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CognitiveSciencexxx(2004)xxx–xxxCharacterizingperceptuallearningwithexternalnoiseJasonM.
Golda,,AllisonB.
Sekulerb,PartrickJ.
BennettbaDepartmentofPsychology,IndianaUniversity,Bloomington,IN47405,USAbDepartmentofPsychology,McMasterUniversity,Hamilton,Ont.
,CanadaReceived9August2002;receivedinrevisedform26September2003;accepted1October2003AbstractPerformanceinperceptualtasksoftenimproveswithpractice.
Thiseffectisknownas'perceptuallearning,'andithasbeenthesourceofagreatdealofinterestanddebateoverthecourseofthelastcentury.
Here,weconsidertheeffectsofperceptuallearningwithinthecontextofsignaldetectiontheory.
Accordingtosignaldetectiontheory,theimprovementsthattakeplacewithperceptuallearningcanbeduetoincreasesininternalsignalstrengthordecreasesininternalnoise.
Weusedacombinationofpsychophysicaltechniques(externalnoisemaskinganddouble-passresponseconsistency)thatinvolvecorruptingstimuliwithexternallyaddednoisetodiscriminatebetweentheeffectsofchangesinsignalandnoiseasobserverslearnedtoidentifysetsofunfamiliarvisualpatterns.
Althoughpracticereducedthresholdsbyasmuchasafactorof14,internalnoiseremainedvirtuallyxedthroughouttraining,indicatinglearningservedtopredominantlyincreasethestrengthoftheinternalsignal.
Wefurtherexaminedthespecicnatureofthechangesthattookplaceinsignalstrengthbycorrelatingtheexternallyaddednoisewithobserver'sdecisionsacrosstrials(responseclassication).
Thistechniqueallowedustovisualizesomeofthechangesthattookplaceinthelineartemplatesusedbytheobserversaslearningoccurred,aswellastestthepredictionsofalineartemplate-matchingmodel.
Takentogether,theresultsofourexperimentsofferimportantnewtheoreticalconstraintsonmodelsofperceptuallearning.
2004CognitiveScienceSociety,Inc.
Allrightsreserved.
Keywords:Perceptuallearning;Internalnoise;Internalsignal;Idealobserver;Classicationimage1.
IntroductionTheobservationthatperformanceinperceptualtaskscanimprovewithpracticehasbeendocumentedforoveracentury(Dresslar,1894;Gibson,1969;Gilbert,1994).
Thisimprove-mentinperceptualdiscriminationswithtrainingisreferredtoasperceptuallearning,anditCorrespondingauthor.
Tel.
:+1-812-855-4635;fax:+1-812-855-4691.
E-mailaddress:jgold@indiana.
edu(J.
M.
Gold).
0364-0213/$–seefrontmatter2004CognitiveScienceSociety,Inc.
Allrightsreserved.
doi:10.
1016/j.
cogsci.
2003.
10.
0052J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxhasbeenfoundtooccurinawidevarietyofperceptualtasks,includingverysimplesensorydiscriminationssuchasvisualandtactileacuitytasks(Fahle,Edelman,&Poggio,1995;Fahle&Morgan,1996;Poggio,Fahle,&Edelman,1992;Sathian&Zangaladze,1998),orientationdiscrimination(Matthews,Liu,Geesaman,&Qian,1999;Schiltzetal.
,1999;Schoups,Vogels,&Orban,1995),motiondiscrimination(Ball&Sekuler,1987;Matthewsetal.
,1999),texturediscrimination(Fine&Jacobs,2000;Karni&Sagi,1991)andauditorypitchdiscrimination(Demany,1985;Recanzone,Schreiner,&Merzenich,1993).
Learningalsohasbeenfoundtooperateoverawiderangeoftimescales,fromwithinaslittleas100trials(Fahleetal.
,1995;Poggioetal.
,1992)toaslongasseveralweeks(Fiorentini&Berardi,1997;Karnietal.
,1998;Karni&Sagi,1993;Sathian&Zangaladze,1998;Schoupsetal.
,1995).
Typically,thelearningisrestrictedtotheexactspecicationsofthestimuliandtaskwheretraininghasoccurred(Ahissar&Hochstein,1997;Ball&Sekuler,1987;Crist,Kapadia,Westheimer,&Gilbert,1997;Fahle&Morgan,1996;Fiorentini&Berardi,1980),andobserversoftendonotrequirefeedbackinordertoexhibitlearning(Ball&Sekuler,1987;Fahleetal.
,1995;Herzog&Fahle,1997,1999).
Thecombinationofthesendings(learningforsimplestimuli,stimulusspecicity,andimplicitlearning)hasbeentakenasevidencethatperceptuallearningoccursatrelativelyearlystagesofsensoryprocessing(Gilbert,1994).
Consequently,muchoftherecentpsychophysicalandphysiologicalworkonthistopichasbeendirectedtowardlocalizingtheneuralsubstrateschangedbyperceptuallearningindifferenttasksandmodalities.
Evidencefromtheseexperimentssuggeststhatperceptuallearningmaymodifyneuralmechanismsatorbeforethelevelofprimarysensorycortex.
Forexample,severalstudieshavefoundonlypartialornointer-oculartransferoflearningforsimplevisualdiscriminationtasks(Ball&Sekuler,1987;Fahleetal.
,1995),suggestingsomeoftheeffectsoflearningforthesetasksoccurinmonocularmechanismsbeforethesiteofbinocularintegration.
Similarly,physiolog-icalstudieshavefoundthatpracticechangestheresponsepropertiesofneuronsinprimarycorticalareasforsimplediscriminationtasks,suchasvisualorientationdiscrimination(Schiltzetal.
,1999;Schoupsetal.
,2001)andauditoryfrequencydiscrimination(Recanzoneetal.
,1993).
Otherphysiologicalstudieshaveinvestigatedthetopographicchangesthattakeplaceinsensorycorticalmapswithpractice(Buonomano&Merzenich,1998;Recanzoneetal.
,1993).
Thesestudieshavefoundsensorycortextobehighlyplastic,withstrikingamountsofcorticalreorganizationandreallocationtakingplaceasaresultofpractice.
However,thereisalsoevi-dencethatsuggestshigher-ordermechanisms,suchasthosefoundintheprefrontalcortex,canchangewithperceptuallearning(Asaad,Rainer,&Miller,2000;Rainer&Miller,2000)(seeGoldstone,1998foraconcisereviewandFahle&Poggio,2002foramoredetailedtreatment).
1.
1.
SignalandnoiseButwhataspectsofperceptualmechanismschangewithlearningOnewaytoapproachthisproblemistoconsidertheeffectsoflearningwithinthecontextofsignaldetectiontheory(Green&Swets,1966).
Signaldetectiontheoryisageneralframeworkdesignedtocharacterizeandquantifyanobserver'sdecisionprocessesandsensitivityinatask.
Oneofthecentraltenetsofsignaldetectiontheoryistheassumptionthatinternalresponsesareprobabilistic,sothataparticularstimulushasonlysomeprobabilityofelicitingaparticularinternalresponse.
ThetheoryalsoassumesanobservermakesdecisionsbycomparingtheinternalresponsetoaJ.
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Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx3criterion.
Oneadvantageofthistheoryisitprovidesanestimateofsensitivity(d)thatisindependentoftheobserver'sresponsecriterion.
Anotheradvantageisthatitcanbeusedtoquantifyoptimal(ideal)decisionprocesses,whichcanbeusedtoestimatetheamountofinformationusedbynon-optimal(e.
g.
,human)observersinagiventask(Geisler,1989;Green&Swets,1966;Tanner,1961).
Tounderstandhowtheseconceptsrelatetoperceptuallearning,considerthetaskoflearningtodiscriminatebetweentwounfamiliarpatterns(e.
g.
,twofaces,etc.
).
Insuchatask,theobserverisbrieyshownoneoftwopatterns(chosenrandomly)andmustdecidewhichpatternappearedintheinterval.
Psychologically,eachtrialwillproduceaninternalresponsewithintheobserverwhichmustbeusedtomakeadecisionaboutwhichsignalwasshown.
Ifweignoreforthemomentanytrial-by-trialvariabilityintroducedbythestimulus(e.
g.
,photonnoise),anidealobserverthatisshownthesameexactstimulusatseveraldifferenttimeswillmakethesamedecisiononeverypresentation.
However,suchcompleteresponseconsistencywouldnotbeexpectedfromahumanobserver.
Unlikeanidealobserver,humanobservershaveinternalvariabilityor'noise'(Barlow,1956,1957;Green,1964).
Thisinternaltrial-by-trialvariabilityisthoughttooriginatefromavarietyofsources,rangingfromthestochasticpropertiesofsensoryneurons(Croner,Purpura,&Kaplan,1993;Tolhurst,Movshon,&Thompson,1981;Tolhurst,Movshon,&Dean,1983;Vogels,Spileers,&Orban,1989)torandomuctuationsinstrategyorresponsecriterion(Burgess,1990;Raghavan,1989).
Asaresult,thesamestimuluswillnotproducethesameinternalresponseoneverypresentation.
Instead,itwillproduceadistributionofresponsesacrossidenticalpresentations,andthevarianceofthisdistributionwillbedeterminedbythemagnitudeoftheinternalnoise.
Asecondwaythathumanandidealobserversdifferisintermsofthegoodnessor'efciency'ofthenon-stochastic(i.
e.
,non-random)aspectsofanycomputationsperformedbetweenstim-ulusencodingandmakingadecision.
Formanytasks(includingthosedescribedintheex-perimentreportedhere),anidealstrategyistousealinearlterortemplatethatismatchedtothespatialandtemporalcharacteristicsofthestimulus.
Anydeviationfromthisstrategyreducesperformancerelativetotheideal.
Addingnoisetothetemplateisonewaytodegradeperformance,butanotherwayistoalterthetemplateinanon-randomfashion.
Forexample,anobserverthatusesjustthebottomhalfofthestimuluswillbesub-optimal(assumingthereisinformationpresentinthetophalfofthestimulus).
Thiswouldbeadeterministicinefciencyratherthanastochasticinefciency.
Anidealobserverusesadeterministiccomputationthatisguaranteedtomakeoptimaluseofalloftheinformationavailableinagiventask.
Unlikeanidealobserver,humanobserversperformsub-optimaldeterministiccomputations.
Theseinefcienciescanarisefrommanysources,rangingfromsub-optimalencodingbysensoryorgans(Banks,Geisler,&Bennett,1987;Banks,Sekuler,&Anderson,1991;Geisler,1989)tothecomparisonofthesensoryrepresentationtoasub-optimalreceptiveeldortemplate(Legge,Kersten,&Burgess,1987).
Intermsofsignaldetectiontheory,thedistancebetweentheunderlyinginternalresponsedistributions(ratherthantheirvariance)isdeterminedbytherelativeefciencyofthedeterministicaspectsoftheobserver'scomputations.
Ultimately,anobserver'ssensitivity(d)isdeterminedbytheratioofsignal(distancebetweenthedistribu-tions)tonoise(standarddeviationofthedistributions)withinthesystem.
Ifwenowconsidertheproblemofperceptuallearningwithinthecontextofsignaldetectiontheory,weseethattheeffectsofperceptuallearning(e.
g.
,improvedperformanceinatask)can4J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxresultfromeitheradecreaseininternalnoiseoranincreaseintheefciencyofthedeterministicaspectsofobservercomputations(orsomecombinationofthetwo).
1.
2.
MeasuringthestrengthofsignalandnoiseItisnotpossibletodiscriminatebetweenchangesininternalnoiseanddeterministicef-ciencybysimplymeasuringsensitivityatdifferentpointsduringlearning.
However,psy-chophysicaltechniqueshavebeendevelopedinrecentyearsthat,whenusedincombinationwithasimplepatterndiscriminationmodel,allowtheeffectsofthesechangestobedisam-biguated.
Thetechniquesarecalledexternalnoisemaskingandresponseconsistency.
1.
2.
1.
ExternalnoisemaskingAstandardtechniqueusedbyelectricalengineerstoestimatetheintrinsicnoiseinanelec-tronicdevice(e.
g.
,anamplier)istorefertheintrinsicnoisetoanexternallyaddedsourceofnoise(Mumford&Schelbe,1968).
Pelli(1981)wasamongthersttoapplyavariantofthistechniquetohumaninformationprocessing.
TounderstandfullyPelli'sapproach,itisusefultorstconsiderhisabstractionoftheinternaltransformationsperformedbyanobserverinapat-terndiscriminationtask.
Pelli's'black-box'modelofanobserverisillustratedschematicallyinFig.
1.
Inthismodel,theobserverreceivesaphysicalstimulus(inthiscase,asignalcorruptedbyanexternallyaddednoise).
Thestimulusisconvertedintoaninternalrepresentation,whereaninternalnoiseofxedvarianceisintroducedandacalculationisperformedontherepresen-tation.
Adecisionisthenmadebasedontheresultinginternalresponse.
Noticethatthemodelassumesthattheinternalnoiseisaddedtotherepresentationandthatboththevarianceoftheinternalnoiseandthecalculationareinvariantwithrespecttothemagnitudeofthestimulus.
Whenthestimuliareachromaticvisualimagesvaryinginluminanceacrossspaceand/ortimeFig.
1.
Ablack-boxmodelofahumanobserverinaperceptualdiscriminationtask(adaptedfromPelli,1981,1990).
Theobserveristreatedlikeablack-boxthatreceivesanoisyexternalstimulus(E+Ne),introducesaxedamountofvariabilitytothestimulus(Ni),performsacalculationthatisreducedtoaninternalresponse,andmakesadecisionbasedonthemagnitudeoftheinternalresponse.
J.
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/CognitiveSciencexxx(2004)xxx–xxx5(astheyareintheexperimentsreportedhere),theinternalnoiseandcalculationarereferredtoascontrast–invariantinternalnoiseandacontrast–invariantcalculation,respectively.
Giventhisframework,theobserver'scontrastthresholdwillbelinearlyrelatedtothecontrastoftheexternalnoise(Leggeetal.
,1987)accordingtotheequationE=k(Ne+Ni)(1)whereEistheenergy(ameasureofsignalcontrast)ofthesignalatthreshold,Neistheexternalnoisepowerspectraldensity(ameasureofnoisecontrast),andkandNiarefreeparameters(seeSection2.
1belowforformaldenitionsofenergyandnoisepowerspectraldensity).
TheparameterNiisoftenreferredtoastheobserver'sequivalentinputnoisebecauseitisequaltotheamountofexternalnoisethatmustbeaddedtothedisplaytodoubletheobserver'snoise-freethreshold.
Theparameterkisameasureofhowrapidlytheobserver'sthresholdincreaseswithincreasingexternalnoise,andisinverselyproportionaltothegoodnessor'efciency'oftheobserver'scalculation.
Anobserver'scalculationefciencyiscomputedbycomparingkforahumanobservertothatanidealobserver,andcanbeinterpretedastheproportionoftheavailableinformationusedbythehumanobserverinthetask.
Giventhismodeloftheobserver,itispossibletoestimatethemagnitudeofanobserver'scontrast–invariantinternalnoiseandtheefciencyoftheobserver'scalculationsbymea-suringthresholdsinvariousamountsofexternallyaddednoise.
Thedifferentialeffectsofcontrast–invariantinternalnoiseandcalculationefciencyonthresholdsinnoiseareillustratedinFig.
2.
ThesolidlineinFig.
2depictsahypotheticalnoisemaskingfunctionforahumanob-server,wherelogthresholdenergy(E)isplottedasafunctionoflogexternalnoisepowerspec-traldensity(Ne).
Noticethatthenoisemaskingfunction(i.
e.
,Eq.
(1))iscurvedwhenplottedinlog–logcoordinates.
Thesolidarrowcorrespondstotheestimateofcontrast–invariantinternalnoise(Ni).
ThedottedlineinFig.
2showstheeffectsofreducingNibyaconstantc.
ChangingNiinthisfashionreducesthresholdsatonlylowexternalnoiselevels,shiftingthekneeofthenoise-maskingfunctiontoalowervalue(thedottedarrowinFig.
2).
ThedashedlineinFig.
2showstheeffectsofreducingtheindexofcalculationefciencykbytheconstantc.
Changingkinthisfashionwillhavethesameeffectacrossexternalnoiselevels,producingauniformshiftintheoverall'height'ofthenoisemaskingfunctionwhenplottedinlog–logcoordinates.
Equivalentinputnoiseandcalculationefciencyhavebeenmeasuredforawidevarietyoftasks,includinggratingdetection(Pelli,1981;Bennett,Sekuler,&Ozin,1999),contrastdiscrimination(Leggeetal.
,1987),letterdiscrimination(Pelli&Farell,1999;Raghavan,1989;Tjan,Braje,Legge,&Kersten,1995),objectrecognition(Tjanetal.
,1995),dividedattention(Dosher&Lu,2000;Lu&Dosher,1998),andmotiondiscrimination(Lu,Liu,&Dosher,2000).
Withlittleexception,theformofthenoisemaskingfunctionshaveconformedwelltothemodeldescribedbyEq.
(1).
However,theremaybeothersourcesofnoiseinthesensorysystemsthatarenotinvariantwithrespecttostimulusmagnitude.
Theeffectsofsuchacontrast-dependentinternalnoiseinPelli'sblack-boxmodelcanbeseenbyincludingasecondindependentnoisesourceinEq.
(1):E=k[Ne+Ni+m(Ne+Ni+E)P](2)wheretheproportionalityconstantmandtheexponentPdeterminethemagnitudeofthecontrast-dependentinternalnoise.
Thereisbothphysiological(Tolhurstetal.
,1983)andpsy-6J.
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/CognitiveSciencexxx(2004)xxx–xxxFig.
2.
Hypotheticalnoise-maskingfunctionsforahumanobserver.
Logofsignalenergythreshold(E)isplottedasafunctionofexternalnoisepowerspectraldensity(Ne).
ThenelydashedlinedepictsareductioninequivalentinputnoiseNibyaconstantfactorcrelativetothesolidline.
Thecoarselydashedlinedepictsanincreaseincalculationefciency(indexedbyk)byaconstantfactorcrelativetothesolidline.
chophysical(Burgess&Colborne,1988)evidencethatthemagnitudeofthecontrast-dependentinternalnoisevariesindirectproportiontothemagnitudeofthestimulus(i.
e.
,theexponentPinEq.
(2)isequaltounity).
TheeffectsofaproportionalnoiseinPelli'sblack-boxmodelcanbeseenbysettingPtounityinEq.
(2):E=k[Ne+Ni+m(Ne+Ni+E)]=k(1+m)1km(Ne+Ni)=k(Ne+Ni)(3)wherekisaconstantequaltok(1+m)/(1km).
AcomparisonofEqs.
(1)and(3)showsthatestimatesofcalculationefciencywillbeaffectedbyboththemagnitudeofanobserver'sproportionalinternalnoise(m)andtheefciencyofanobserver'sdeterministiccomputations(k),confoundingthesetwofactorsinthecontextofPelli'sblack-boxmodel.
Pelli(1990)isexplicitaboutthisaspectofthemodel,andassumesthatanyproportionalnoisestemsfromthestochasticpropertiesofthecontrast–invariantcalculation(i.
e.
,randomchangesinthecal-culationacrosstrials).
However,proportionalnoisemayarisefromsourcesotherthananoisycalculation(Lillywhite,1981).
Thus,weconsidercontrast-dependentnoiseasaseparatesourceofinternalnoiseinthemodel.
1.
2.
2.
ResponseconsistencyGreen(1964)andothers(Burgess&Colborne,1988;Spiegel&Green,1981)devisedamethodofmeasuringinternalnoisethatisindependentofthedeterministicoperationsoftheobserver.
Thetechniqueiscalledresponseconsistency,andittakesadvantageofthefactthatJ.
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/CognitiveSciencexxx(2004)xxx–xxx7internalnoisewillcausetrial-by-trialvariabilityinanobserver'sinternalresponsestoidenticalstimuli.
Consideragainataskwhereobserversmustidentifyasignalpresentedinexternalnoise.
Ifthesignalandnoiseshownoneverytrialoftheexperimentwererecorded,andthentheexactsametrial-by-trialsequencewasshownasecondtime,thetaskwouldbephysicallyidenticalinbothpassesthroughtheexperiment.
Theresponsesofanoiselessobserverwouldalsobeidenticaloneachcorrespondingtrialinthesequence,regardlessofwhetheraresponsewascorrectorincorrect.
However,foranobserverwithinternalnoise,therewouldberesponseinconsistencybetweencorrespondingidenticaltrialsinthetwopasses,andthedegreeofinconsistencydependsontheratioofinternaltoexternalnoiseatthelevelofthedecisionvariable(Burgess&Colborne,1988).
Althoughbothcontrast–invariantandcontrast-dependentnoisewillproduceresponseinconsistency,thecontrast-dependentcomponentofanobserver'sinternalnoisecanbeestimatedbymeasuringresponseconsistencyunderconditionsofhighexternalnoise(wherethecontributionofthecontrast–invariantinternalnoisewillbenegligible;seeFig.
2).
Inthecontextofperceptuallearning,responseconsistencyoffersawayofmeasuringchangesincontrast-dependentinternalnoiseasafunctionoflearningindependentlyofchangesinthedeterministicaspectsofanobserver'scalculations.
1.
3.
MeasuringthecalculationOneofthelimitationsofthetechniquesdescribedaboveisthattheyonlyprovideagrossindexofthegoodnessofobservers'calculations,leavingtheparticularnatureoftheircompu-tationsunspecied.
Forexample,twoobserversthatexhibitthesamedegreeofimprovementincalculationefciencywithpracticemaybasetheirdecisionsonverydifferentaspectsofthesamestimulus.
TheresponseclassicationtechniquedevelopedbyAhumadaandhiscol-leagues(Ahumada&Lovell,1971;Beard&Ahumada,1998;Watson&Rosenholtz,1997)offersawayofaddressingthisproblem.
Consideronceagainanidenticationtaskwhereanobservermustidentifyanoisystimulusasoneoftwopossiblepatterns,S1orS2.
Onsometrials,anobserverwillincorrectlyclassifythestimulus.
Forexample,theobservermayre-spondthatthesignalwasS1when,infact,S2wasshown.
Ifthesignalwasembeddedinalargeamountofexternalnoise,therearetwopossiblereasonsforthismistake.
Onepossi-bilityisthatinternalcontrast-dependentnoisewashigh,causingtheobservertomisclassifythestimulus.
AsecondpossibilityisthattheexternalnoisewasdistributedinsuchawaytomakethestimuluslookmorelikeS1thanS2.
Aslongastheinternalcontrast-dependentnoiseisnotexcessivelyhigh,theexternalnoisewillleadtomisclassicationonmanytrials.
Thenoiseeldsshownoneachtrialcanberecordedandsortedintoa2*2stimulus–responsematrix.
Aftermanytrials,thesenoiseeldscanbeaveragedineachsignal–responsecate-goryandsummedacrosscategoriesinsuchafashionastoproduceaclassicationimage.
Theclassicationimageisamapthatshowsthelocationsinthestimulusthathaveaffectedanobserver'sresponsesduringtheexperiment.
Morespecically,itshowsthecorrelationbetweenthenoisemagnitudeateachlocationinthestimulusandanobserver'sresponsesthroughouttheexperiment.
Ithasalsobeenshown(Ahumada,2002;Murray,Bennett,&Sekuler,2002)thattheclassicationimagewillbeproportionaltothelineartemplate(i.
e.
,linearcalcula-tion)usedbyanobserverinmanytasks(includingtheidenticationtaskdescribedabove).
Itisnotuncommontomodelthedeterministicoperationsperformedbyanobserverinatask8J.
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/CognitiveSciencexxx(2004)xxx–xxxasalinearlterortemplatethatiscomparedtothestimulus(Abbey,Eckstein,&Bochud,1999;Ahumada,2002;Bochud,Abbey,&Eckstein,2000;Eckstein,Ahumada,&Watson,1997;Leggeetal.
,1987;Levi,Klein,&Carney,2000;Lu&Dosher,1998,1999,2001a;Murrayetal.
,2002;Watson,1998).
Suchtemplatematchingmodelshavebeensuccessfulincharacterizinghumanperformanceinawidevarietyofvisualpatterndiscriminationtasks.
Theresponseclassicationtechniquehasbeenappliedsuccessfullytoawiderangeofau-ditoryandvisualdiscriminationtasks(e.
g.
,Abbeyetal.
,1999;Abbey&Eckstein,2002;Ahumada,1996;Ahumada&Lovell,1971;Eckstein,Shimozaki,&Abbey,2002;Gold,Murray,Bennett,&Sekuler,2000;Levi&Klein,2002;Neri,Parker,&Blakemore,1999;Neri&Heeger,2002;Sekuler,Gold,Gaspar,&Bennett,2001;Watson&Rosenholtz,1997).
Inthecontextofperceptuallearning,theresponseclassicationtechniqueoffersawayofspecifyingthenatureofthechangesthatoccurinanobserver'scalculationoverthecourseoftraining.
1.
4.
OverviewThegoaloftheworkpresentedherewastoapplythesignaldetectionmodelandexternalnoisemethodsdescribedabovetoilluminatetheeffectsofperceptuallearning.
Experiments1and2involvedusingexternalnoisemasking(Experiment1)andresponseconsistency(Ex-periment2)topartialouttherelativecontributionsofstochasticandnon-stochasticfactorstoperceptuallearningintwovisualpatterndiscriminationtasks:humanfaceandabstracttextureidentication.
Humanfaceswereusedasstimulibecausetheyreectacomplex,real-worldperceptuallearningproblemthatthevisualsystemmustsolvethroughoutthelifespan.
Thera-tionaleforusingabstracttexturepatternsinadditiontofacesstemsfromrecentworkregardingthemechanismsthatmediatehumanfaceperception.
Someevidencesuggeststhattherearecorticalmechanismsspecicallydevotedtofaceperception(Kanwisher,McDermott,&Chun,1997;Perrett,Hietanen,Oram,&Benson,1992),whereasotherevidencesuggeststheapparentspecialstatusoffacesisrootedinexpertise(Gauthier,Tarr,Anderson,Skudlarski,&Gore,1999;Gauthier,Skudlarski,Gore,&Anderson,2000;Gauthier&Tarr,1997).
Iflearningtorecognizenovelfacesismediatedbyface-specicmechanisms,theeffectsoflearningfoundwithfacesmaynotapplytootherkindsofpatterns.
Thetexturepatternswerechosentoaddressthisissuebecausetheyaredissimilartofaces.
TheresultsofExperiment1showedthatcalculationefciencyincreasedbyasmuchasafactorof4acrosslearningsessions,withnocorrespondingdecreasesinequivalentinputnoise.
TheresultsofExperiment2showedlittleornosignicantchangesinresponseconsistencyinhighlevelsofexternalnoise.
Takentogether,theseexperimentsindicatedthatlearningofbothfacesandtextureswasmediatedbypurelydeterministicchangesintheefciencyofobservers'calculations.
Experiment3usedtheresponseclassicationtechniquetoexplorethespecickindsofchangesthattookplaceinobservers'calculationswithlearning,andshowedthatobservers'strategiesbecamemoresimilartotheidealstrategyaslearningtookplace.
2.
Experiment1ThemaingoalofExperiment1wastodiscriminatebetweenincreasedcalculationefciencyanddecreasedcontrast–invariantinternalnoiseaspossiblesourcesoftheimprovementsinJ.
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/CognitiveSciencexxx(2004)xxx–xxx9performancethattakeplacewithlearninginapatterndiscriminationtask.
Wemeasuredsignalidenticationenergythresholdsinarangeofexternalnoisepowerspectraldensities,acrossaseriesoflearningsessions.
Measuresofcontrast–invariantinternalnoiseandcalculationefciencywerethenderivedfromthesedata,whichallowedustotracethechangesinthesequantitiesaslearningtookplace.
2.
1.
Method2.
1.
1.
ApparatusStimuliweredisplayedusingaMacintoshG3computerona13in.
ApplehighresolutionRGBcolormonitor.
Themonitordisplayed640*480pixels,whichsubtendedavisualangleof12.
9*9.
6fromtheviewingdistanceof100cm,ataframerateof67Hz(non-interlaced).
LuminancecalibrationswereperformedwithaHagnerOptikonuniversalspotphotometer,andthecalibrationdatawereusedtobuilda1779-elementlook-uptable(Tyler,Chan,Liu,McBride,&Kontsevich,1992).
TheexperimentwasconductedintheMATLABprogrammingenvironment(version5.
1),usingin-housesoftwareandtheextensionsprovidedbythePsy-chophysicsToolbox(Brainard,1997)andtheVideoToolbox(Pelli,1997).
Luminanceonthedisplayrangedbetween0.
3and80.
2cd/m2.
Averageluminancewas28.
8cd/m2.
2.
1.
2.
SignalsThereweretwoclassesofsignalsusedintheexperiment.
Therstclassofsignalswasdigitalimages(256*256pixelsinsize)ofhumanfacesthatwereconstructedusingAdobePhotoshop(version3.
0)andMATLAB.
Thesecondclassofsignalswasrandomlygeneratedband-passlteredGaussiannoiseelds(also256*256pixelsinsize)generatedusingMATLAB(seebelowfordetailsaboutthefaceandtextureimages).
Thevaluesineachimagerepresentedthecontrast(ci)atpixellocationi,denedbyEq.
(4):ci=liLL(4)whereLisaverageluminanceandliistheluminanceoftheithpixel.
Eachimagelewasnormalizedsothatroot-mean-square(RMS)contrastoftheimageequaled1.
RMScontrastisdenedascRMS=1nni=1c2i(5)wherenisthenumberofpixelsintheimage.
2.
1.
2.
1.
Faces.
ThefacestimuliconsistedoftheimagesofvemaleandvefemaleCaucasianfacesthatweusedinaprevioussetofexperiments(seeGold,Bennett,&Sekuler,1999aforamoredetaileddiscussionofhowthefaceimageswereconstructed).
Theimageswerecroppedtoshowonlytheinnerportionofeachface,eliminatingnon-facialcuessuchashairandears.
Theshapeofthevisibleregionofeachfacewaselliptical,andthesizeandheight:widthratiowereconstantacrossallstimuli(198:140pixels;4.
0*2.
9).
Thefaceswerecenteredwithina256*256pixel(5.
25*5.
25)backgroundofaverageluminance(seeleftsideofFig.
3).
10J.
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/CognitiveSciencexxx(2004)xxx–xxxFig.
3.
Theface(left)andtexture(right)stimuliusedinExperiments1and2.
2.
1.
2.
2.
Textures.
Thetexturepatternswereproducedbyrandomlygenerating10Gaussianwhitenoiseeldsthatwerethesamesizeasthefacestimuli(256*256pixels).
Thenoiseeldswereconvertedintothespatialfrequencydomain,andlteredwitha2–4cycleperimage(c/image)circularlysymmetricrectangularlter.
Theamplitudeofallfrequenciesoutsideofthe2–4c/imagepass-bandweresettozero,andtheamplitudewithinthepass-bandremainedunchanged.
Theimageswerethenconvertedbackintothespatialdomain.
Thelteringpro-ducedasetof10uniqueblob-liketextures,shownontherightsideofFig.
3.
J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx112.
1.
3.
DisplaynoiseTheexternalnoiseaddedtothesignalateachpixelwascreatedbydrawingarandomsamplefromanindependentGaussiandistributionofcontrastvalues,withameanof0andavarianceasrequiredbythecondition.
Thenoiseoneachtrialwasstatic(i.
e.
,didnotvarytemporallyduringthecourseofatrial),white(i.
e.
,pixelswereindependentofeachother)andwasmatchedtothesizeofthesignal(256*256pixels).
Valuesbeyond±2standarddeviationsfromthemeanwerediscardedandreplacedbyrandomsamplesfromtheremainingcontrastvalues.
Ineachtask,thresholdsweremeasuredinvedifferentlevelsofexternalnoisepowerspectraldensity.
Forthefacetask,theexternalnoisepowerspectraldensitylevelswere:0.
04,0.
20,1.
02,5.
11and25.
55*106degree2(seeSection2.
1.
7andEq.
(7)foradenitionofnoisepowerspectraldensity).
Pilotstudiessuggestedthatequivalentnoisewashigherforthetextureidenticationtask,sothelowestexternalnoiselevelwasremovedandreplacedbyahighernoiselevelof51.
10*106degree2.
Auniquenoiseeldwasgeneratedoneverytrial.
2.
1.
4.
ViewingconditionsViewingwasbinocularthroughnaturalpupils,andaheadrest/chinreststabilizedtheobserver'sheadthroughoutthesession.
Thecomputermonitorsuppliedtheonlysourceofilluminationduringtheexperiment.
2.
1.
5.
HumanobserversAllparticipantshadnormalorcorrected-to-normalvisualacuity(self-reported).
Twoob-serversparticipatedineachtask,withoneobserver(AMC)participatinginbothtasks.
Oneobserverwasanauthor(JMG)andtheremainingtwoobserverswerenaivetothepurposesoftheexperiment.
2.
1.
6.
TasksandprocedurePerformancewasmeasuredusingasingle-interval,1-of-10identicationtask.
Thesignalenergyandpowerspectraldensityofthenoisewerevariedaccordingtotheproceduresdetailedinthethresholdestimationsectionbelow.
Observerswerefamiliarizedbrieywithhighcontrastversionsofthestimulibeforethebeginningoftheexperiment.
Atthestartofeachtrial,asmallxationpointappearedatthecenterofthescreen(3*3pixelsinsize),andabrieftoneindicatedatrialcouldcommencewithamouseclick.
Afterthemousewasclicked,thestimulus(signal+noisecombination)appearedfor34frames(approximately500ms).
Next,thedisplaywassettoaverageluminance,andafterabrief100mspause,100*100pixelhighcontrastthumbnailversionsofthe10possiblesignalsappearedonthescreensurroundingtheregionwherethestimulushadbeendisplayed.
Observersidentiedthestimulusbyclickingthemouseontheappropriateimage.
Onceanimagewaschosen,thedisplayswereclearedandsettoaverageluminance.
Auditoryfeedbackwasgivenaftereachtrialtoindicatetheaccuracyoftheresponse.
2.
1.
7.
ThresholdestimationIdenticationthresholdsateachofthevelevelsofexternalnoiseweremeasuredbyvaryingsignalenergyacrosstrials.
Signalenergywasmanipulatedaccordingtothemethodofconstantstimuli.
Pilotstudiesidentiedseveralsignalenergylevelsthatspannedthethresholdrange12J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxforatypicalunpracticedobserverineachlevelofexternalnoisepowerspectraldensity.
SignalenergyEisdenedasE=(cRMS)2na(6)wherenisthenumberofimagepixelsandaistheareaofasinglepixel,indegreesofvisualanglesquared.
NoisepowerspectraldensityNisdenedasN=σ2a(7)whereσisthestandarddeviationofthenoise,expressedinvaluesofcontrast.
Signalenergylevelswereadjustedaftereachsessionforeachobserverasrequiredbytheirrateoflearning.
Beforeeachtrial,asignalwaschosenrandomlytoappearwithinthestimulusinterval.
Therewere31trialsperstimulusenergylevelwithineachsession,yieldingatotalof775trials(31trials*5signallevels*5noiselevels).
Thelevelofexternalnoise,signalenergyandtheidentityofthesignalwerechosenrandomlyoneachtrial.
Eachsessionwascompletedwithoutbreaksandlastedabout1h.
Onlyonesessionwascompletedeachday.
Observersinthefaceidenticationtaskcompletedsixsessionswithin10days.
Observersinthetextureidenticationtaskcompletedfoursessionswithin7days.
Psychometricfunctionswereestimatedbymaximum-likelihoodtstoaWeibullfunction.
Thresholdwasdenedasthesignalenergyyielding50%correctresponses.
Condenceinter-valsforthethresholdestimateswerecalculatedbybootstrapsimulations(Efron&Tibshirani,1993).
Eachsimulationconsistedofatleast500simulateddatasets.
2.
1.
8.
IdealobserverTheidealdecisionruleforthe1-of-Mpatternidenticationtaskdescribedaboveisargmaxj=1,.
.
.
,mni=1TijRi(8)wheremisthenumberofpossiblesignals,nisthenumberofpixelsineachsignal,Tijistheithpixelinthejthnoise-freenormalizedsignal,andRiistheithpixelinthenoisystimulus.
Thisruleamountstochoosingthesignalthatyieldsthehighestcross-correlationbetweenthestimulus(i.
e.
,signal+noisecombination)andeachoftheMpossiblenoise-freesignalmatrices(Green&Swets,1966;Tjanetal.
,1995).
1IdealobserverthresholdswereobtainedinallconditionsthroughMonteCarlosimulations,inwhicheachnoise-freesignalmatrixwascomparedtothestimulusatarangeofsignalenergyvaluesforeachcorrespondingnoiseleveltestedwithhumanobservers.
Idealthresholdswereestimatedfrompsychometricfunctionsthatwerettothedata(usingtheproceduredescribedabove)fromatleast10,000simulatedtrials.
2.
1.
9.
EquivalentinputnoiseandcalculationefciencyRecallthatanobserver'sequivalentinputnoiseandcalculationefciencyareestimatedbymeasuringsignalidenticationenergythresholdsacrossarangeofexternalnoisepowerspectraldensitylevels.
Eq.
(1)isttothethresholds,withthenegativex-intercept,Ni,astheestimateofcontrast–invariantinternalnoiseandtheslope,k,asanindexofefciency.
Aswiththeexternalnoise,Niisexpressedinunitsofpowerspectraldensity.
J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx13Itcanbeshown(Tjanetal.
,1995)thattheidealobserver'ssignalidenticationenergythresholdinthetasksdescribedaboveisalinearfunctionofnoisepowerspectraldensity,i.
e.
,Eideal=kidealNe(9)whereNeisthepowerspectraldensityoftheexternalnoise.
Theslopeparameterkidealvarieswiththesetofsignalsandisdirectlyrelatedtotheintrinsicdifcultyofthetask(i.
e.
,thesimilarityofthesignalmatrices).
Thehumanobserver'scalculationefciencyηisdenedasη=kidealk(10)Lineartstobothidealandhumannoisemaskingfunctionswereestimatedbymaximum-likelihoodminimization,andbootstrapsimulations(Efron&Tibshirani,1993)providedcon-denceintervalsforthettedparameters(minimum500simulatedexperiments).
2.
2.
ResultsanddiscussionFig.
4showsthe50%correctthresholdsignalenergylevelsasafunctionofexternalnoisepowerspectraldensityforeachobserverintheface(toppanels)andtexture(bottompanels)identicationtasks.
Eachsymbolcorrespondstoasinglethreshold.
Thelledsymbolscor-respondtothersthalfoftheexperiment,theopensymbolsthelasthalfoftheexperiment.
Ther2valuesforeachobserver'smaximumlikelihoodttoEq.
(1)ineachsession(smoothcurves)aresummarizedinTable1,alongwithFstatisticsthatindicatethesignicanceofthets.
Theaverager2valuesacrosssessionsforallobserverswereabove.
95,indicatingthattheeffectofnoiseonthresholdsinthistaskwaswellcharacterizedbyEq.
(1).
Table1StatisticsfortheEq.
(1)tstothethresholddatafromExperiment1SessionAMCCGBr2Fpr2FpFaceidentication1.
968191.
073.
92e05.
99882,500.
411.
06e082.
9968951.
751.
18e07.
99903,043.
446.
46e093.
9969976.
751.
10e07.
9962789.
401.
86e074.
9955664.
082.
88e07.
99811,649.
092.
98e085.
999923,762.
603.
80e11.
99731,110.
798.
00e086.
9957703.
092.
50e07.
9926407.
199.
74e07Mean.
99224,524.
896.
66e06.
99701,583.
392.
15e07Textureidentication1.
9874234.
583.
82e06.
972671.
051.
90e042.
99944,913.
761.
95e09.
9969954.
281.
17e073.
99761,248.
095.
98e08.
9716106.
572.
67e054.
927338.
283.
14e4.
897626.
307.
51e04Mean.
97791,608.
687.
94e05.
9599289.
552.
42e0414J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxFig.
4.
Noisemaskingfunctionsfortheobserversintheface(toppanels)andtexture(bottompanels)identicationtasksfromExperiment1.
Eachpanelplotsanindividualobserver'ssignalenergythresholdsasafunctionofexternalnoisepowerspectraldensity.
Thelledsymbolscorrespondtothesessionsinthersthalfoftheexperiment,theopensymbolsthelasthalfoftheexperiment(seelegend).
Solidlinescorrespondtomaximum-likelihoodtstoEq.
(1).
Errorbarsoneachsymbolcorrespondto±1standarddeviation.
Often,theerrorbarsaresmallerthanthesymbols.
InspectionofFig.
4revealsacleartrendacrosssessionsforallobservers.
Namely,theheightofthefunctionsshiftsdownuniformlyacrossexternalnoiselevelswithpractice.
However,thekneeofthefunctionsremainsconstantacrosssessions.
Thiskindofshiftinthenoisemaskingfunctionisconsistentwiththeeffectsofincreasedcalculationefciencywithnochangeincontrast–invariantinternalnoise.
TheseeffectscanbeseenmoreclearlyinFig.
5,whichplotscalculationefciency(leftpanel)andequivalentinputnoise(rightpanel)foreachobserver,asafunctionofpractice.
Althoughoneobserver(AMC)didshowasmallbutstatisticallysignicantdecreaseintheestimateofcontrast–invariantinternalnoisefromthersttothelastsessioninthefaceidenticationtask(z=2.
89,p<.
01),therewerenostatisticallysignicantchangesacrosssessionsforanyoftheotherobserversineithertask.
Incontrast,allobserversshowedhighlysignicantincreasesincalculationefciencyacrossJ.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx15Fig.
5.
Calculationefciency(left)andequivalentinputnoise(right)asafunctionofexperimentalsessionfortheobserversintheface(circles)andtexture(triangles)identicationtasksinExperiment1.
Errorbarsoneachsymbolcorrespondto±1standarddeviation.
sessions(z=4.
3,p<1.
0e05fromthersttothelastsessionforallobservers),withefciencyincreasingbyaboutafactorof4inthefaceidenticationtask(AMC:4.
4;CGB:4.
3)andbyaboutafactorof2–3inthetextureidenticationtask(AMC:2.
3;JMG:2.
9).
Theabsolutelevelsofcalculationefciencyinthenalsessionsrangedbetween1and2%,valuesthataresimilartopreviousmeasuresofefciencyforfamiliarfaceidentication(Goldetal.
,1999a).
Thus,theresultsfromExperiment1indicatethatlearningservedtoincreasecalcula-tionefciencybuthadvirtuallynoeffectuponcontrast–invariantinternalnoise.
However,recallthatthequantityofcalculationefciencyincorporateslimitationsimposedbyboththeefciencyofthedeterministicaspectsofobservers'calculationsandthemagnitudeofanycontrast-dependentinternalnoise.
ThedatafromExperiment1donotallowustodis-criminatebetweenthesetwoconstraints.
TeasingthesefactorsapartwasthesubjectofExperiment2.
3.
Experiment2InExperiment2,weuseddouble-passresponseconsistencyinahighlevelofexternalnoisetoisolatetheeffectsofcontrast-dependentinternalnoiseonlearninginourfaceandtextureidenticationtasks.
Ifdecreasesincontrast-dependentinternalnoisecontributedtotheincreasesincalculationefciencyfoundinExperiment1,observersshouldbecomemoreconsistentaslearningtakesplace.
16J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx3.
1.
Methods3.
1.
1.
StimuliThestimuliusedwerethesamesetsof10facesand10texturesusedinthepreviousexperiments.
Ineachtask,signalenergyidenticationthresholdsweremeasuredinthehighestlevelsofexternalnoiseusedinthepreviousexperiments(faces:25.
55*106degree2;textures:51.
10*106degree2).
N/2uniquenoiseeldsweregeneratedforeachexperimentalsession,whereNisthenumberoftrialswithinagivensession.
Thesequenceofsignalidentities,signalenergylevels,andseedsusedtogeneratethenoiseeldsduringthersthalfofeachsessionweresavedbeforeeverytrialtoallowfortheexactreproductionofthesamesequenceofstimuliduringthesecondhalfofthesession.
3.
1.
2.
ProcedureSignalenergywasvariedacrosstrialsduringthersthalfofeachsessionaccordingtotwointerleavedadaptivestaircases.
Staircaseswereusedtoremovetheneedtomanuallyadjustthecontrastlevelsofthesignalsinbetweensessions(aswasthecasewiththemethodofconstantstimuliusedinExperiment1).
Twointerleavedstaircaseswereusedtoobtainmeasurementsthatspannedtherangeofthepsychometricfunction.
Signalenergylevelswerechosenthatcoarselysampledarangeofseverallogunitsaroundaroughthresholdestimate(instepsizesofapproximately0.
2logunits).
Thestaircaseshiftedthroughtheselevelsaccordingtotheaccuracyoftheobserver'sresponses.
Oneofthestaircasesuseda1-up-1-downruleandtheotherstaircaseuseda1-up-2-downrule.
Thestaircasesmaintainedthisprocessthroughoutthersthalfofeachexperimentalsession,consistingof200trialsperstaircase(400trialstotal).
Thesecondhalfofthesessionconsistedofanexactreplicationofthersthalfofthesession(i.
e.
,anexactpixel-by-pixelreproductionofthesequenceoftrialsshownduringthersthalfofthesessionwasshownagainduringthesecondhalfofthesession),yieldingatotalof800trialspersession.
Notethatthisrepetitionmeantthatthesignalenergywasalteredacrosstrialsinamannerthatwascontingentupontheobserver'sresponseduringtherstbutnotthesecondhalfofeachsession.
Despitethisdifference,noneoftheobserversreportedbeingawareofthisaspectoftheexperiment.
Thresholdswereestimatedbyttingpsychometricfunctionstothecombineddatafromthetwostaircases.
Eachsessionwascompletedwithoutbreaksandlastedabout1h.
Onlyonesessionwascompletedeachday.
Eachobservercompletedsixsessionswithin10days.
Theobserverswereunawarethattherstandsecondhalvesofeachsessionwereidentical.
3.
1.
3.
ObserversTwoobserversparticipatedinthefaceidenticationtaskandtwointhetextureidenticationtask.
Allobservershadnormalorcorrected-to-normalvisualacuityandwerenaivetothepurposesoftheexperiment.
NoneoftheobservershadparticipatedinExperiment1.
3.
2.
ResultsanddiscussionSignalenergythresholdsforallobserversintheface(leftpanel)andtexture(rightpanel)identicationtasksareplottedasafunctionofsessioninFig.
6.
Thesedatashowthattherewasaclearimprovementacrosssessionsforallobservers.
Infact,threeofthefourobserversexhibitedJ.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx17Fig.
6.
Face(leftpanel)andtexture(rightpanel)identicationthresholdsplottedasafunctionofsessionforthetwoobserversinExperiment2.
TheexternalnoisepowerspectraldensitywassettothehighestlevelsusedinthecorrespondingtasksfromExperiment1.
Errorbarsoneachsymbolcorrespondto±1standarddeviation.
amarkedlyhigheramountofimprovementacrosssessionsthanwasfoundinExperiment1:observerJSW'scontrastenergythresholdsdecreasedbyafactorof8.
2inthefaceidenticationtask,andobserversLAP'sandSKH'scontrastenergythresholdsdecreasedbyfactorsof14and12.
5,respectively,inthetextureidenticationtask.
AkeydifferencebetweenExperiments1and2wasthatthepresentationofthedifferentnoiselevelswascompletelyrandomizedacrosstrialsinExperiment1,yetwasxedthroughoutExperiment2;similarly,signalenergywascompletelyrandomizedacrosstrialsinExperiment1,yetwascorrelatedacrosstrialsinExperiment2byvirtueoftheuseofadaptivestaircasesandtherepetitionoftrialsinthesecondhalfofthesession.
Thus,onepossibilityisthatgreaterstimuluscertaintytendedtopromotegreaterlearninginExperiment2.
RegardlessofthedifferenceinabsoluteperformancebetweenExperiments1and2,allobserversshowedlargeimprovementswithpractice.
Fig.
7showsthecorrespondingresultsoftheconsistencyanalysesfortheface(toppanels)andtexture(bottompanels)identicationtasks.
Eachpanelplotsthepercentageofcorrectresponsesateachstimuluslevelasafunctionofthepercentageofagreementbetweencor-respondingresponsesmadeinthetwopassesthroughthesessionforanindividualobserver.
Eachsymbolcorrespondstoasinglestimuluslevelwithinagivensession.
Inallpanels,theclosedsymbolscorrespondtotherstthreesessions,theopensymbolsthelastthreesessions.
Tounderstandhowtointerprettheseplots,considertheperformanceofanobserverwithnointernalnoise.
Theresponseofsuchanoiselessobserverwouldbeexactlythesameifastim-uluswererepeatedtwice.
Similarly,theresponsesofanoiselessobserverwouldbeperfectlycorrelatedifitmadetwoidenticalpassesthroughanexperiment.
Asaresult,anoise-freeobserver'sdatainourtaskwouldfallalongtherightmostsideoftheplot—performancewouldvarywithsignalenergy(causingpercentcorrecttovary),butthepercentageofagreement18J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxFig.
7.
Responseconsistencyplotsforallobserversintheface(toppanels)andtexture(bottompanels)identicationtasksfromExperiment2.
Eachpanelplotspercentcorrectasafunctionofpercentagreementforeachstimulusleveltested.
Thelledsymbolscorrespondtotherstthreesessions,theopensymbolsthelastthreesessions(seelegend).
Errorbarsoneachsymbolcorrespondto±1standarddeviation.
Thelargevariationinerrorbarmagnitudeisduetotheunequalnumberoftrialsateachdatapoint(aresultoftheuseofastaircaseprocedure).
Solidlinesineachplotcorrespondtotheperformanceofasimulatedobserverwithaninternal/externalnoiseratioapproximatelyequaltotheaverageoftheestimatedinternal/externalnoiseratiosacrosssessions.
betweencorrespondingresponsesinthetwopassesthroughtheexperimentwouldalwaysbe100%foreverystimuluslevelshown.
Nowconsidertheperformanceofanobserverlimitedbyinternalnoise(e.
g.
,ahumanobserver).
Theresponsesofanoisyobserverwillnotbeperfectlyconsistentforrepeatedtrials.
Instead,percentagreementwillvarywithpercentcorrectandtheratioofthestandarddeviationsoftheinternal(σi)andexternal(σe)noises(σi/σe)(Burgess&Colborne,1988;Green,1964).
Therelationshipbetweenpercentcorrect,percentagreementJ.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx19andσi/σeiswelldescribedbyastraightlinethatpassesthroughthetoprightcornerofaplotsuchasthoseshowninFig.
7.
Thislinefollowstheform:pc=mlog10pa100+100(11)wherepcispercentcorrectperformanceatagivenlevelofsignalenergy,paisthepercentagree-mentbetweenthetwopassesthroughtheexperiment,andmisafreeparametercorrespondingtotheslopeoftheline.
Theparametermwillvarysystematicallywithσi/σe.
Specically,asσi/σedecreases,paforanygivenlevelofpcwillincrease,causingthedatainplotslikethoseshowninFig.
7toshifttotheright.
Thus,wewouldexpecttoseearightwardshiftinthedataacrosssessionsinFig.
7ifcontrast-dependentinternalnoisewereresponsibleforsomeorallofthedecreaseinthresholdsseeninFig.
6.
However,theconsistencyanalysesshowthattherewerenosystematicchangesinpercentagreementacrosssessionsforanyoftheobservers.
Instead,foreachobserverallofthedataappeartofallalongasingleline.
ThisconclusionwasveriedbyttingthedatafromeachsessiontoEq.
(11)todeterminetheslopeparameterm.
MonteCarlosimulationswerethenusedtoconvertmintoσi/σeestimatesacrosssessionsforeachobserver.
Specically,asimulatedobserver2builtwithdifferentratiosofσi/σewasimplementedtodeterminetherelationshipbetweenσi/σeandm.
Inthetasksreportedhere,thisrelationshipiswelldescribedbyafunctionoftheformσiσe=α+γ1eβ1m+γ2eβ2m(12)whereα,γ1,γ2,β1andβ2arettedparameters.
Maximum-likelihoodminimizationwasusedtotEq.
(11)tothehumandata,andbootstrapsimulations(Efron&Tibshirani,1993)wereusedtoproducecondenceintervalsform(minimum500simulatedexperiments).
Thettedparameters3forEq.
(12)werethenusedtocalculateσi/σe.
Theσi/σeestimatesforeachobserverareplottedasafunctionofsessioninFig.
8.
Thesedatashowthatσi/σerangedbetween1and1.
5forthetextureidenticationtaskandabout1–4forthefaceidenticationtask.
Thereareseveralnoteworthyaspectstothesedata.
First,thefactthatσi/σewassignicantlygreaterthanzeroinhigh-contrastexternalnoiseisconsistentwithpreviousresults(Burgess&Colborne,1988;Green,1964;Spiegel&Green,1981)andimpliesthatitisappropriatetoincludeacontrast-dependentinternalnoisecomponentintheblack-boxmodeldescribedabove.
Second,althoughthedataaresomewhatnoisy,thereappeartobenosystematicdecreasesinσi/σewithpracticeforanyoftheobservers.
Ofcourse,itcouldbethat,giventherelativelysmallnumberoftrialsineachsession,theresponseconsistencymeasureissimplynotsensi-tiveenoughtodetectthemagnitudeofchangenecessarytoproducetheobservedchangesinperformancethattookplacewithlearning.
Totestthispossibility,werstestimatedhowmuchofadecreaseinσi/σewouldhavebeennecessarytoproducethetotalchangesweobservedintheslopesofthenoisemaskingfunctionsacrosslearningsessionsmeasuredinExperiment1.
Thatis,weestimatedhowmuchσi/σewouldhavehadtohavedecreasedifalloftheimprove-mentsinperformancewithlearningwereduetochangesininternalcontrast-dependentnoise.
ThisestimatecorrespondstomeasuringthemultiplicativeconstantmfromEq.
(3).
However,recallthatthecontributionofthecontrast–invariantinternalnoise(Ni)willbenegligibleunderconditionsofhighexternalnoise.
Inaddition,thecontributionofthesignalenergyEwillalso20J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxFig.
8.
Internal/externalnoisestandarddeviationratio(σi/σe)estimatesforallobserversintheface(leftpanel)andtexture(rightpanels)identicationtasksfromExperiment2.
Eachpanelplotsσi/σeasafunctionofsession.
Thedifferentsymbolscorrespondtodifferentobservers(seelegend).
Errorbarsoneachsymbolcorrespondto±1standarddeviation.
benegligibleunderthresholdsconditions.
So,undertheseconditions,Eq.
(2)canbesimpliedtothefollowing:E=k(Ne+mNe)=k(1+m)Ne=kNe(13)wherek=k(1+m),theslopeofthenoisemaskingfunction(or,equivalently,m=k/k1).
Letka,kaandmaequalk,kandmafterlearning,andkb,kbandmbequalk,kandmbeforelearning.
Then,ma=kaka1(14a)mb=kbkb1(14b)Ifweassumethatallofthechangesinkaresolelyduetochangesinm,thenka=kb.
SolvingforkbinEq.
(14b)andsubstitutingforkainEq.
(14a),wehavema=(mb+1)kakb1(15)Becausembisgivenbythesquaredempiricalestimateofσi/σeobtainedintherstsession(recallthatσi/σewasexpressedasaratioofstandarddeviations,whereasmisexpressedasaratioofvariances),kaandkbcanbecomputedfromthethresholdestimates(E)obtainedintherstandlastsessions,usingtherelationshipk=E/Ne,asdenedinEq.
(13).
J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx21Table2ParametersandstatisticsfortheresponseconsistencysensitivityanalysesfromExperiment2ka/kbmbmbSEEmpiricalmaEmpiricalmaSEPredictedmaEmpiricalvs.
predictedma(p-value)FaceidenticationJSW0.
365311.
25872.
54004.
10991.
14180.
46190.
0014LCK0.
119318.
19113.
09928.
74152.
05486.
01120.
1839TextureidenticationLAP0.
07102.
01130.
31181.
06400.
35390.
78631.
71e07SKH0.
07962.
89650.
22071.
88730.
42920.
68971.
93e09WeusedEq.
(15)tocomputethepredictedvalueofmaforeachobserverineachconditionofExperiment2.
Wethencomparedthesevaluestotheempiricalestimatesobtainedinthelastsessionoftheexperiment.
Ifresponseconsistencywasnotasensitiveenoughmeasuretodetectchangesininternalnoise,theempiricalandpredictedvaluesofmashouldnotbesignicantlydifferentfromoneanother.
TheresultsofthisanalysisareshowninTable2.
Foreachobserver,therstcolumninTable2showstheratioofka/kb.
Thenextfourcolumnsshowtheempiricalvaluesofmaandmb,alongwiththecorrespondingestimatesofstandarderrorforeachparameter(computedusingbootstrapsimulationsasdescribedpreviously).
Thenaltwocolumnsshowthepredictedvaluesofmaandtheresultsofastatisticalcomparisonbetweenthepredictedandempiricalvaluesofma.
Forallobservers,thepredictedvaluesofmawerelowerthantheempiricalestimates.
Thedifferencesweremoststrikinginthetextureidenticationtask,wherethepredictedvaluesindicatedthat,intheabsenceofachangeinefciency,internalnoisewouldhavehadtohavebeennegativebythenalsessioninordertoproducetheobservedimprovementsinperformancewithlearning!
Withtheexceptionofoneobserverinthefaceidenticationtask(LCK),thedifferencesbetweenthepredictedandempiricalvaluesofmawerestatisticallysignicant.
4Theseresultssuggestthatresponseconsistencyissensitiveenoughtoruleoutthepossibilitythatreductionsincontrast-dependentinternalnoiseaccountforallofthelearninginourfaceandtextureidenticationtasks.
Ofcourse,smallchangesincontrast-dependentinternalnoisemayhavecontributedtotheimprovementsthattookplacewithlearning.
Nevertheless,ourresultssuggestthatasignicantportionofthelearningeffectsinourexperimentswereduetodeterministicincreasesintheefciencyofobservers'calculations.
Withinthecontextoftheblack-boxmodeloutlinedabove,theseresultsimplythatpracticeshouldincreasethesimilaritybetweenthecalculationsusedbyhumanandidealobservers.
ThispredictionofthemodelhasbeenexaminedmorecloselyinExperiment3.
4.
Experiment3ThepurposeofExperiment3wastoextendtheresultsofthersttwoexperimentstoincludeamoredetaileddescriptionofthechangesthattakeplacewithperceptuallearning.
Experiments22J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx1and2showedthattheeffectsofperceptuallearninginourpatternrecognitiontaskscanbeattributedtoagradualincreaseintheefciencyofthedeterministicaspectsofobservers'calcu-lations.
However,theexactnatureofthosechangesremainsunspecied.
Experiment3wasde-signedtoaddressthisquestionbyusingtheresponseclassicationtechnique.
AswasmentionedintheIntroduction,theclassicationimagewillbeproportionaltotheobserver'stemplate,orthecalculation,usedbyalinearobserverinaoneoftwopatternidenticationtask(seeAbbeyetal.
,1999;Murrayetal.
,2002).
Inthecontextofperceptuallearning,responseclassicationcanbeusedtracethechangesinanobserver'scalculationsaslearningtakesplace.
Inaddition,humanandidealclassicationimagescanbecomparedtotestthepredictionthatanobserver'scalculationsshouldbecomemoresimilartotheidealaslearningoccurs(Murrayetal.
,2002).
4.
1.
Methods4.
1.
1.
StimuliThesignalsweretwonewfacesandtextures(Fig.
9),generatedinthesamefashionasdescribedinExperiment1.
Thesignalswereembeddedinthehighestlevelsofexternalnoiseusedinthepreviousexperiments(faces:25.
55*106degree2;textures:51.
10*106degree2).
4.
1.
2.
ObserversTwoobserversparticipatedinthefaceidenticationtaskandtwointhetextureidentica-tiontask.
Alloftheobserverswerenaivetothepurposesoftheexperimentandhadnormalorcorrected-to-normalvisualacuity.
Oneobserver(LCS)participatedinboththefaceandFig.
9.
Theface(top,lefttwopanels)andtexture(bottom,lefttwopanels)stimuliusedinExperiment3.
Therightmostpanelineachrowisthedifferencebetweenthetwopanelstotheleft.
Thisimageisalsotheclassicationimageforanidealobserverafteraninnitenumberoftrials(seetextfordetails).
J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx23textureidenticationtasks.
Thesecondobserver(SKH)inthefaceidenticationtaskhadalsoparticipatedinthetextureidenticationtaskinExperiment2.
4.
1.
3.
ProcedureIneachtask,signalenergywasmanipulatedacrosstrialsaccordingtoa1-up-2-downstair-casetomaintainroughlyconstantpercentcorrectperformancethroughoutthesession.
Auniquenoiseeldwasgeneratedbeforeeachtrial.
Thesequenceofsignalidentities,observerresponsesandseedsusedtogeneratethenoiseeldsweresavedaftereverytrialtoallowforsubsequentcomputationoftheclassicationimages.
Eachsessionconsistedof800trialsthatwerecom-pletedwithoutbreaksoverthecourseofabout1h.
Onlyonesessionwascompletedeachday.
Eachobserverparticipatedinatotalof12sessionsoverthecourseof16days.
4.
2.
ResultsanddiscussionSignalenergythresholdsfortheface(leftpanel)andtexture(rightpanel)identicationtasksareplottedasafunctionofsessioninFig.
10.
Thesedatashowacleareffectoflearning,withthemajorityoflearningoccurringwithintherst4–6sessions.
Overthecourseoftheentireexperiment,thresholdsdeclinedbyaboutafactorof3inthefaceidenticationtask(LCS:3.
1;SKH:2.
7)andaboutafactorof5inthetextureidenticationtask(AJR:5.
2;LCS:5.
1).
NotethatthemagnitudeoflearninginExperiment3wassimilartothatfoundinExperiment1,despitetheuseofastaircaseprocedureandsinglenoiselevelasusedinExperiment2.
ThisresultsuggeststheincreasedlearningeffectsfoundinExperiment2werenotduetothesetwomanipulations.
However,themagnitudeoflearningmayalsodependuponthenumberofFig.
10.
Signalenergythresholdsplottedasafunctionofsessionintheface(leftpanel)andtexture(rightpanel)identicationtasksfromExperiment3.
TheexternalnoisepowerspectraldensitywassettothelevelsusedinthecorrespondingtasksfromExperiment2.
Errorbarscorrespondto±1standarderror.
24J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxalternatives(whichwassignicantlysmallerinExperiment3),makingitdifculttocompareresultsacrossExperiments.
Thecorrespondingclassicationimageswerecomputedforeachobserverineachses-sion.
Theseimageswerecomputedbyaveragingthenoisematricesaddedtothesignalspoint-by-pointacrosstrialsaccordingtoeachsignal–responsecombination.
Inthecaseofonlytwosignals,therearefoursignal(S)–response(R)combinations:S1R1,S1R2,S2R1andS2R2.
Thesefourmatricesmaybecombinedtoformtheobserver'srawclassicationimage,C,asfollows:C=(S1R2+S2R2)(S1R1+S2R1)(16)Ithasbeenshown(Ahumada,2002;Murrayetal.
,2002)thatthismethodofcombiningnoiseeldsisstatisticallyoptimalintermsofmaximizingthesignal-to-noiseratiointheclassicationimageforanunbiasedlinearobserver.
Thecontrastofapixelintheresultingclassicationimagecorrespondstothecorrelationbetweenthecontrastofthenoiseatthatpixelacrosstrialsandtheobserver'sresponses.
Whendisplayedasapicture,pixelsthatarebrighterthanmeangrayindicateapositivecorrelationbetweenthenoisecontrastatthatpixelandtheresponse'image#2,'andthosedarkerthanmeangrayindicateanegativecorrelation.
(N.
B.
:Thechoicetopositivelycorrelatethenoisewiththeresponse'image#2'isaresultofarbitrarilychoosingthesecondimageinthesettocorrespondtoS2andR2inEq.
(16).
)Unfortunately,theclassicationimagescomputedfromindividualsessionsweretoonoisytorevealanyvisiblefeatures.
Themajorreasonforthisabsenceofsalientfeaturesisthesmallnumberoftrialsusedtocomputeeachimage.
Typically,severalthousandtrialsareneededtoproducevisiblefeaturesinrawclassicationimages(Abbeyetal.
,1999;Beard&Ahumada,1998;Goldetal.
,2000).
However,themajorityoflearninginthesetasksoccurredwithintherstsixsessions,enablingustocollapsethedataacrosssessionswithintherstandsecondhalvesoftheexperiment.
Collapsingacrosssessionsinthisfashionincreasesthenumberoftrialsineachclassicationimagebyafactorof6to4,800.
Inaddition,statisticalanalyseswereusedtotestforglobalchangesintheclassicationimages.
Specically,theresultsfromExperiments1and2implythattheobserver'scalculationsbecomemoresimilartotheidealcalculationwithlearning(thusincreasingcalculationefciency).
Giventheseresults,astrongpredictionofalinearversionofourblack-boxmodelisthattheobserver'sclassicationim-agesshouldbemoresimilartotheidealobserver'sclassicationimageafterlearninghastakenplace.
Foraoneoftwoidenticationtask,theidealobserver'sclassicationimage(template)issimplythedifferencebetweenthetwopossiblesignals(Noreen,1981).
The'goodness'ofahumanobserver'sclassicationimagecanbecomputedbycross-correlatingitwiththeidealclassicationimagewhenbotharenormalizedtounitenergy(Murray,2002).
Theresulting'normalizedcross-correlation'scorewillrangebetween1(perfectnegativecorrelation)and1(perfectpositivecorrelation).
TheidealclassicationimagesfortheoneoftwofaceandtextureidenticationtasksareshownintherightmostpanelsofFig.
9.
Theidealobserver'sclassicationimagecanbethoughtofasan'informationmap,'withthecontrastateachpixelcorrespondingtoitsrelativeinformativeness.
Ifvisualprocessingcanbeapproximatedbyalineartemplate,andiflearningimprovestheefciencyofthisprocessing,thenitfollowsthatthecalculationsarebecomingmoresimilartothoseusedbytheidealobserver.
Thisshouldresultinanincreaseinthesimilaritybetweenthehumanandidealclassicationimages.
J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx25Fig.
11.
Theraw(toprow),smoothed(middlerow)andthresholded(bottomrow)classicationimagesforbothobserversinthefaceidenticationtaskfromExperiment3,pooledacrosseithertherstorlasthalfoftheexperiment.
Inthethresholdedimages,allofthepixelsthatfellwithinarestrictedrangerelativetomeangrayinthesmoothedclassicationimageweresettoasinglelowvalueandtheremainingpixelsweresettounity.
Theresultingimagewasthenmultipliedbytheidealtemplate(seeFig.
9).
ThetoppanelsofFigs.
11and12showtheclassicationimagesfromtherstandsecondhalvesoftheexperimentforobserversinthefaceandtexturediscriminationtasks,respectively.
Althoughtheimagesstillappearnoisydespitecollapsingacrosssessions,closeinspectionrevealssmall'hotspots'emergingfromthebackgroundofnoise.
Fig.
13showstheresultsoftheidealcorrelationanalysisforeachoftheclassicationimages.
Theleftpanelshowstheresultsforthefaceidenticationtask,therightpanelthetextureidenticationtask.
Theheightofeachhistogrambarcorrespondstothecorrelationforanindividualobserverineithertherst(blackbars)orlast(graybars)halfoftheexperiment.
Thesedatashowthattherewasanincreaseinthecorrelationwiththeidealtemplatefromthersttothelasthalfoftheexperimentforeachobserverineachcondition.
Az-testwasperformedtodeterminewhethereachcorrelationdifferedsignicantlyfromwhatonewouldexpectiftherewasnostructurepresentintheimageatall(i.
e.
,comparedtozerocorrelationwiththeidealtemplate).
Theresultsofthisanalysisforeachimageareshownaboveeachbar,expressedasp-values.
Thesedatashowthattherewasasignicantamountofstructurepresentwithinmostofthecombinedclassicationimage,andtheamountofstructureincreasedduringthesecondhalfoftheexperimentforalloftheobservers.
Asecondz-testwasperformedtodeterminewhethertheincreaseincorrelationfromthersttothelasthalfoftheexperimentwasstatisticallysignicantforeachobserverineach26J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxFig.
12.
Theraw(toprow),smoothed(middlerow)andthresholded(bottomrow)classicationimagesforbothobserversinthetextureidenticationtaskfromExperiment3,pooledacrosseithertherstorlasthalfoftheexperiment.
condition.
Theresultsofthisanalysisareshownasp-valuesbeloweachobserver'sdata.
Thesedatashowthattheincreaseincorrelationwasstatisticallysignicantforallbutoneobserver(LCS)inthetextureidenticationtask.
Despitethestatisticalsignicanceoftheanalysesdescribedabove,theabsolutelevelsofcorrelationwerequitelow.
However,suchlowcorrelationsarenotsurprising,giventhatthecorrelationanalysisincludestheentirestimulus;unliketheidealobserver,humanobserversundoubtedlyuseonlyaportionoftheavailablestimulusinformation.
Asaresult,theunusedregionsonlycontributenoise,whichservestoreducethecorrelationbetweenthehumanandidealtemplates.
Therefore,itisofinteresttoexaminetheeffectsofremovingsomeofthenoisefromtheclassicationimagesthatobserverswereunlikelytohaveusedduringthetask.
Inthecaseofthefaceidenticationtask,previousworkhasshownobserverstendtomakeespe-ciallyefcientuseofspatialfrequencieswithina2-octavewidebandcenteredaround6c/face,withefciencygraduallydecliningaboveandbelowthecenterfrequency(Goldetal.
,1999a;Nasanen,1999).
Thisndingsuggestsanappropriateltertoapplytothefaceclassicationimageswouldbea2-octavewidelog-frequencyltercenteredat6c/face.
Inouranalysis,welteredtheclassicationimages(bothhumanandideal)byalog-Gaussianspatialfrequencyltercenteredat6c/imagewithabandwidthof2octaves(±1octaveathalf-height).
Inthecaseofthetextureidenticationtask,thesignalwashighlylocalizedinthefrequencydomain(2–4c/image).
AnoptimalstrategyinthistaskistorelyonlyuponfrequenciesthatfallwithinJ.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx27Fig.
13.
Normalizedcross-correlationbetweenthehumanandidealclassicationimagesintherstandlasthalvesoftheface(leftpanel)andtexture(rightpanel)identicationtasksfromExperiment3.
Theresultsofastatisticaltestcomparingthecorrelationscoretothescoreforastructurelessclassicationimageisshownaboveeachbar.
Theresultsofastatisticaltestcomparingthecorrelationsfortherstandlasthalvesoftheexperimentareshownbeloweachobserver'sdata.
thislimitedrange,andsoanappropriatelterforthetextureclassicationimageswouldbeonethatismatchedtothesignalband.
However,unlikewithfaces,thereisnopreviousworksuggestingobserversmightfollowsomethingsimilartothisoptimalstrategy.
Becausewhitenoiseisuncorrelatedinboththespatialandthespatialfrequencydomains,wecancomputeaclassicationimageinthespatialfrequencydomaininthesamemannerasaspatialclassi-cationimagetoseeifamplitudeinthesignalbandwascorrelatedwithobservers'responses.
ThisamountstosimplycomputingtheFouriertransformofthespatialclassicationimages.
Theresultsofthisspatialfrequencyanalysisforoneobserver(LCS)inthetextureidenti-cationtaskareshowninFig.
14(theresultsforthesecondobserverweresimilar).
Thisgureshowsthepowerspectrum(i.
e.
,theenergyateachfrequency)oftheclassicationimagesfromtherst(leftpanels)andlast(rightpanels)halvesoftheexperiment,plottedinpolarcoordinates.
Intheseplots,theorigincorrespondstotheDCcomponent(i.
e.
,averagelumi-nance),thedistancefromtheoriginspatialfrequency,andtheanglemadebyavectordrawnfromtheorigintoagivenpointorientation.
Forclarity,onlyfrequenciesbelow40c/imageareshown.
Thebottompanelsareblurredversionsofthetoppanels,whichremovedsomeofthemaskingproducedbyimagepixelization.
5Theseguresshowahighcontrastregionwithina2–4c/imageringaroundthecenterofeachplot,indicatingthisobserverreliedalmostexclusivelyuponthesefrequenciestoperformthetask.
Thisresultindicatesthatremoving28J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxFig.
14.
Amplitudespectraofoneobserver'sclassicationimages(LCS)intherst(leftcolumn)andlast(rightcolumn)halvesofthetextureidenticationtaskfromExperiment3,plottedinpolarcoordinates.
Intheseplots,theorigincorrespondstotheDCcomponent,thedistancefromtheoriginspatialfrequency,andtheanglemadebyavectordrawnfromtheorigintoagivenpointorientation.
Onlyfrequenciesbelow40c/imageareplotted.
Thebottomtwopanelsareblurredversionsofthetoptwopanels(convolvedwitha5*5pixelkernel).
frequenciesoutsideofthesignalbandwouldgreatlyincreasethesignal-to-noiseratiointhetextureclassicationimages.
ThemiddlepanelsofFigs.
11and12showtheclassicationimagesforthefaceandtextureidenticationtasks,respectively,afterbeingsmoothedbytheappropriatelters.
Thebottomrowineachgureshowsthesmoothedimagesafterbeingthresholdedtoremoveallbutthehighestcontrastpixels.
Specically,allofthepixelsthatfellwithinarestrictedrange6frommeangrayweresettoasinglelowvalueandtheremainingpixelsweresettounity.
Thesethresholdedimageswerethenmultipliedbytheidealclassicationimage,resultinginanimagewherehighcontrastregionsindicatetheinformativelocationsusedbytheobservertoperformthetask.
Inaddition,thenormalizedcross-correlationbetweenthehumanandidealsmoothedclassicationimageswascomputedforeachobserverineachcondition.
Asbefore,theresultsofthecross-correlationanalysisshowedthatthesimilaritybetweenhumanandidealclassicationimagesincreasedwithlearning(AJRtextures:.
39–.
52;LCStextures:.
36–.
41;LCSfaces:.
04–.
09;SKHfaces:.
04–.
06).
Thesmoothingprocessalsoservedtoincreasetheabsolutelevelsofcorrelationbetweenthehumanandidealtemplates,especiallyinthetexturediscriminationtask:oneobserverinthetexturetask(AJR)showedanincreaseofJ.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx29afactorof21.
7relativetotheunlteredcorrelationinthesecondhalfoftheexperiment.
Theseeffectscanalsobeseenbyvisuallyinspectingthesmoothedandthresholdedclassicationimages,wherehighlyvisiblefeaturesnowemergefromthebackground.
Inthefaceexperiment,bothobserversconcentratedupontheregionaroundthelefteyeandeyebrow.
Inthetextureexperiment,AJRusedanelongatedregioninthecenter,whereasLCSusedamorelocalizedblobinthetopleftcorner.
Thethresholdedclassicationimagesrevealthatalloftheselocationswerehighlyinformative.
Closerinspectionoftheimagesrevealsthatalloftheobserversappeartohaveusedslightlylargerregionsinthesecondhalfoftheexperimentthanintherst.
Thischangeinregionsizewasveriedbycountingthepercentageofpixelsthatexceedthresholdineachhalfoftheexperiment.
Thepercentageofpixelsinthethresholdedimagesincreasedbyaboutafactorof1.
3forbothobserversinthefaceidenticationtaskandbyaboutafactorof1.
8forbothobserversinthetextureidenticationtask.
Althoughthisincreaseinregionsizeprobablycontributedtotheincreaseincalculationefciency,theuseofmorepixelsdoesnotnecessarilyinsurethatthecalculationwillbemoreefcient.
Forexample,anobservercouldusetwiceasmanypixelsinthesecondhalfoftheexperiment,butifnoneofthepixelswereinformative,thecalculationwouldbelessefcient.
Itisthereforelikelythatalargeportionoftheimprovementresultedfromare-weightingofpixelsusedinbothhalvesoftheexperiment.
Anaspectoftheclassicationdataworthemphasizingisthestrikingdifferencebetweenthetemplatesusedbythetwoobserversinthetexturediscriminationtask.
InspectionofthesmoothedandthresholdedclassicationimagesinFig.
12showsthatthetwoobserversadoptedradicallydifferentstrategiestodiscriminatebetweenthetextures:oneobserver(AJR)primarilyrelieduponaregioninthecenterofeachtexture,whereastheotherobserver(LCS)primarilyre-lieduponaregioninthetopleftcornerofeachtexture.
Despitethesedifferences,Fig.
10showsthattheperformanceofthetwoobserverswasactuallyquitesimilarthroughoutthelasthalfoftheexperiment.
Theseresultsdemonstratehowtheresponseclassicationanalysisisabletorevealadditionalinformationnotcapturedbythemoregrossmeasureofthresholdorefciency.
DotheobservedchangesinclassicationimagesaccountforthemagnitudeoflearningWeexaminedthisquestionbycalculatingfaceandtexturediscriminationthresholdsforasimulatedobserverthatusedthelteredclassicationimagesastemplates.
Ifchangesincalculationimageswereassociatedwithperceptuallearning,thentheratioofthresholdsmeasuredinsessions1–6and7–12oughttobesimilarinsimulatedandrealobservers.
Infact,simulatedandobservedthresholdratioswerequitesimilar:averagedacrosstasksandobservers,thresholdsinsessions1–6were2.
1and1.
8timeshigherthaninsessions7–12insimulatedandrealobservers,respectively.
Thisresultisconsistentwiththeclaimthatthelearningeffectcanbeattributedlargelytochangesinthelinearcalculationsthatunderlieresponseclassicationimages.
Didchangesincontrast-dependentinternalnoiseaffectourresultsAlthoughthepreviousexperimentsshowedthatinternalnoisedoesnotdecreasewithlearningina1-of-10identi-cationtask,itstillremainsapossibilitythatthelearningeffectsfoundinExperiments1and2donotextendtoaoneoftwoidenticationtask.
Theimportanceofthispossibilityismademoreapparentbynotingthatthechangesintheclassicationimagesobservedinthepresentexperimentarealsoconsistentwiththeeffectsofadecreaseincontrast-dependentinternalnoise:suchareductionwouldincreasethesignal-to-noiseratiointheclassicationimages,causingthemtoconvergemorequicklyandthusincreasethecorrelationwiththeidealtemplateandthenumberofsignicantpixels.
30J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxFig.
15.
Calculationefciency(leftpanel),equivalentinputnoise(middlepanel),andσi/σeestimates(rightpanel)asafunctionofexperimentalsessionforallobserversintheoneoftwoface(opensymbols)andtexture(closedsymbols)identicationtasksfromExperiment3.
Inallgures,errorbarsoneachsymbolcorrespondto±1standarddeviation.
Thispossibilitywastestedbymeasuringcalculationefciencyandinternalnoise(bothcontrast–invariantandcontrast-dependent)fortheoneoftwoidenticationtasks.
Nearlyallpreviousstudies(includingExperiment1)havefoundalinearrelationshipbetweensignalenergythresholdandexternalnoisepowerspectraldensity(Pelli&Farell,1999).
Basedontheuniformityofpreviousresults,weassumedthatthenoisemaskingfunctionsinouroneoftwoidenticationtaskswouldbelinear,allowingustomeasurethresholdsinonlytwolevelsofexternalnoise.
Thissimplicationincreasedthenumberoftrialsateachlevelofexternalnoise,allowingustoconcurrentlymeasureequivalentinputnoise,calculationefciencyandresponseconsistencyinthesameexperiment.
WeusedthehighestandlowestexternalnoiselevelsfromExperiment1,andtherstandlasthalvesofeachsessionwereidenticaltoallowforthemeasurementofresponseconsistency.
7Eachsession(totalof6)consistedof200repeatedtrialsperexternalnoiselevel,foratotalof800trials.
Twonewobserversparticipatedinboththefaceandtexturediscriminationtasks.
TheresultsofthisexperimentareshowninFig.
15,whichplotscalculationefciency(leftpanel),equivalentinputnoise(middlepanel)andhighnoiseσi/σeforeachobserver,asafunctionofpractice.
Thesedataareverysimilartothe1-of-10identicationtaskdataofExperiments1and2,withallobserversshowingincreasedcalculationefciencywithlittleornochangeinequivalentinputnoiseorσi/σe.
TheonenotableexceptiontothistrendwasobserverLEWinthetextureidenticationconditionwhoseequivalentnoiseremainedapproximatelyconstantduringsessions1–5butincreasedduringsession6.
However,theoverallresultswereconsistentwiththeconclusionthatanincreaseintheefciencyofthedeterministicaspectsofobservers'calculationsmediatedthechangeswefoundintheclassicationimageswithlearning.
5.
GeneraldiscussionManyexperimentshavedemonstratedthatpracticeinperceptualtasksproducessubstantialimprovementsinperformance.
Withinthecontextofsignaldetectiontheory,theseimprove-J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx31mentscouldbeduetoeitherincreasedsignalstrengthordecreasedinternalnoise(orsomecombinationofthetwo).
Todiscriminatebetweenthesetwopossibilities,weusedacom-binationofpsychophysicaltechniquesinvolvingtheadditionofexternalstimulusnoiseinconjunctionwithablack-boxmodelofthehumanvisualsystem.
Experiment1wasdesignedtodiscriminatebetweentheeffectsofcontrast–invariantinternalnoiseandtheefciencyofinternalcalculationsasobserverslearnedtoidentifytwounfamiliarsetsofpatterns,humanfacesandband-passlterednoisetextures.
Performanceinbothtaskswasconsistentwiththemodel'spredictionthatsignalenergythresholdshouldbelinearlyrelatedtothepowerofanex-ternallyaddednoise.
Thisrelationshipalsoallowedustoinfertheamountofcontrast–invariantinternalnoiseandtheefciencyofthecalculationaslearningtookplace.
TheresultsofEx-periment1indicatedthatcontrast–invariantinternalnoiseremainedxedwhilecalculationefciencyincreasedsystematicallywithpractice.
Experiment2allowedustodisambiguatepurelydeterministicchangesinobservercalcu-lationsfromchangesincontrast-dependentinternalnoise,bothofwhichcouldcontributetochangesincalculationefciency.
Double-passresponseconsistencywasusedtomeasurein-dependentlyanycontributionsmadebycontrast-dependentinternalnoisetotheincreasesincalculationefciencyfoundinExperiment1.
Eventhoughperformanceimprovedwithprac-tice,theconsistencyofresponsesmadebetweenidenticalpassesthroughtheexperimentdidnotincreasesignicantlywithlearning,indicatingcontrast-dependentinternalnoisedidnotplayalargeroleintheincreasesincalculationefciencyfoundinExperiment1.
Thethirdandnalexperimentallowedustospecifydirectlysomeofthechangesthattakeplaceinanobserver'scalculationsasperceptuallearningoccurs.
Responseclassicationwasusedtomeasurethelinearcomputationsemployedbyobserversastheylearnedtorecognizeunfamiliarfacesandtextures.
Theresultingclassicationimagesrevealedsomeofthestrategiesusedbyobserversinthedifferenttasks,aswellassomeofthegrosschangesthattookplacebetweentherstandlasthalvesoftheexperiment.
Theclassicationimagesalsowereusedtoverifythepredictionthatobservers'classicationimagesshouldbemoresimilartotheidealclassicationimageafterlearninghastakenplace.
Theresultsoftheseexperimentshaveimportantimplicationsforcurrentmodelsofpercep-tuallearning.
Inthenexttwosections,wediscusstheconstraintsourresultsplaceontheneuralmechanismsunderlyingperceptuallearningandontheoriesofperceptuallearningingeneral.
5.
1.
RelationtoneuralmechanismsHowislearningrepresentedinthebrainPerceptuallearningaffectsvirtuallyeverylevelofprocessinginthebrain,fromsynapticconnectionstoglobalpatternsofbloodow(forrecentreviews,seeFahle&Poggio,2002;Gilbert,Sigman,&Crist,2001).
Atthelevelofsinglecellresponses,severalresearchershavefoundchangesinrelativeringratesforvisualstimuliasafunctionofexperience(e.
g.
,Logothetis,Pauls,&Poggio,1995;Rainer&Miller,2000;Rolls,Baylis,Hasselmo,&Nalwa,1989;Schoups,Vogels,Qian,&Orban,2001).
Forexample,RainerandMiller(2000)trainedmonkeystorecognizecomplexobjectsdegradedbynoise.
Afterseveraldaysofpractice,themonkeysperformedsignicantlybetterwithhighlevelsofstimulusdegradation.
Individualneuronsintheprefrontalcortexshowedarelatedeffectofexperience.
Afterpractice,fewerneuronsrespondedtothetargetstimuli,buttheneurons32J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxthatdidrespondweremoreselectivelytunedforthosestimuli,andrespondedwelloverlargerrangesofdegradation.
Althoughthedetailsoftheirstudydifferfromtheexperimentsreportedhere,thesortsofchangesinneuralcodingidentiedbyRainerandMillercouldberespon-siblefortheimprovementswefoundinhumanobserverswithlearning—themodicationofresponseselectivityofneuronsisonewaythevisualsystemcould"netune"aperceptualtemplate.
Ofcourse,itislikelythatourresultsaredrivenbyresponsechangesatthelevelofcellpopulationsratherthanindividualneurons.
Schoupsetal.
(2001)recentlydescribedhowsuchapopulationlevelanalysismightaccountforperceptuallearning.
Monkeysweretrainedtodiscriminatetheorientationofasmallgratingfromothersimilarorientations.
Overthecourseoftraining,themonkeys'orientationdiscriminationthresholdsdecreasedsignicantly,butonlyforgratingswithsimilarorientationasthetrainedstimulus.
Afterlearning,Schoupsetal.
foundthattheslopeofthetuningfunctionmeasuredatthetrainedorientationincreased,butonlyforthesub-populationofV1neuronsmostlikelytocodetheorientationdifferencedetectedbythemonkey(i.
e.
,thoseneuronswithpeakactivity12–20fromthetrainedorientation).
Thischangeinneuronalcodingwasapparentonlywhenneuronsweretestedwiththetrainedorientation.
Nosucheffectwasfoundwhentuningfunctionsweremeasuredatanuntrained,orthogonalorientation,evenwhenthatuntrainedorientationhadbeenpassivelyviewedforanequalnumberoftimesasthetrainedorientation.
Apopulationmodeloftheobtainedresponsechangesaccountswellforthebehaviorallearningeffects.
Schoupsetal.
'sresults,likethoseofRainerandMiller(2000)andthepresentsetofexperiments,areconsistentwiththeideathatperceptuallearningismediatedbyanincreaseintheefciencyofperceptualencoding.
Inthecaseofneuronalresponses,increasedefciencyisobtainedthroughchangesinthetuningfunctionsofvisualneurons.
However,ourresultsalsospecicallyleadtothepredictionthatnochangeinnoiseaccompaniesthechangeinneuronaltuningfunctions.
ThispredictionwastesteddirectlybySchoupsetal.
Theyfoundthatnoise(asindexedbythenormalizedresponsevarianceofneurons)didnotchangeasafunctionoflearning,supportingtheideathatperceptuallearningchangesthediscriminativesignalbutnotinternalnoise.
5.
2.
MechanismsoflearningForourtasks,theidealcalculationistocorrelatetheinputwithcopiesofthestimulusalternatives(i.
e.
,templates),andtoselecttheitemyieldingthehighestcorrelation.
Forexample,inthetwo-itemtaskinExperiment3theidealruleistotransformtheinput,s,intoadecisionvariable,v,usingtherulev=iw1isiiw2isi=i(w1iw2i)si=iwdisi(17)wherew1andw2aresetsofweightscomprisingthetemplatesforthetwostimulusalternatives,wdisthedifferencebetweentheweights,andthesummationistakenoverallipixelsinthestimulus.
Foridealperformance,thew1andw2arethestimulithemselves—sowdissimplythedifferencebetweenthestimuli—andtheidealruleistoselectalternative1ifvisgreaterthan0andalternative2,otherwise.
Thetaskoflearningistoselectasetofweightsthatisassimilaraspossibletowdtomaximizeresponseaccuracy.
J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx33Theproblemofhowtosearchaweight-spaceforanoptimalsolutiontoacategorizationproblemhasbeenstudiedextensivelyintheneuralnetworkliterature,resultinginawiderangeofnetworkarchitecturesandweight-settingproceduresthatareusefulindifferentsituations(Arbib,1995).
Also,severalneuralnetworkmodelshavebeenproposedspecicallytoaccountforperceptuallearning(Estes,1994;Herzog&Fahle,1998;Mato&Sompolinsky,1996;McLaren,Kaye,&Mackintosh,1989;Poggioetal.
,1992;Weiss,Edelman,&Fahle,1993).
Ourexperimentsusedsimplelearningtasks,soitislikelythatmost,ifnotall,ofthesemodelscouldaccountforthedecreaseinthresholdthatoccurredasafunctionofpractice.
Indeed,thestimuliinExperiment3arelinearlyseparable,soevenasimpleperceptron(Rosenblatt,1958)couldlearntoperformoptimallyinthattask.
However,anotheraspectofourresults—namely,thatresponseconsistencydoesnotchangewithlearning—doesposeachallengetosomeneuralnetworkmodels.
Consider,themodeldescribedbyMcLarenetal.
(1989).
Inthismodel,stimuliarerepresentedasalargenumberofelementsthataresampledrandomlyoneachtrial.
Duringrepeatedpresentation,associationsareformedbetweenelementsthatco-occur,andinhibitoryconnectionsareformedbetweenel-ementsthatdonotco-occur.
The(positive)associationsensurethatthecompletesetofelementsrepresentingastimulus—ratherthanasmall,variablesubset—willbeactivatedeachtimethatitemispresented,andtheinhibitoryconnectionsensurethatthesetsofelementsactivatedbydifferentstimuliwillbecomemoredistinct.
Thus,themodelaccountsforthebasicndingthatpracticeimprovesstimulusdiscrimination,aswellasavarietyofmoresubtleeffects(Jones,Wills,&McLaren,1998;McLaren,1997;Wills&McLaren,1998).
Notice,however,thatthemodelalsopredictsthatresponseconsistencyshouldincreasewithlearning.
IntheframeworkillustratedinFig.
1,therandomstimulussamplingthatoccursatthestartoflearningwouldmanifestitselfasaninternalnoise.
If,asposited,randomstimulussamplingisasignicantsourceofinternalvariability,theneliminatingitbyformingassociationsamongco-occurringelementsshouldreduceinternalvariability,andthereforeincreaseresponseconsistency.
Wefoundnoevidencethatresponseconsistencychangedwithlearning,suggestingthatrandomstimulussamplingdidnotdiminishwithpractice,orthatitdoesnotcontributesignicantlytointernalvariability.
Moregenerally,theconsistencydataforceustoconsiderhowsourcesofnoisetintoamodel'sarchitecture.
Theconsistencydataimplythattheinternal/externalnoiseratioatthelevelatwhichthedecisionvariableiscomputedremainsconstant.
ForthemodelillustratedinFig.
1,internalnoiseisaddedattherstprocessingstage,andthesubsequentcalculationcannotdistinguishbetweentheexternalandinternalcomponentsofthetotalnoise.
Therefore,anyadjustmentofthetemplateweightsinEq.
(17)willleavetheinternal/externalnoiseratio(andresponseconsistency)unchanged.
Thisstatementistruebothforcontrast-dependentandcontrast-independentinternalnoise.
Nowconsideramodelinwhichinternalnoiseisaddedtothedecisionvariableafterthecalculation.
Thetemplatesthatyieldoptimalperformanceintheearly-noisemodelwillalsoyieldoptimalperformanceinthelate-noisemodel.
However,inthelate-noisemodel,externalnoiseislteredthroughthetemplatebeforeitiscombinedwithinternalnoise,andsoalteringthetemplatemightaltertheinternal/externalnoiseratioandresponseconsistency.
Forthenoiseusedinourtasks,itiseasytoshowthattheproportionofnoisevariancepassedbythelineartemplateequalsthesumofthesquaredweights(Bracewell,2000).
Toaccountfortheconsistencydata,thelate-noisemodelthereforemustassumethat34J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxeither(a)theinternalnoiseisentirelycontrast-dependent,or(b)thattheweightsofthetemplateareadjustedsothatthesumofsquaredweightsisconstantwithlearning.
Theimportantpointisnotwhethersuchconstraintsonthenoiseorweight-adjustmentalgorithmsareeasytoimplement,butratherthattheybecomenecessaryonlywhenoneconsiderstheeffectsoflearningonresponseconsistency.
Mostnetworkmodelshavemultiplelayersconsistingofnoisy(i.
e.
,probabilistic)non-linearelements,anditisdifculttopredicthowtheconsistencyofsuchnetworkschangeswithlearning.
Measuresofresponseconsistencymaybeusefulindistinguishingamongcompetingneuralnetworkmodels.
5.
3.
PerceptualTemplateModelDosherandLu(1998,1999)conductedasetofexperimentsverysimilartoourExperiment1.
Theirtaskinvolvedtrainingobserverstojudgetherelativeorientationofaperipherallypresentedsinusoidalgratinginvaryingamountsofexternalnoise.
Theirresultswerenearlyidenticaltothosewefoundwithourfaceandtextureidenticationtasks—practiceproducedauniformshiftdowninlogthresholdasafunctionoflogexternalnoise.
However,DosherandLuinterpretedtheirresultswithinthecontextofaslightlydifferentblack-boxmodel,whichtheytermthePerceptualTemplateModelorPTM(Lu&Dosher,1998).
ThePTMissimi-lartotheblack-boxmodeldescribedintheintroduction.
However,onekeydifferenceisthatitplacesalloftheobserver'sinternalnoise(bothcontrast–invariantandcontrast-dependent)afterthecalculation,ratherthanbeforeit.
8InthePTMframework,threeprocessescanpo-tentiallycontributetoperceptuallearning.
Oneprocess,termedsignalenhancementor,equiv-alently,additivenoisereduction,correspondstoeitherareductionincontrast–invariant(i.
e.
,additive)internalnoiseoranincreaseinthegainoftheperceptualtemplate(i.
e.
,thesumofthesquaredweightsofthetemplate,asdescribedinEq.
(17)).
LuandDosher(1998)haveshownthatthesetwoprocessesaremathematicallyequivalentinthePTMframework,andareempiricallyindistinguishable.
Signalenhancementandadditivenoisereductionwilllowerthresholdsprimarilyinconditionswhereexternalnoiseislow.
Thesecondprocessistermedexternalnoiseexclusion.
DosherandLuillustratethisprocessbyconsideringtheef-fectsoflearningonaspatial-frequencytunedlter,ortemplate,thatismaximallysensitivetothetargetspatialfrequencyandhasabandwidththatnarrowswithlearning.
Forsuchacalculation,learningwouldnotalterthelter'sresponsetothetargetfrequency,butwouldreducethelter'sresponsetotheexternalnoise.
Therefore,externalnoiseexclusionreducesthresholdsprimarilyinconditionswhenexternalnoiseishigh.
Forthenarrow-bandtargetsusedintheirexperiments,narrowingthelter'sspatialfrequencybandwidthalsomakesthetemplate'sshapemoresimilartotheshapeoftheideallter.
Therefore,onewayofcharac-terizingthisprocessistosaythatexternalnoiseexclusioncorrespondstoacombinationof(i)anincreaseintheefciencyofthetemplate(i.
e.
,itssimilaritytotheideal,asdescribedinthecorrelationanalysisfromExperiment3);and(ii)areductioninthetotalpowerpassedbythetemplate(i.
e.
,areductioninthetemplate'sgain,henceexcludingexternalnoisepowerthatpreviouslypassedthroughthelter).
Notethatexternalnoiseexclusionincorporatestwomechanisms,andthatoneofthem(reducedtemplategain)hastheoppositeeffectofsignalenhancement.
Thethirdprocessisinternalnoisereduction.
Thisprocesscorrespondstoare-ductionincontrast-dependentinternalnoise.
Asintheearly-noisemodel,theeffectofinternalJ.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx35noisereductionistoreducethresholdsuniformlyatalllevelsofexternalnoiseinlog–logcoordinates.
IfwenowconsidertheeffectsofperceptuallearningfromExperiments1and2intermsoftheseprocesses,thePTMappearstoofferadifferentaccountofthedata.
OnewaythePTMcouldaccountforourresultsisintermsareductionincontrast-dependentinternalnoise.
However,thispossibilityisruledoutbytheresponseconsistencyanalysisinExperiment2.
DosherandLu(1999)alsoruleoutthispossibilityfortheirlearningdatabycomparingnoisemaskingfunctionsmeasuredattwodifferentdcriterionlevels.
AsecondwaythePTMcouldaccountforourresultsisintermsofacombinationofsignalenhancement(reducingthresholdsatlowlevelsofexternalnoise)andexternalnoiseexclusion(reducingthresholdsathighlevelsofexternalnoise).
Ifthesetwomechanismschangedbyexactlythesameamount,thresholdswouldbereduceduniformlyacrossalllevelsofexternalnoise.
ThisexplanationisequivalenttotheoneadoptedbyDosherandLutoaccountfortheirresults,anditappearstodiffersignicantlyfromourclaimthatonlythecalculationchangeswithlearning.
However,recallthatexternalnoiseexclusionisacombinationofincreasedtemplateefciencyandreducedtemplategain.
Thissecondcomponentofexternalnoiseexclusion(reducedtemplategain)is,inmathematicalterms,theinverseofsignalenhancement.
Thus,theclaimthatlearningincreasesbothexternalnoiseexclusionandsignalenhancementisformallyequivalenttotheclaimthatlearningincreasesonlytemplateefciencybutdoesnotchangetemplategain.
Inotherwords,iftemplateefciencycanbechangedwithoutalteringtemplategain,thenthePTMcouldaccountfortheeffectsoflearningintermsofasinglemechanism—increasedtemplateefciency.
Althoughitispossiblethatallthreemechanisms(contrast–invariantinternalnoise,templategain,andtemplateefciency)changedinsuchawayastomimictheeffectsofanincreaseintemplateefciency,itismoreparsimonioustoconcludethatpracticeproducedchangesinjustonemechanism:templateefciency.
However,LuandDosher(2001b)haverecentlyobtainedotherresultsthatcannotbeac-countedforsolelyintermsofincreasedtemplateefciency.
Specically,theyfoundthatper-formanceinafovealgratingorientationdiscriminationtaskimprovedwithpractice,butonlyinhighlevelsofexternalnoise.
IntermsofthecurvesshowninFig.
2,thiswouldproduceadownwardshiftinthresholdsonlyinhighlevelsofexternalnoise,producingarightwardshiftinthekneeofthenoisemaskingfunction.
Inbothmodels,thisresultcanbeaccountedforintermsofacombinationofincreasedtemplateefciencyandincreasedcontrast–invariantinternalnoise.
However,alatenoisemodel(likethePTM)canalsoaccountfortheresultsintermsofacombinationofincreasedtemplateefciencyanddecreasedtemplategain(i.
e.
,externalnoiseexclusion).
Thus,foralatenoisemodel,anaturalaccountofLuandDosher'sfovealgratingdiscriminationdataisthatlearningservedtonarrowthebandwidthofanini-tiallybroadbandtemplatetobettermatchthenarrow-bandgratingpattern.
Incontrast,anearlynoisemodelisforcedtoaccountforthisresultintermsofincreasedtemplateefciencycou-pledwithincreasedcontrast–invariantinternalnoise.
Althoughitispossiblethatinternalnoisemightincreasewithperceptuallearning,itismoreparsimoniousinthiscasetoassumethatthedominantinternalnoiseoccurredafterthecalculationandthatinternalnoiseremainedxed.
OnewayofreconcilingthedifferentndingsistoassumethatthegratingdiscriminationtasksusedbyLuandDosher(2001b)tapmechanismsatrelativelyearlystagesinvisualprocessing,whereasourfaceandtextureidenticationtaskstaplaterstages,andthatthedominantinternal36J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxxnoiseexistsbetweenthetwostages.
Intheend,untilitispossibletodeterminethelocationofinternalnoiserelativetothetemplateinagiventask,choosingbetweenanearlyandlatenoisemodeltodescribeone'sdataultimatelyamountstoaquestionofparsimony.
6.
ConclusionsandfuturedirectionsInconclusion,ourndingssuggestthatperceptuallearningisassociatedwithchangesinthelinearcalculationsthatobserversperformonastimulus.
UsingtheframeworkdepictedinFig.
1,learningisassociatedwithchangesincalculationefciency,ratherthanreductionsininternalnoise.
Oneobviousquestionthatremainsiswhetherthisresultgeneralizestootherstimuliandtasks.
Weobtainedsimilarresultsusingfacesandrandomtextures,bothofwhicharespatiallycomplex,butitispossiblethatdifferentresultsmightbeobtainedifobserverswereaskedtoidentifysimplerstimuli.
Also,weusedidenticationtasks,anditisanopenquestionwhetherlearningindetectiontasksorvariouskindsofdiscriminationtasks(e.
g.
,same–differentorAB-Xtasks)exhibitssimilarcharacteristics.
Theclassicationimagedatademonstratedthatobserversbasetheirresponsesontheinfor-mationconveyedbyonlyasmallsubsetofpixels.
Similarndingshavebeenreportedinotherstudiesusingspatiallycomplexpatterns(e.
g.
,Goldetal.
,2000;Sekuleretal.
,2001).
Thisresultraisesthequestionofwhatdeterminesthepartsofastimulusthatareusedbyanobserver.
Ourresultsareconsistentwiththeideathatobserverslearntousepixelsthatareinformative,butitisinterestingtonotethattheinformativepartsofthefacesandtexturesinourexperimentsalsowererelativelyhighincontrast.
Itispossiblethatpartsofthestimuluswereselectedonthebasisofdetectabilityorsalience,ratherthantheamountofdiscriminativeinformationtheycontained.
Additionalexperimentsareneededtodeterminetherelativecontributionsofthesefactorstofeatureselection.
Notes1.
Itisworthnotingthatcross-correlationistheidealruleherebecauseallofthealternativesineachtaskhavethesamecontrastenergy.
SeeGreenandSwets(1966)forfurtherdiscussionofidealdecisionrulesforothertasksandstimuli.
2.
Thesimulatedobserversusedtheidealdecisionrule(Eq.
(8)).
Theuseofthisruleinsuredtheperformanceofthesimulatedobserverwouldeventuallyreach100%correctathighlevelsofsignalenergy.
3.
ThevaluesoftheparametersttoEq.
(12)inExperiment2wereα=.
114,γ1=4.
07,β1=.
013,γ2=75.
907andβ2=.
036.
4.
ThefailuretoreachstatisticalsignicanceforLCKwasmostlikelyduetotheparticularlylargeerrorsassociatedwiththeempiricalestimatesofmforthisobserver(seeFig.
8).
Thegreatervariabilitystemsfromthisobserver'shighlevelsofcontrast-dependentinternalnoiseincombinationwiththenonlinearityofEq.
(12):lowervaluesofm(whichcorrespondtogreaterinconsistency)arelocateduponprogressivelysteeperportionsofthefunction,whereσi/σeestimatesarelessreliable.
J.
M.
Goldetal.
/CognitiveSciencexxx(2004)xxx–xxx375.
Theplotswereconvolvedwitha5*5convolutionkernel(thematrixproductof[12321]withitselftransposed).
6.
Specically,theexpectedvarianceofastructurelessclassicationimageblurredbytheappropriateconvolutionkernelwascomputed,andallpixelsthatfellwithin±2.
57standarddeviationsoftheexpectedmean(zerocorrelation)weresettoasingle,lownumber.
7.
Newsimulationswereconductedtomeasuretheslopesofthenoisemaskingfunctionsfortheidealobserverintheoneoftwofaceandtextureidenticationtasks.
AdditionalsimulationswereconductedtoobtainthevaluesoftheparametersforEq.
(12)inaoneoftwoidenticationtask(α=.
0727,γ1=1.
076,β1=.
003,γ2=141.
523,β2=.
031).
8.
ThePTMalsoincludesanoptionalnon-linearityafterthetemplatebutbeforetheinternalnoise.
Theinclusionofthenon-linearityisoftenimportantformodelingvariousaspectsoftheshapesofpsychometricfunctions(Ecksteinetal.
,1997;Leggeetal.
,1987;Lu&Dosher,1999;Watson&Solomon,1997).
However,thequalitativedifferencesbetweenthetwomodelsintermsofthechangesthattakeplaceintheshapeofthenoisemaskingfunctionwithchangesinthedifferentmechanismsarethesamewithorwithoutthekindofnon-linearitydescribedinthePTM.
Forexample,inthecontextofthePTM,includingthenon-linearitydoesnotaltertheconclusionthatlearningreducesinternalnoiseandincreasesexternalnoiseexclusion(andsimilarly,includinganon-linearitybeforetheinternalnoiseinthemodelillustratedinFig.
1wouldnotaltertheconclusionthatlearningincreasescalculationefciencybuthasnoeffectoncontrast–invariantinternalnoise).
Whatthenon-linearitydoesdoisenablethePTMtoaccountforthefactthattheamountofinternalnoisereductionandexternalnoiseexclusionthatisobservedduringlearningdependsuponthecriterionusedtodenethreshold(Lu&Dosher,1999).
AcknowledgmentsWewouldliketothankRichardMurrayandthreeanonymousreviewersforhelpfulcom-mentsduringearlierstagesofthiswork.
PartsofthisresearchwerepublishedpreviouslyinNature(Gold,Bennett,&Sekuler,1999b),submittedtotheUniversityofTorontoinpartialful-llmentofaPh.
D.
inpsychology(Gold,2001),andpresentedattheAssociationforResearchinVisionandOphthalmology(Gold,Bennett,&Sekuler,1999c)andVisionSciencesSociety(Gold,Bennett,&Sekuler,2002).
ThisresearchwassupportedbyNSERCgrantsOGP105494andOGP0042133toP.
J.
B.
andA.
B.
S.
,andaUniversityofTorontoHealthaward.
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