentedinmatrixformas Ixix+Iyiy=Ixix0i+Iyiy0i I0xx+I0yy=I0xx0+I0yy0 (4) HereIsisthen-by-2matrixwhichincorporatesthedataintheIxiandIyi,and~Cisthe n-dimensionalvectorwithcomponentsixix0i+iyiy0i.thevector~xdenotestheintersection Is~x=~C (5) leastsquare(ls)estimationisgivenby~x=(itsis)?1its~c pointwhosecomponentsarexandy.thesolutiontotheintersectionpointusingstandard ydirectionsisassumedtobethesame,letitbe2s,andalsotheexpectedvaluesofhigher- twoothernoisytermsandthusthestatisticsaresomewhatdierent. themeasurementmatrixaisbiased.thestatisticsoftheestimationhavebeenstudiedforthe Itiswellknown[10]thattheLSsolutiontoalinearsystemoftheformA~x=~bwitherrorsin caseofi.i.d.noiseintheparametersofaand~b.inourcase~bistheproductoftermsinaand Tosimplifytheanalysis,thevarianceofthenoiseinthespatialderivativesinthexand (6) ~xisfoundbydeveloping(6)intoasecond-ordertaylorexpansionatzeronoise.itconverges (thansecond)ordertermsareassumedtobenegligible.intheappendixtheexpectedvalueof ~x0istheactualintersectionpointand~x0="1npni=1x0i inprobabilityto where M0=Is0tIs0="Pni=1I02xiPni=1I0xiI0yi n!1e(~x)=~x0+nm0?1(~x0?~x0)2s plim Pni=1I0xiI0yiPni=1I02yi# 1nPni=1y0i#isthemeanofthe~x0i. (7) (~x0?~x0).vector(~x0?~x0)extendsfromtheactualintersectionpointtothemeanpositionof eigenvectorsareorthogonaltoeachother.m0?1hasthesameeigenvectorsasm0andinverse shiftedbyatermwhichisproportionaltotheproductofmatrixm0?1andthedierencevector whichdependsonlyonthespatialgradientdistribution,isarealsymmetricmatrixandthusits theedgeelements.thusitisthemasscenteroftheedgelsthatdeterminesthisvector.m0, eigenvalues.thedirectionoftheeigenvectorcorrespondingtothelargereigenvalueofm0?1is dominatedbythenormaltothemajororientationoftheimagegradientsandthustheproduct Using(7)allowsforaninterpretationofthebias.Theestimatedintersectionpointis ofm0?1withvector(~x0?~x0)ismoststronglyinuencedbythisorientation.thustheeect 15

17 thisgureisapparentlynotthecontinuationofthelowerportionontheright,butistoohigh. interactiveversionsee[16]).theupper-leftportionoftheinterrupted,tiltedstraightlinein AnotherversionofthisillusionisshowninFigure11b.Hereitappearsthatthemiddleportion ofmoregradients.forthecaseoftwointersectinglinesthismeansmoredisplacementofthe ofm0?1ismorebiasinthedirectionoffewerimagegradientsandlessbiasinthedirection AversionofthePoggendorillusionasdescribedbyZollnerisdisplayedinFigure11a(foran oftheinclined(interrupted)lineisnotinthesamedirectionasthetwoouterpatterns,butis intersectionpointinthedirectionperpendiculartothelinewithfeweredgeelements. turnedclockwisewithrespecttothem. ThebestknownillusionsduetointersectinglinesarethePoggendorandZollnerillusions. illusionsastheresultoftheperceptualenlargementofacuteangles.ithasbeenfoundinextensivestudiesthatacuteanglesareoverestimated,andobtuseanglesareslightlyunderestimated Famousvisionscientists[13,34]sawthePoggendoraswellastheZollnerandotherrelated (a) Figure11:Poggendorillusion. (b) (althoughregardingthelattertherehasbeencontroversy).theeectislargerwithshortline segmentsandseemstodiminishwithlongintersectinglines.ourmodelpredictsthatthese phenomenaareduetothebiasintheestimationoftheintersectionpoint. lineismoveddownandtotheright.asaresultthetwolinesegmentsappeartobeshiftedin ismovedupandtotheleft,andtheintersectionpointoftherightverticalwiththelowertilted oppositedirectionsandnottolieonthesamelineanymore. plottheestimatedbiasinthex-andy-directionsasafunctionoftheanglebetweentheline elements.inthisplotthedataisaverticallinewhichmakesananglewithatiltedline,and intheacuteangleandreacheszeroat90degrees.ourmodelpredictsthis.figures12bandc ReferringtoFigure11a,theintersectionpointoftheleftverticalwiththeuppertiltedline thenumberofedgeelementsontheverticallineistwicethatonthetiltedline,asillustrated thediagonallinesinfigure13bareallparallel,buttheylookconvergentordivergent.inthese infigure12a. Fromparametricstudiesitisknownthattheillusionaryeectdecreaseswithanincrease Figure13showstwoversionsoftheZollnerillusion.TheverticalbandsinFigure13aand 16

18 φ Figure12:(a)Thetiltedlineintersectstheverticallineinthemiddleatanangle.Thereare vertical.(c)biasparalleltothevertical. twiceasmanyedgeelementsontheverticalasonthetiltedline.(b)biasperpendiculartothe (c) bias in x bias in y tiltedbars. samedirectionasperceivedbythevisualsystem.figure14showstheestimationofthetilted lineelementsforapatternsuchasinfigure13awith45degreesbetweentheverticalandthe tilted,asillustratedinfigure13c.inasecondcomputationalstep,longlinesarecomputedas anapproximationtothesmalledgeelements,andthisgivesrisetotiltedlinesorbarsinthe linesegmentscausetheedgeelementsalongthelongedgesbetweenintersectionpointstobe patternsthebiasesintheintersectionpointsofthelonglines(oredgesofbands)withtheshort phi phi thelineelementsbetweenintersectionpointstobetilted. Figure13:(a),(b)Zollnerpatterns.(c)Thebiasintheintersectionpointsoftheedgescauses increasinglyacuteanglebetweenthemainlineandtheobliques,whichcanbeexplainedas beforeusingfigures12bandc(or,similarly,figure17).thevaluewherethemaximumoccurs Inexperimentswiththisillusionithasalsobeenfoundthattheeectdecreaseswithan (b) (c) variesamongdierentstudies.itissomewherebetween10and30degrees;belowthat,some 17

19 counteractingeectsseemtotakeplace. Figure14:EstimationofedgesinZollnerpattern.Thelineelementsarefoundbyconnecting twoconsecutiveintersectionpoints,resultingfromtheintersectionofedgesoftwoconsecutive timesmoreelementsonthevertical) acongurationsimilartotheoneinfigure17. consistsofedgeelementsuniformlydistributedontheverticalandonthetiltedline(with1.5 barswiththeedgeoftheverticalbar(oneinanobtuseandoneinanacuteangle).thedata strongestwiththeparallellinesverticalorhorizontal.thezollnerillusion,ontheotherhand, wasfoundtobemaximalwhenthejudgedlinesareat45degrees,asinfigure13b. ofthepoggendorandzollnergures.somestudiesfoundthatthepoggendorillusionis Otherparametricstudieshavebeenconductedontheeectofalteringtheorientations lines,thatis,moreedgeelementsareestimatedintheseandnearbydirections.(inthepatterns zontaldirections:wegenerallyseebetterintheseorientations.anatomicalstudies[15]have tations[21].basedonthesendingsweassumethereismoredataforhorizontalandvertical foundorientation-selectivecellsinthecortex;dierentcellsrespondtodierentangles.ithas alsobeenfoundthatthereismoreactivityforhorizontalandverticalthanforobliqueorien- consideredtheremaybemoreactivitybecauseofmoreeyemovementsalongthesedirections, asthevisualtasksrequirejudgmentsalongthesedirections.) Psychophysicistshavecoinedtheterm\spatialnorms"torefertotheverticalandhori- distributiononthebiasisstrongestinthedirectionoftheeigenvectorcorrespondingtothe inthedirectionsofthespatialnorms.inadirectionwheretherearemoreestimatesthereisa largersignaltonoiseratio,andthisresultsingreateraccuracyinthisdirection. Asimplestatisticalobservationexplainsthehigheraccuracyintheestimationofentities y-directionincreases.thepoggendorillusionisstrongerwhentheparallellinesarevertical verticaltoobliqueelementsincreases,thebiasinthex-directiondecreasesandthebiasinthe largereigenvalueofm0?1andweakestintheorthogonaldirection.changingtheratioof measurements(edgels)alongthedierentlineschangesthebias.figure15illustrates,forthe andy-directionsasafunctionoftheratioofedgeelements.itcanbeseenthatastheratioof caseofaverticallineintersectinganobliquelineatanangleof30degrees,thebiasinthex- Expressedinourformalismof(7),thistakesthefollowingform.Theeectofthegradient andthemainlinesaretilted,asinthiscasethebiasperpendiculartothemainlines(alongthe islarger,andthezollnerillusionisstrongerwhenthesmalllinesarehorizontalandvertical orhorizontal,becauseinthiscasethebiasparalleltothelines(alongthey-axisintheplot) 18

20 x-axisintheplot)islarger. Figure15:(a)Averticallineandatiltedlineintersectingatanangle.(b)Biasperpendicular totheverticalasafunctionoftheratioofedgeelementsontheverticalandtiltedlines.(c) ExamplesaretheOrbisonguresandthepatternsofWundt,Hering,andLuckiesh.Inthese Biasparalleltothevertical. Manyotherwell-knownillusionscanbeexplainedonthebasisofbiasedlineintersection. (b) patternsgeometricalentitiessuchasstraightlines,circles,trianglesorsquaresaresuperimposedondierentlytiltedlines,andtheyappeartobedistorted.thisdistortionresultsfrom theerroneousestimationofthetiltinthelineelementsbetweenintersectionpointsandthe approximatedbystraightlines,andtheratioofelementsbetweenthecircleandthebackground andtwototheouterones.arcsonthecirclebetweentwoconsecutivebackgroundlineswere edges,andwecomputedtheintersectionbetweenanybackgroundedgeandcircleedge(using withastraightlinefourintersectionpoints,twocorrespondingtotheinneredgesofthecircle edgewas2:1(resultinginacongurationsimilartothatoffigure17withthearccorresponding congurationssuchasthoseinfigure15a).thisprovidedforeveryintersectionofthecircle subsequentttingofcurvestotheselineelements. tothevertical).consecutiveintersectionpoints oneoriginatingfromanobtuseandonefrom Figure16illustratestheestimationofthecurveintheLuckieshpattern.Eachlinehastwo theouterlinesegments(onlyininterestingplaces).thisresultedinacurveliketheonewe perceive,withthecirclebeingbulbedoutontheupperandlowerleftandbulbedinonthe anacuteangle wereconnectedwithstraightlinesegments.thenbeziersplineswerettedto upperandlowerright. illusionswillresultintheoverestimationofacuteanglesforshortlinesegments.theunderestimationofobtuseanglescanbeexplainedifweassumeanunequalamountofedgeldataonthe thebiasinthey-directionincreases(withincreasingangle)forobtuseangles,andthisresults elementsontheverticalthanonthetiltedline.thebiasinthex-directionchangessignand Nextletuslookattheerroneousestimationofangles.Thebiasdiscussedfortheabove inasmallunderestimationofobtuseangles. twoarcs.figure17illustratesthebiasforacuteandobtuseanglesforthecaseofmoreedge verticallinesarethesamelength,buttheoneontheleftwithterminalspointingoutward isthemuller-lyerillusion.theoriginalmuller-lyergureisshowninfigure18.thetwo appearsconsiderablylongerthantheoneontherightwithinward-pointingterminals.inthis gure,theestimatedintersectionpointsoftheterminalsandthestraightlineareoutwardin theleftandinwardintherightgure.thiscertainlycouldbeacontributingfactorinthe Theillusionwhichprobablyhasbeenstudiedthemostinthepsychophysicalliterature 19 φ bias in x, phi= Pi/6 3 Weight bias in y, phi =Pi/6 3 Weight

21 (a) (b) Figure16:EstimationofLuckieshpattern:(a)thepattern:acirclesuperimposedonabackgroundofdierentlyarrangedparallellines,(b)ttingofarcstothecircle,(c)magniedupper leftpartofpatternwithttedarcssuperimposed,(d)intersectionpointsandttingofsegments (c) Figure17:(a)Biasperpendiculartothevertical.(b)Biasparalleltotheverticalforlines toouterintersections.phi intersectingatangle(asinfigure15);theratioofverticaltotiltededgeelementsis bias in x, weight = bias in y, weight =3 1.5 phi 2 2.5

22 illusion,buttheeectofthebiasdoesnotseemtobelargeenoughtofullyexplainthisstrong illusion.itislikelythatthereareadditionalcausesforthisillusion,whichareduetohigherlevelinterpretationprocesses.theillusionalsooccurswithotherterminals;itseemstohappen whenaspatialextentisboundedbycontoursthatareeitheroutwardlyconvexorconcave, eithercurvedorangular.forotherterminalsonewouldhavetotcurvestotheedgelsalong theterminalsandcomputetheirintersectionswiththestraightlines. 4Errorsinmotion Whenprocessingimagesequencessomerepresentationofimagemotionmustbederivedas arststage.itisbelievedthatthehumanvisualsystemcomputestwo-dimensionalimage measurementswhichcorrespondtovelocitymeasurementsofimagepatterns,calledoptical Figure18:Muller-Lyerillusion. perpendiculartolinearfeaturesarecomputedfromlocalimagemeasurements.inthecomputationalliteraturethisone-dimensionalvelocitycomponentisreferredtoas\normalow"and theambiguityinthevelocitycomponentparalleltotheedgeisreferredtoasthe\aperture ow.theresultingeldofmeasurements,theopticaloweld,representsanapproximation problem."inasecondstagetheopticalowisestimatedbycombining,inasmallregionof totheprojectionoftheeldofmotionvectorsof3dscenepointsontheimage. isbiased. theimage,normalowmeasurementsfromfeaturesindierentdirections,butthisestimate Opticalowisderivedinatwo-stageprocess.Inarststagethevelocitycomponents velocityofanimagepointinthex-andy-directionsby~u=(u;v),thefollowingconstraint isobtained: derivativesoftheimagegrayleveli(x;y;t)byix;iy,thetemporalderivativebyit,andthe isthatimagegrayleveldoesnotchangeoverasmalltimeinterval.denotingthespatial Weconsideragradient-basedapproachtoderivingthenormalow.Thebasicassumption thedirectionofthegradient[14].weassumetheopticalowtobeconstantwithinaregion. Eachofthenmeasurementsintheregionprovidesanequationoftheform(8)andthuswe Thisequation,calledtheopticalowconstraintequation,denesthecomponentoftheowin Ixu+Iyv+It=0 21

23 obtaintheover-determinedsystemofequations whereisdenotes,asbefore,thematrixofspatialgradients(ixi;iyi),~itthevectoroftemporal derivatives,and~u=(u;v)theopticalow.theleast-squaressolutionto(9)isgivenby Asanoisemodelweconsiderzero-meani.i.d.noiseinthespatialandtemporalderivatives.As ~u=?(itsis)?1its~it: Is~u+~It=0; (9) thetwodirectionsandweassumethathigherthansecond-ordernoisetermscanbeignored. expectedvalueoftheow,usingasecond-ordertaylorexpansion,isderivedintheappendix; intheprevioussection,weassumeequalvariance2sforthenoiseinthespatialderivativesin itconvergesinprobabilitytoplim Thestatisticsof(10)arewellunderstood,astheseareclassicallinearequations.The ThebiasisproportionaltothenegativeproductofM0?1withtheactualow.Inthecase where,asbefore,theactualvaluesaredenotedbyprimes. ofauniformdistributionofimagegradientsintheregionwheretheowiscomputed,m0 thecomputedopticalow;thereisanunderestimation.inaregionwherethereisaunique (andthusm0?1)isamultipleoftheidentitymatrix,leadingtoabiassolelyinthelengthof Equation(11)isverysimilarto(7)andtheinterpretationgiventhereapplieshereaswell. n!1e(~u)=~u0?n2sm0?1~u0; gradientvector,m0willbeofrank1;thisistheapertureproblem.inthegeneralcase,the eigenvectorwiththelargereigenvalueisinthedirectionoffewermeasurements;thusthere ismoreunderestimationinthisdirectionandlessunderestimationinthedirectionofmore measurements.theestimatedowthereforeisbiaseddownwardinsizeandbiasedtowardthe majororientationofthegradients. thebiascanbeunderstoodrathereasily.theeigenvectorsofm0?1areinthedirectionsof entationsurroundinganinnerring.smallretinalmotions,orslightmovementsofthepaper, manynormalowmeasurementsinonedirectionastheother.forsuchagradientdistribution oflengthtowidthisfour,leadingtoagradientdistributioninasmallregionwithfourtimesas causeasegmentationoftheinsetpattern,andmotionoftheinsetrelativetothesurround. oftworectangularcheckerboardpatternsorientedinorthogonaldirections abackgroundori- Thetilesusedtomakeupthepatternarelongerthantheyarewide;inFigure19theratio Figure19showsavariantofapatterncreatedbyHajimeOuchi[27].Thepatternconsists Figure20b,whichhasitsminimumat0anditsmaximumat=2(thatis,when~u0isaligned withthemajorgradientdirection).theerrorinangleisgreatestfor=4(thatis,when~u0is opticowandthedominantgradientdirection.theplotsarebasedontheexactsecond-order Figure20showstherelationshipbetweentheopticowbiasandtheanglebetweentheactual thetwogradientmeasurements,withthelargereigenvaluecorrespondingtofewergradients. nearlyeliminatedwhenitisalignedwiththeow(thatis,intheouchipattern,whenthelong edgeoftheblockisperpendiculartothemotion).thebiasforanglesbetween=2and exactlybetweenthetwoeigenvectorsofm0?1)anditis0for0and=2(figure20c).overall, Taylorexpansion. thismeansthebiasislargestwhenthemajorgradientdirectionisnormaltotheowandis As~u0=(0;1),thenoisetermin(11)leadstoabiasinlength,asshownbythecurvein 22

24 Figure19:ApatternsimilartotheonebyOuchi. 23

25 fewer measurements (a) y u 0 θ more measurements x betweentheexpectedowandtheactualow.theerrorhass=0:15. andoflength1.(b)expectederrorinlength.(c)expectederrorinangle,measuredinradians, Figure20:(a)16measurementsareinthedirectionmakinganglewiththepositivexaxis and4measurementsareinthedirection+=2.theopticalowisalongthepositiveyaxis (c) theta theta 24

26 errorintheangle. gradientdirectiondierintheinsetandthesurround,sotheregionalvelocityestimatesare isobtainedfromtheaboveplotsbyreectingthecurvesin=2andchangingthesignofthe forallanglesthedierencebetweentheerrorvectorintheinsetandtheerrorvectorinthe biasedindierentways.when,insteadoffreelyviewingthepatternoffigure19,thepage ismovedindierentdirections,weobservethattheillusionarymotionoftheinsetismostly surroundingareaprojectedonthedominantgradientdirectionoftheinsetisinthisdirection. anglewiththemotionofthepaperislessthan90.usingfigure20,itcanbeveriedthat aslidingmotionorthogonaltothelongeredgesoftherectangleandinthedirectionwhose IntheOuchipattern,therelativeanglesbetweentherealmotionandthepredominant aperceivedmotiontotheright.ifthemotionofthepaperisupward,thedierencevectoris downward;itsprojectiononthemajorgradientdirectionoftheinsetiszero,andthushardly anyillusionarymotionisperceived.figure21shows,forasetoftruemotions,thebiasesin Forexample,whenthemotionisalongtherstmeridian(totherightandup),theerrorin theinsetisfoundinthegraphatangle==4andinthesurroundat=3=4.thetwo theperceivedmotion. motionofthepaperistotheright,thedierenceinerrorvectorsisduetolength,resultingin estimatedmotionvectorsareofthesamelength,eachinthedirectiontowardthegradients ofthelongeredges,andtheprojectionoftheresultingdierencevectoristotheright.ifthe s=t=0:1;in(c),s=t=0:2. thetruemotion.thenoiseisgaussianandthespatialgradientmagnitudeis1.in(a)and(b), betweentheerrorintheinsetandtheerrorinthesurround,andprojecttheresultingvector thetruemotionandthecalculatedmotion.toderivetheslidingmotion,computethedierence onthedominantgradientdirectionintheinset.thelinefromthecenterisinthedirectionof Figure21:Theregionalmotionerrorvectoreld.Thevectorsshownarethedierencesbetween (b) (c) easilyexperienced. oftheinsetforalargerangeofangles(thatis,directionsofeyemovements),theillusionis underfreeviewingconditions,thetriggeringmotionisduetoeyemovements,whichcanbe vectorsoftheinsetandsurroundhasasignicantprojectiononthedominantgradientdirection whichiswhyaclearrelativemotionoftheinsetisseen.whenexperiencingtheouchiillusion approximatedthroughrandom,fronto-paralleltranslations.sincethedierenceinthebias In[34]HelmholtzdescribesanexperimentwiththeZollnerpatternwhichcausesillusionary Weassumethatinadditiontocomputingow,thevisualsystemalsoperformssegmentation, motion.whenthepointofaneedleismadetotraversezollner'spattern(figure13a)hori

27 bandsoccurs.therst,third,fthandseventhblackbandsascend,whilethesecond,fourth, zontallyfromrighttoleft,itsmotionbeingfollowedbytheeye,aperceptionofmotioninthe andsixthdescend;itisjusttheoppositewhenthedirectionofthemotionisreversed. opticalowfromlefttoright.foreachbandtherearetwodierentgradientdirections,i.e., therearetwodierentnormalowcomponentsineachneighborhood.fortheoddbandsthe y-component),andthiscomponentalongthey-axisisperceivedasupwardmotionofthebands twonormalowcomponentsareinthedirectionoftheowanddiagonallytotherightandup. Thustheestimatedowmakesapositiveanglewiththeactualow(thatis,ithasapositive (seefigure22).forevenbandstheestimatedowisbiaseddownward,causingtheperception ofdescentofthebands.similarly,ifthemotionoftheeyeisreversedtheestimatedowhasa Thebiasexplainsthiseectasfollows:Amotionoftheeyesfromrighttoleftgivesriseto negativey-componentintheoddbandsandapositivey-componentintheevenbands,leading toareversalintheperceivedmotionsofthebands. Figure22:Theeyemotiongivesrisetoowu0andnormalowvectorsn1andn2.The theband. 5Theinherentproblem Animportantquestionarises.Istherebiasbecauseofthearchitectureofthevisionsystem? estimatedow,u,hasapositivey-component,whichcausesillusionarymotionupwardin Isthebiasduetothelinearestimationonly?Coulditbecorrectedusingmoresophisticated variables,butanymethodofcompensatingforthebiasrequiresknowledgeofthestatisticsof statisticaltechniques,orcoulditbeavoided? oftenitisnotpossibletoobtainaccurateenoughestimatesofthenoiseparameterstoimprove majorproblem,however,liesintheacquisitionofthestatisticsofthenoise.wearguethat thenoise.inthenoisemodelsconsideredintheprevioussections,thisamountstoknowledge ofthecovariancematrixofthenoise.ifthiswereavailable,inverselterscouldbeappliedto reconstructthegraylevelsignal,andthecorrectedleastsquaresestimatorcouldbeusedto removetheasymptoticbiaswhensolvinglinearsystemsonthebasisofimagederivatives.the Itiswellknownthatlinearestimationisbiasedifthereareerrorsinallthemeasurement thesolution. arefrequencydomainandcorrelationmodels,butcomputationallytheyarenotverydierent. dicultiesoccur,andnoisyestimatesleadtobias.foranextensivediscussionofthestatistics Inallthemodelsthereisastageinwhichsmoothnessassumptionsaremadeandmeasurements withinaregionarecombinedtoobtainmoreexactmeasurements.atthisstagestatistical Thereexistothermodelsforcomputingopticalow.Besidesgradient-basedmodelsthere 26 n 1 n 2 u u

28 ofopticalowestimationsee[8]. A~x=~b,thereiserrorinthevariablesinmatrixAinadditiontotheerrorinthevariablesin hasreceivedalotofattention.theproblematicbiasarisesbecauseinthesystemofequations position,butinformationaboutnoiseratiosisdiculttocompute.itcanbeobtainedonly ables,thatis,theratiosofthetwospatialandtemporalderivativenoisetermsorthenoisein distributed.thismeansthatwehavetoknowtherelativeamountsofnoiseintheerrorvaricallyunbiasedsolutionforsuchsystems,ifthenoisevariablesareindependentandidentically vector~b.thenonlineartotalleastsquaresestimatorhasbeenshowntoprovideanasymptoti- Inrecentyearsthetechniqueoftotalleastsquaresforsolvingsystemsoflinearequations thistechnique:thevarianceislarger.totalleastsquaresisknowntoperformverypoorlyif fromthevariationintheestimatedvariablesovertheimage.thereisanotherproblemwith outliersarepresent,andthesearediculttodetectfromafewmeasurements. acquireenoughdatatocloselyapproximatetheseparameters,butusuallythenoiseparameters longedgesandbarsneedtobedetected,andinthecaseofmotion,discontinuitiesdueto changesindepthanddierentlymovingentitiesneedtobedetectedandthescenesegmented. goodnoisestatisticalotofdataisrequired,sodataneedstobetakenfromlargespatialareas acquiredoveraperiodoftime,butthemodelsusedfortheestimationcanonlybeassumedto holdlocally.thustointegratemoredata,modelsofthesceneneedtobeacquired.specically, Ifthenoiseparametersstayedxedforextendedperiodsoftimeitwouldbepossibleto Whyisitsodiculttoobtainaccurateestimatesofthenoiseparameters?Toacquirea theobjectsbeingviewed,theorientationoftheviewerin3dspace,andthesequenceofeye movementsallhaveinuencesonthenoise.asidefromallthesefactors,inordertoestimate derivatives(ortocomputefouriertransforms)thesystemneedstointerpolate.theaccuracy ofinterpolationcandependincomplexwaysonthepatternofgraylevelsintheimage. othersourcesofnoisebesidessensornoise.thelightingconditions,thephysicalpropertiesof donotstayxedlongenough.sensorcharacteristicsmaystayxed,buttherearemany whichallowittoimprovetheestimates. example,whenviewingastaticpatternwithcontrolledeyemovementsorwhenxatingona thepatternsweakensafterextendedviewing,inparticularwhensubjectsareaskedtoxate. Inthiscasethevisualsystemmayacquirereasonableapproximationsofthenoiseparameters pointintheimageforsometime.thismayexplainwhytheillusionaryperceptioninsomeof Undercontrolledconditionsitmaybepossibletokeepthenoiseparametersxed,for 6DiscussionandSummary Scholarshaveclassiedpatternsasgeometricalopticalillusionsonthebasisofhowtheyare formed(theyarelinedrawings)andtheoreticianshavebeenchallengedtoprovideexplanations forallofthem.thereis,however,noreasontoexpectallillusionaryeectsinvolvingblack largenumberofgeometricalopticalillusions.otherillusionsinthiscategorywhichwehaven't andwhitepatternstobeduetothesamecause. dealtwithare:thecontrasteect(thatis,theoverestimationoflargeextentsnexttosmall guresenclosedbetweenconverginglines(thegurenearertheintersectionofthelinesis intervalseparatedbylinesintosubintervals);theponzoillusions,whichinvolvetwoequal anunlledextent(anexampleistheoppel-kundtillusion anunlledintervalnexttoan unlledextents,thatis,theeectthatalledextentisoverestimatedwhencomparedwith onesandtheunderestimationofsmallextentsnexttolargeones);theillusionsoflledand Inthispaperweadvancedatheoryaboutbiasinthevisualsystemwhichaccountsfora perceptuallyenlarged);thevertical-horizontalillusion,thatis,thephenomenonofjudginga 27

29 areofgreatimportanceinsomeoftheseillusions.thecontrasteectoccursonlyforacertain verticallinetobelongerthanahorizontallineofthesameextent;theframingeect,thatis, (withamaximumofabout2:3),andafterthataninectionoccurs,thatis,forlargerratiosthe oneperceptuallydecreasedandtheinneroneincreased occursuptoarangeofabout1:2 rangeofsizeratios.forexample,thedelboeufillusion twoconcentriccircleswiththeouter atteningofarcs,thatis,theeectthatshortarcsareperceptuallyattened.sizeparameters is,whencertaingureswhichencloseidenticalareasarecompared,andpartoftheboundaryof theoverestimationofframedobjects,andtheeectofoverestimatingunboundedgures,that smallercirclechangesfrombeingoverestimatedtobeingunderestimated.similarly,framing leadstounderestimationofthesizeoftheframedobjectforratiosoftheobjecttotheframe oneismissing,thedimensionwhichistherebyundenedatoneendisoverestimated;andthe closeto1:1. explainedtheillusionnamedafterhimintermsoftheelongationofthearmsofanobtuseangle ascomparedwiththoseofanacuteangle,whilebrentano'sexplanation[2]wasintermsofthe nowadaysasaddinglittletoameredescriptionoftheillusion.thusmuller-lyerhimself theoriesareaimedonlyatonespecicillusionandmostoftheearlytheorieswouldbeseen perceptualenlargementofacuteanglesandthereductionofobtuseangles. Theoriesaboutillusionshavebeenformulatedeversincetheirdiscovery.Manyofthe mathematicalnaturebasedonthegeometryandstatisticsofcapturinglightraysandcomputing ofthebrain(forexample,lateralinhibition).incontrast,thetheoryproposedhereisofa basedonphysicalanalogies,orgeneralobservationsaboutthephysiologyandarchitecture system,articialorbiological.however,onemightndthatourtheoryresemblessomeexisting theirproperties(grayvaluesandspatiotemporalderivatives),andthusitappliestoanyvision toexplaintheworkingsofhumanvisualperception.thesemechanismsareeitherhypothetical, morenoticeinrecentyearsarebasedonaspecicsortofmechanismwhichhasbeensuggested Theorieswhichattempttoexplainabroadspectrumofillusionsandwhichhaveattracted theblurringanddiusingeectofthemediumandtheconstructionoftheeye.theperceived locationsofthelinesareatthepeaksoftheirdistributions;thustwocloselinesinuenceeach fallingontoascreenfromtwoslitsourcesinthestudyofdiractionpatterns.thisisbecauseof theoriesiftheyareputinastatisticalframework. togetheraectoneanother.helikenedthelightcomingintotheeyefromtwolinestothelight other'slocationsandbecomeonewhenthesumoftheirdistributionsformsasinglepeak. Thisleadstoanoverestimationofacuteangles,andprovidesanexplanationoftheZollner, Poggendor,andrelatedillusionsaswellastheMuller-Lyereect.Thediractionamountsto Chiang[6]proposedaretinaltheorythatappliestopatternsinwhichlinesrunningclose data.chiang'smodel,however,ismuchmorelimited.itonlyappliestolines,andconsiders thewholeprocesstotakeplaceintheretina. uncertaintyinthelocationoftheperceivedgraylevelvalues,oritcanbeinterpretedasnoisy inuencethenoisedistributionandthusthebiasperceived,butthereareothernoisesources theyarearelevantsourceofnoise.theparticulareyemovementsmadeinlookingatapattern besideseyemovements,andthisexplainstheexistenceofillusionaryeectsforsomepatterns evenunderxationortachistoscopicviewing. causativefactorinillusions.ourtheoryalsoproposesthateyemovementsplayarolebecause Thereareanumberoftheoriesinwhicheyemovementsareadvancedasanimportant overunlledthanlledelements.inthemuller-lyergurestheeyesmovemorefreelyoverthe healsoexpresseddoubtthattheycouldbethemainsource,asotherillusionsarenotinuenced bythem.carr[5]proposedthattheeyesreacttoaccessorylinesandasaresultpassmoreeasily Helmholtzsuggestedin[34]thatocularmovementsareofimportanceinsomeillusions,but 28

30 gurewithoutgoingnsthanovertheonewithingoingns,andinthepoggendorandzollner guresdeectionsandhesitationsintheeyemovementsareassociatedwiththeintersections distortion. arelinearandrectilinear,horizontalorvertical.whenviewinglineswhichlieothevertical orhorizontal,aneyemovementcorrectionmusttakeplaceandthiscangiverisetoperceptual theeld.virsu[33]suggestedthatatendencytoeyemovements,thatis,instructionsforeye movements,hasaperceptualeect.hesuggeststhattheeyemovementsmostreadilymade Centrationonpartoftheeldcausesanoverestimationofthatpartrelativetotherestof ofthelonglineswiththeobliques.piaget[28]proposeda\lawofrelativecentrations."by \centration"hereferstoakindofcenteringofattentionwhichisverymuchrelatedtoxation. projectionsofthree-dimensionaldisplays[31,32].themostdetailedandmostpopularsuch aftereectsappliedtoillusions[11,18].inthesetheoriesinterferencebetweennearbylines occursbecauseofsatiationinthecortexorlateralinhibitionprocesses.therearealsotheories basedontheassumptionthattheperceptualsysteminterpretsillusionarypatternsasat theoryisduetogregory[12],whoinvokes\sizeconstancyscaling,"whichcanbetriggeredin twodierentways,eitherunconsciouslyorbyhigher-levelawareness,toexplainillusions. Otherfamoustheoriesincludetheorieswhosemainobjectivewastheexplanationofgural aseriousproblemforearlyvisualprocessesandunavoidablyleadstobias.anartifactofthe biasisillusionaryperceptionsinvolvingpatternswherethebiasishighlypronounced.noise intheimagedata thatis,theimagegraylevelanditsspatialandtemporalderivatives causes andtemporalintegrationofdatathatmovingsystemsareconfrontedwith,andduetothe ispresentinanyvisualdata.itisduetothesensingprocess,andinparticularthespatial operationsinvolvedincomputingderivatives,orinestimatingandlocatingcertainfrequency componentsofthesignal.theproblemisthatthenoiseparametersusuallycannotbeestimated Inthispaperwehavediscussedamajorhurdlethatvisionsystemshavetodealwith.Noise anyreferencetotheparticularsettingsthathumanvisionsystemsarefacing,neitherwhat datatobecollected. well,astheychangewiththelightingandviewingconditions,oftentoorapidlytoallowenough smallamountofnoise,andthisallowedustoexplainalargenumberofthegeometricaloptical oftheamountofnoiseandtherelationshipofeyemovementstothenoise.wehaveassumeda thespecicprocessesareandhowtheyareimplementedintheneuralhardware,norinwhich illusions.otherillusionscouldbeexplainediflargeamountsofnoisewerepresent,forexample, usingpsychophysicalandphysiologicalmethodsitshouldbefeasibletoconductinvestigations rangestheuncertaintyinthedatalies.theseareissuesthatneedtobeexamined.inparticular, Thebiashasbeendemonstratedforverygeneralmathematicalmodels.Wehaven'tmade theponzoillusionsorthedelboeufcircles. ofsmoothing,willcausebothcirclesinthemiddleoffigure23atocontract.ifweincrease outercirclewillcontractandtheinnercirclewillexpand,andeventuallythetwocircleswill collapseintoone.thusaratherlargeamountofnoise largeenoughtoaccountformutual theamountofnoisesothatthetwocirclesaecteachotherunderthesmoothing,thenthe inuencesincirclesofsmallersizeratios,butnotlargeenoughtoaccountformutualinuences incirclesoflargersizeratios explainsthendingsinparametricstudiesofthedelboeuf circles.thatis,suchnoisecausesanoverestimationoftheinnercircleforsmallsizeratiosand ConsidertheDelboeufcirclesinFigure23a.Smallamountsofnoise,thatis,smallamounts lineclosertotheintersectionofthelines,causesanexpansionofthisline. seenthatnoiselargeenoughtocausesmoothing,whichaectsoneofthelinesandthevertical anunderestimationforlargerratios.similarly,fortheponzoillusioninfigure23b,itcanbe Theimageformationprocessshouldnotgiverisetolargeamountsofnoise,buttheremay 29

31 informationtoneuronsoflargerreceptiveelds,andviceversa.itisclearthatthehigher-level resolution.thereisevidence[35]thatwithinthehumanvisualsystemtherearehierarchiesof leadsustobelievethattheremustexistsomeprocessesthatcombineinformationfromlocal processesofsegmentationandrecognitionneedinformationfromlargepartsoftheimage.this employingmultiplelevelsofimageresolutioncouldaccountforafairamountofnoise;even neuronsofincreasinglylargereceptiveelds,withneuronsofsmallerreceptiveeldsproviding beothernoiseduetotheprocessinginthevisualsystem.forexample,ascalespaceframework Forexample,tointerpretanyoftheillusionarypatternsinFigure23requiresthatthewhole neighborhoodsintorepresentationsofglobalinformation,andtheseprocessescarryuncertainty. verysmallamountsofnoiseatlowimageresolutiontranslatetolargeamountsofnoiseathigh patternbeunderstoodasjudgmentsofthesizesofdierentpartsofthepatternaremade. Thereisstillmuchresearchtobedonealongtheselines.Visualrecognition,ofcourse,isnot understoodandscalespaceanalysishasbeenlookedatonlywithrespecttothetransformation ofimagefeatures,notwithrespecttoitsutilitarianfunctioninfacilitatingvisualcomputations. Opticalillusionsmighthelpinprovidinginsightsintotheseprocesses. fromphysicsthatthereisuncertaintyinanymeasurement,butvisionresearchershavenot itcausesbias.thusthelocationofalineintheimagemaynotproperlyreectitslocation realizedthattheuncertaintyinthevisualdatahasmajorconsequencesforvisualprocesses; Thendingsofthispaperarealsoofimportanceforpracticalapplications.Itiswellknown Figure23:(a)Delboeufillusion.(b)AversionofthePonzoillusion. (b) inthe3dworldandthelocationoftheintersectionoftwolinesdoesnotcorrespondtoa physicalintersectioninspace.asaconsequence,thereisgreatuncertaintyintheestimation AppendixAExpectedValueoftheLeastSquaresSolution ofcorrespondingfeatures;thisshouldbecleartoallpractitionersofvisionwhoconsiderpoint correspondencesastheirstartingpointforfurthercomputation. Inthissectionasecond-orderTaylorexpansionoftheexpectedvaluesoftheleastsquares solutionforboththeintersectionpoint(section3)andtheow(section4)isgiven. Asthenoiseisassumedtobeindependentandzero-mean,therst-ordertermsaswellasthe IsisthematrixconsistingofthespatialderivativesIxi;Iyi.~CinSection3isthevectorof temporalderivativesitiand~cinsection4isthevectorwhoseelementsareixix0i+iyiy0i. Let~ybethevectortobeestimated,thatis,eithertheintersectionpoint~xortheow~u. second-ordertermsinthenoiseofthetemporalderivatives(orthepositionalparameters)vanish. TheexpectedvalueE(~y)oftheleastsquaressolutionisgivenby E(~y)=E(ItsIs)?1(Its~C) 30

32 Thismeansthatitisonlythenoiseinthespatialderivativeswhichcausesbiasinthemean. TheexpansionatpointsN=0(i.e.,Ixi=Iyi=Iti=0orIxi=Iyi=x0i=y0i=0) canbewrittenase(~y)=~y0+xi Fornotationalsimplicity,wedene 2! Tocomputethepartialderivatives,theexplicittermsofthematrixMare andthetermsof~bare M=264PiI0xi+IxiI0yi+Iyi PiI0xi+Ixi2 ~b=24pii0xi+ixii0ti+iti PiI0yi+IyiI0ti+Iti35 PiI0xi+IxiI0yi+Iyi PiI0yi+Iyi2 375 and~b=264pii0xi+ixi2(x0oi+xoi)+i0xi+ixii0yi+iyi?y0oi+yoi wendtherst-orderandsecond-orderderivativestobe @2~y @xq?1?2m?1"2ixiiyi?m?1"20 Iyi0#M?1"2IxiIyi 00#M?1~b 31

33 SinceweassumeE(I2xi)=E(I2yi),theexpansioncanthusbesimpliedto @I2yiN=02s 2 +Pi(M0?1 "2Ixi0Iyi0 Iyi00#M0?1"2Ixi0Iyi0 Iyi00# +"0Ixi0 Ixi02Iyi0#M0?1"0Ixi0 Ixi02Iyi0#!~y0?M0?1 "2Ixi0Iyi0 +"0Ixi0 wherewehaveunderlinedthetermsthatdonotdependonn(wherenisthenumberof measurementsbeingcombinedinaregion).thesetermswillgiveaconsistent,statistically constantresponse.therestofthetermsdiminishproportionallyto1n.informalexperiments showthatthesetermsbecomenegligibleforn>5,anumberclearlysmallerthanthenumber 0# 2y0i# andthemaintermsintheexpectedvalueoftheintersectionpointe(~x)are E(~x)=~x0?nM0?1~x02s+M0?1"x0i y0i#2s or E(~x)=~x0+nM0?1(~x0?~x0)2s; Iti#; andthesecond-orderderivativesvanish.theexpectedvalueoftheowe(~u)simpliesto E(~u)=~u0?nM0?1~u02s: References [1] [2]F.Brentano.UbereinoptischesParadoxen.Z.Psychologie,3:349{358,

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