,/'r'b,i Rer. Vol.24, No.4. pp. 333 340, 1984 Printedin Great Brirain. All righrs reserved 0042,6989i 84 53.00+ 0.00 CopyrishtO 1984Pergamon PressLrd PHASE REVERSAL DISCRIMINATION* DAVID J. FrELD and JAcoB NACHMTAS Departmentof Psychology,Universityof Pennsylvania, Philadelphia,PA 19104,U.S.A. (Recei,ed 24 August 1982l.in fnal reuisedfonn 23 August 1983) Abstract-In the Fourierrepresentation of space,the parameterofphaseplaysa crucialrole. In this study, severalexperimentswerc peformed involvingdisctiminationof variousphaserelationsof fundamental (2 c/deg)to secondharmonic(4 c/deg)at low contrastlevels.The resultswere consistentwith a model involving four "channels".each optimally sensitiveto one of the following phaserelations: +cosine (bright bar), -cosine (dark bar), +sine (left edg€),and sine (right edge). Phase Discrimination Contrast Channel In the past two decades,Fourier theory has been applied to the study of spatial vision with considerablesuccess. Thereis now abundantevidgncethat the visual system contains rangesof detectorsor "channels" se[sitive to different spatial frequencies (Campbell and Robson, 1968;Graham and Nachmias, l97l; Wilson and Bergen, 1979). Although there is still some disagreementas to the exact parameters(e.g. bandwidth) of thesechannels,and evenas to what their output is actuallyusedfor, one can concludethat the visualsystemprocesses spatial information in a way that is at least somewhat analagousto that of Fourier analysis. Fourier analysistranslatesinformation from the space domain into information in the frequency domain with parametemof both amplitude and phase.All spacedomain information is retainedonly if both parametersare considered.With few exceptions (e.g.Burr, 1980;Atkinson and Campbell,1974; Stromeyerand Klein, 1974i Tolhurst and Dealy, 1975),the researchto datehaslargelyignoredphase. This neglectmight be justified if there wereevidence that the visual systemcould not usephaseinformation. However,this is clearly not the case. As an extremeexampl€,considera bright bar on a gray backgroundand a dark bar of equalcontrast on a similar gray backgound. They have the same amplitude spectra and differ only in their phas€ spectra. Yet, such bars are clearly discriminable. Edgescomposea similar classof stimulus.A left edg€ (i.e.bright on the left and dark on the right) can have the sameamplitudespectrumasa right edge(i.e.dark on the left and bright on the right) and differ only in the phase spectm. Yet, such edgesare also easily discriminable.Hence,phaseinformation is available to the visual system. *A portion of this work was prcsentedat the European Conference on VisualPerceptioh,University of Sussex, Bdghton, England,September,1980. Detection of a complex waveform appearsto be independentof the relativephaseof its components. Grahamand Nachmias(1971)haveshownthat for a waveform consisting of a fundamental and third harmonic,the probability of detectionis not affected by the components'relative phase,evgn though a changein the phasesignificantlyaltersthe amplitud€ of the complex waveform.This result is consistent with the hypothesisthat detectionis mediatedby narrow band mechanismswhich are selectivelysensitive to eachcomponent.Detectionoccursonly when one or both ol the componentsindependentlyactivate their most sensitivechannels. Onelimitation of this hypothesisis that it provides no explanationfor how we discriminateon the basis of phasedifferencesalone.To perform suchdiscriminationsthe systemmust somehowrelatethe relative position information of both frequencycomponents. Nachmiasand Weber(1975)studiedphasesensitivity with complexwaveformsconsistilgof a fundamental and third harmonic. They found that when both componentswere only slightly above threshold for detection,phasediscriminationwas difficult or impossible. However, once the components were suffciently above detection threshold, phase differencescould be discriminatedeasily. In other words,they found that therewasa rcgionwhereboth componentscould be d€tectedbut their relativeposition or phase could not. Nachmias and Weber interpretedtheir findings as suppod for the theory that the mechanismsresponsiblefor phasediscrimilation are broad-bandand lesssemitivethan those responsiblefor detection.They speculatedthat these broad-bandmechanismsmight includethe edgeand bar detectorspostulatedby previous investigators. BuII (1980)has also shown that the thresholdfor phasediscrimination,expressed in termsof the phase anglebetweentwo compolrents,is constantacrossa largerangeof spatialfrequencies.Burr believesthat thesefindingsprovide further supportfor the notion that the discrimiration is mediatedby broad-band mechanismsrather than som€mechDhase-selective DAvrD J. FIILD and JAcoB NACHMTAS anism selectiveto the absolute positiol of local detectors,on th€ otherhand,havea ceotralinhibitory region flanked on each sid€ by weaker excitatory features. Shapleyand Tolhurst (1973)and Kulikowski and regions[Fis. l(b)]. As noted earlier,a bright bar and a dark bar of King-Smith (1973)developedthe conceptsof edg€ and bar-selectivedetecto$ to account for certain equalsizeand contrasthavethe sameFourier ampliaspectsof their detectionresults.Theseinvestigators tude spectra,but diffe! in their phasespectra.In the measuredsensitivityto linesand edgesin th€ presence caseof a bright bar, all componentsarc in +coslne of subthresholdlines,edgesand gratings.They found phaseat the centerof the bar; in the caseof the dark that the sensitivityto suchstimuli could be predicted bar, all componentsare in -cosin€ phase at the by hypothesizingthat the visual system contains centerof the bar. Sincebright and dark bar detectoN detectorssensitiveto edgesof specificodentation, can distinguishbetweencomponentsin cosineand positionand potarity{phase)and bar-selective de- -cosine phase,they are consideredphasesensitive. tecton sensitiveto lines or bars of specificorien- In partiaular,bright bar detectomare +cosine sensitation, poladty and size. Graham (1980)however, tive and dark bar detecton are -cosine s€nsitive. At edge-selectiue detector1scharacterizedby an haspointedout that suchresultscanbe accountedfor receptive field profile as illustmtedin (naffow-band spatial odd symmetric detector by only one classof frequencychannel)if probability summationis taken Fig. l(c) ard (d). Like bright and dark bals, a left into account. These results, therefore, cannot be edgeand a right edgemay difer only in their phase taken as definitivein their support of phase-speciflc spectra.In the caseof a right edge,all components are in +sine phaserclative to the edge,while in the detectors. Sincethe edge and bar detectorconcept will b€ caseof a left edge,all componentsare in -sine phase. usedthroughoutthis paper,a briefdigressionmust be Sinceleft and right edgedetecto$ are differentially madeinto som€terminology.D€tectorsarc classified sensitiveto left and right edges,they can be considright or phase-selective: by their receptivefield responseprofile, namely the ered to be phase-sensitive and left detector'sresponseto unit amountsoflight falling in edge detecto$ are +sine phase-sensitive, different parts of the receptivefreld. Bat-selectbe edgedetectorsare -sine phase-sensitive. The term channelwrll refer to an array of detectors detectorshave even symmetryas illustratedin Fig the sameclass.In this case,a channelis an aray profile of of one class 1(a) and (b). The receptivefield of bar-selectiv€detector consistsof a centml ex- of detector with the same phase sensitivity. For "weaker" example,we may speak of dark-bar, or left edgecitatory region flanked on each side by inhibitory regions[Fig. l(a)]. Theseare calledbright selectivechannels,or channelsoptimally sensitiv€to bar detectors;they will produce a greater positive a 45 degphaserelation.Finally, a.r),J/enwill rcfer to responseto a bright bar situated in its optimal a pair of channels.The cosine (bar) system, for position (centralrcgion) than they will to a dark bar example,consistsof bright and dark bar-selective situated in its optimal position (flanks). Dark bar channelsand the sine (edge)systemconsistsof left channels. and right edge-selective It is importantto note that the namesrcfer only to the type of stimuli that produce a diflbrential peak rcsponsemagnitudein each pa of channels,Bafselectioedeteclorsmay respondquite efectfuely to optimally placed edges,Howeoer,at an edge,the peak responseof a bright bar channel will not diJlerfrom that of a dark bar channel;and.thereforethe difetence in the chqnnels'peak rcsponsesptoDidesno information about the polatity of the edge. The theory that therc are four classesof detector sensitiveto four different phase dations [cosine (bright bar), -cosine (dark bar), sine(left edge),and -sine (dght edge)lwill be referred to as the four channelmodelthroughoutthis paper.It is possibleto postulateone or more additional classesof detector with neitherevennor odd symmetry.Suchdetectors may not respondoptimallyto eithercosineor sine phases,but ruther to someintermediatephaseangle. The lesponseprofile of one suchdetectoris shownin Fig. 1(e).Sucha detectormay respondoptimally to a combination of left edge and bright bar morc effectivelythar to an edgeor bar alone. One line of evidencethat therc are at least four Fig. l. Receptivefield responseprofiles of various phase selective-detectors. classesofphase specificdetectoris found in the work Phas€reversaldiscdmination of Tolhurst and Dealy (1975).Using a forced-choice procedure, they compaled the probability of detecting a low contrast dark or bright bar with the probability of identifying its polarity. They found that the probability af d€tectionwas only slightly higherthan th€ probability of identification.That is, if a subject could detect the presenceof a bar, he could also identify whether the bar was dark or bright. The same result was found with edges:the probability of detectingan edge was only slightly higher than the probability of correctly identifying whetherit was a lefr or righl edge.The authors concluded that if it were true that only a single detector is usually active at threshold, then thes€ resultsprovide support for at least four classesof phasespecificdetector:two edgedetectorsand two bar detectors. The difficulty in using edgesand bars as stimuli is that such stimuli have broad frequencyspectrawith fixedphaseand amplituderelations.Smallvariations 1n phase cannot be investigatedwith such stimuli. Consequently,we have chosento use stimuli with only two fr€quencycomponents:the fundamental and second harmonic. With these components,a largearray of phaseanglesand relativecontrastscan be produced, including those that uniquely correspond to the phaseand contrast relationsof edges and bars. The luminanceprofile of our gratings, Z(r), is given by the equation 180' _ 270' -sine phaserelationsof Fig. 3- Polarreprcsentation of possible fundamental lo second harmonic, are the sameas the phaseanglesthesecomponents havein a left and a right edge,respectively. Similarly, the 0 aod l8odeg phaseanglescorrepondsto those in a b ght bar and dark bar rcspectively.For that reason,a 0-180 discriminationwill be referred to bright bar-dark bar discrimination while a 90 270deg discrimination corresponds to a l€ft edgFright edgediscrimination.In all the expedments to be reported here, the subjectswere required to make discriminationsbetweenstimuli consistingof L (x) : Ls| + acos(2rfx)+ b cos(2r2fx + 0)l fundamentaland secondharmonic that differed in where Z is meaDluminance,I is the fundamental relativephaseby 180deg.That is, in any givenblock frequency(2 c/deg),and d is the phaseangle of the of trials, the observerhad to discriminatebetween secondharmonic.When 0 is 0 deg, lor example,the two phaseangles,dr and 0r, where0r is the basephase fundamentaland secondharmonic are in +cosine angle,and 0, : 0r + 180 deg. phase.This corrcspondsto the phase relation in a Figure 3 is a convenientrepresentationof the bright bar. At 180deg, the components are in various phase relations betweenfundamental and -cosine phase.This correspondsto the phase r€- second harmonic that were investigatedin these lation of a dark bar. qxperiments. Any my represents one particularphase Figure 2 shows the luminance proliles and the relation, whose value is the angle betweenthat ray componentsfor these phase angles. The 90 and and th€ horizontalpositiveaxis.Thus in any block of 270deg phaseanglesin our two-compolentgratings tdals, our subj€cts had to disc minate between phasesthat are representedby rcys pointing in di ametricallyoppositedircctions. o'\'\A^A-\.$ !-\oeirlue 18o"f,AtATt v+#rl?\ GENf,RAL Mf,THODS Al stimuli usedin theseexperiments werceithersingle vertical sinusoidal $atings or a sum of two such ,d^-.o^"^v^-vrA?zr'.',,logratings.The two componentsconsistedof a fundamental (2 c/deg) and its secondharmonic (4 c/deg). Stimuli weregeneratedby a PDP 1U10computerand displayedby z-axis modulation of a high frequency rcster on the faceof a Tektronix 604 CRT with P31 phosphor,at a frame mte of 200Hz. Details of the display systemare describedby Watson (1979). The observer,whoseheadwasstabilizedby a chin Fig. 2. Examplesof stimuli used in this seriesof experi- rest,viewedthe screenbinocularlyfrom a distanceof 228cm with natural nuoils. The screensubtended ments, zzo"\"{-ff \* r.r'^;6 tffir\tr ,,u'fte\A\ ,"fl#/v_flr 336 DAvrD J. FIELD and JAcoB NACHMIAS A tumaround refem to a decrcment in contrast following a prcvious increment,or vice versa.After the fust two turnarounds,the stepsiz€was changed to 2 dB (a factor of 1.26).After the l2th turnaround, the block was terminatedand the m€an contmst at the last 10turn aroundswasrccorded.This meanwas taken as the estimateof the contrast level yielding 79% respons€scorect on that block. Excluding - -2A warm-up, a block consistedof about 50 t als. The experimentswere completedover a period of approximately 3 months. Although both subjects were expedencedin detection and discrimination experiments,each was given practice runs at each condition until their data reachedthe asymptotes 26 repoded here. o 9 0 Data were discardedwhen the presentedcontrast (deg ) BosePhoseongle exceeded th€ display's linear range or where the Fig. 4- Discriminationthresholdsfor 180deg phase a predifference as a functionof basephaseangle Error bam variance in turnaround contrast exceeded rerun. Data were : value. These blocks determined (r? 4). Seetextfor details. rcpresent t I SE points in subsequentfigures representthe mean of one or more blocks.When more than two blocks on 2.5deghorizoltally, and 1.9degverticallyat the ey€ a givencondition wererun, the standarderror of the It wassurroundedby an 8 deg dia circularsurfaceof mean is shown. a b o u rt h e s a m ec o l o ra n d l u m i n a n c(el 5 c d r m ' ) . There were two obseNers throughout the inT EXPERIMtrNT to normal \estigalion.Both had normalor corrected The first experimentinvestigatedthe subject'sgenvision. D.F. was the first author, and L B. was an purposes of the ability to discriminate180degphasedifferenc€s eral undergraduatewho was naiveto the with the contrast ratio of fundamentalto second experiment. thrcsholdestimateswere obtained harmonics€tat I : 1.On a giventdal, two stimuli (e.g. Psychophysical in all experimentswith a versionof the two temporal representingphaseanglesof 45 and 225deg),one in interval, forced choice (2AFC) staircaseprocedure. each inte al, were presentedto the subject. The correspondingto Each trial was initiated by the observerand con- subject'stask was to pressa button "correct" stimulus. The "correct" stimulus alsistedof two 250msecobservationintervalsmarked the by tones. Contrast was modulated t€mporally ways had a phaserelation found in th€ upper two r\ith a raised cosine profile to minimiTe any quadmnts of Fig. 3. The incorrect stimulus had a on-off transients. Contrast is defined by phaserelation found in the bottom two quadrants. (f-"* - a.i"y(f-,, + l.i") where,-", and Z.,n repre' The minimum amount ol contrast required to dissent respectivelythe peaksand troughsof the com- criminate the two stimuli was determinedusing the ponentwaveforms.From trial to tdal, the positionof two alternativeforced choicestaircasemethod. the gratingrclativeto the screenwasvariedrandomly Resuhs over a range of 0.5deg (one period) to prevent Figure 4 showsthe resultsof Experimelt 1. The discriminationon the basisof absolutephase. Depending on th€ €xpedment, stimuli were first point to be noted,is that 180deg discrimination presentedeith€r in one or in both obseryationint€r- is at leastpossibleat all phaseargles.And although "correct" stimulusor 90-270 (edge) discrimioation requires somewhat vals.The subjectidentifiedthe interyalby pressingone of two keys.After eachtrial, more contrast than 0-180 (bar) discrimination,the auditory feedbackwasprovidedas to the corectness thresholdsas a function of basephaseangle appear result€din a roughly comparable.The data also appear symof the r€sponse.Threecorrectr€sponses response metrical about 90 and 180deg.The intent of this inco[ect while one reduction in contrast rgsultedin an equivalentincreasein contrast.The test experiment,however,was to set a baselinefor later stimulus,whosecontrastwascontrolledby the stair- experiments,so we postponefurther discussionsof case, was mndomly presentedin one of the two thesedata. The observers'phenomenologicalreports ilr this observationintervals. A block of trials proceededas follows.The subject experimentarc also of interest.Thcy reportedto be began in a warm-up mode with incrementsand usingone oftwo strategiesfor all the discriminations. decrcmentsset at 8 dB (changingcontrast by lactor Although thesereports should not be given undue of 2.5). After the subjectwas familiarizedwith the weight, it is interestingto note that thesestrategies task, the subjectpresseda button startingthe exped- roughly correspondto what one might expectfrom a "Mullerian"notion of edge and bar-selectivedemeotalblock.The block consistedof 12turnarounds. Phasercversal discrimination tectors (Watson and Robson, 1981).That is, the reportsparallelwhat onemight expectifthe phenomenal appearanceof the barely discriminablepatterns dependson the relativ€activity of thesedetectom.ln the neighborhoodof the 90-270discriminatiol, subjects reportedusing a strategybasedon the relative left right asymmetryofthe stimuli (edgestrategy).In other regions,subjectsmaintaired that they used a strategybasedon the "lightness"ofthe wide bar (bar strategy).The intriguing point is that the observers reported using only one or the other of thesetwo strategiesfor eachdiscdmination. EXPERIMf,NT 2 The secondgxperimentwas an attemptat isolating the differentprocesses believedto be responsiblefor the data of Experiment1. If only a limited number of "detectors"wereinvolved,then it might be possible to separatethe activity of one set from that of another. Our working assumptionis that there ar€ only four classesof detectorinvolvedin thesephase discriminationtasks (cosine,-cosine, sine, -sine). Thesemechanismsrespond optimally to the phase relationslabeledA, E, C, and G in Fig. 3. On theseassumptions,considerthe task when the subjectis requiredto discriminatestimuli in brightbar phase(0deg or +cosine) from stimuli in darkbar phase(180degor -cosine). Such a discrimi nation would best be performed by comparingthe output of channelsoptimally sensitiveto thos€phase relations. However, with only the four channels describedabove,therewould be two waysof making a 45-225 discrimination. One could look at the output of the cosine (bar) system;a 45 deg phase relation stimulusproducesmore output in a bdghtbar channel (+cosine) while 225deg stimulus produces a greater output in a dark-bar channel ( cosine).Or one can look at the output of the sine (edge)system;a 45 deg stimulus producesa greater output in the right-edgechannel (+sine) and a 225deg stimulus producesa greater output in the left-edgechannel( sine). A similar argumentcan be made for a 135-315 discrimination.Either systemcan make the discrimination. However,if we interlacethesetwo types of discrimination(45 225 and,135 315),we may force the observerto basehis decisionson the information in one systemalone. Considerthe casein which either pair of stimuli can occur randomly on a given trial. That is, on a giventrial eitherthe 45 225pair or the 135 315pair are equally likely to occur throughout the staircase. Also assumethat the 45 stimulusand the 135stimulus are designatedas the "conect" stimuli in their pair. The sinesystemwill haveno diffculty respective making this discrimination.The corect stimulusof each pair produces the same output in the sine system. Therefore, if discdmination of 45 225 rs ,t. activity of the sine system,adding the "t:,:.:: 331 second set of stimuli should not be detrimental. However,for th€ cosinesystem,this interlaceddiscriminalionuould be impossibler thereis no consistent information available. The 45 deg "correct" stimulus produces the same output in the cosine systemas the "incorrect" 315 stimulus.The cosine systemcannot differentiat€betweenthesestimuli and can thereforenot differentiat€consist€ntlybetweena corr€ctand an incorrectstimulus. With suchan interlacedstaircase, whereeitherpair of stimuli is equally likely, we should be able to see the activity of the sine systemunconfoundedby the activity of the cosinesystem.In a similar way, we should be able to mak€ the sine information irrelevent.To do this, we simplychangethe correctstimuli from 45 and 135 to 45 and 315. Now the cosine systemshould find no dificulty with the discrimination, but the sine systemwill find it impossible. If the visual systemco[tains only the four typesof phasesensitivechannels,then a procedurethat requiresa discriminationof 90 I 0 from 270j 0 should allow one to isolat€the activityof the sinesystemand a procedurethat requiresa discriminationof 01d from 180:l d shouldisolatethe activity of the cosine system.Furthermore,the combinedactivitiesof the two systems(sineand cosine)shouldaccountfor all the data from Experimentl. Blockswith interlacedsetsof stimuli suchas those describedabove were pr€sentedto the two subjects under the sam€conditionsas thoseof Experiment1. Each setof four stimuli was chosenso that eachpair was symmetricabout the sine and cosineaxis. For example,the complementof the 100 280deg pair is the 80 260deg pair. The contrast ratio of fundamental to secondharmonicwas kcpt at I tl, and the staricasevaried the contrast of both pairs simultaneously. That is, a total of three successive correct responses on eitherpair of stimuli produceda decrement in contrastfor both pairs. As shownin Experiment 1, contrast thresholdswere rath€r symmetic about base anglesof 90 and 180deg.Making the contrastsof both pairc equivalentshould therefore havelittle or no effecton the outcomeof Experiment 2. Results The trianglesin Fig. 5 show the resultsfor the conditionswhen sineinformation is madeirrelevant. On our assumptions,th€sedata should reflect the activity of the cosine(bar) syst€m.The new data are plottedalongwilh lhoseof the preriouserperimenl (dashedlines).The symmetryof the trianglesaround a base phase of 0 deg is due to the nature of the experiment.Each pair of points representsthe contrast threshold for discriminatingphase angles of (0 t 0) from (180t 0) phaserelatiors[e.g.(45,315) from (225,135)1. The points are repeat€dto allow for comparisonwith those of Expedment l. The circles show th€ resultswhen cosineinformation is irrelevant, which theoreticallyshouldrcvealjust the activ- 338 DAvrD J. FrELD and JAcoB NAcHr!,tras -26 o 90 t80 Bose phoseongle{deg} Fig. 5. Discriminationthresholdsin ExperimentIL Dashed lines representthe results of Exp€rimentI. Circles and triangles representcosine i elevant and sine ilrelevant conditions.respectively. SE averagedlessthan 0.051o9units for both subjects(fl = 4). Here we attempt to seeif mechanismsother than thoseof our four channelmodel are involvedin the discriminationof phasedifferences. This experiment is similarto Experiment1 in that it involvesa 180deg phase-shiftdiscriminationfor a numberofbasephase angles,eachbaseanglebeingpresentedin a separate staircase. The differenceis that in this experiment,the contrastof the fundamentalis kept constant(at l% contrast),while the staircasecontrolsthe contrastof the secondharmonicalone.In other words, her€the thresholdrcprcsentsthe minimum contrastof second harmonic required to make a 180deg phase-shift discrimination. For the conditions of this expedment,our four channelmodelmakesquite simplepredictions,which are illustratedin Fig. 6(b). Thesepredictionscan be understoodmost easilyif w9 representthe contrast and phase of the second harmonic in our twocomponentstimuli on polar coordinatessimilar to thoseof Fig. 3. To understandthe coordinat€sin this and subsequent figures,recallthe trigonometricidentity cos(, - ,) : (cosu)(cosa) - (sinu) (sinr). From this identity, it follows that a secondharmonic of contrast D and phase a\gle 0, b cos(2\t2fx+ 0), can be decomposedinto orthogonalcosineand sine compon€nts,[cos(2lt2fx) and,sirQn2fx)] of contmsts D cos0 and -6 sin 0, respectively.Thesecosineand sinecomponentsare the "x" and ')" coordinatesof the polar plots in Figs 6 and 8. On thesecoordinates, the l€ngth of the line joining ary point in the graph to the origrn represents the contrast,,, of the second harmonic,and its phaseangle,d, is givenby the angle the line makeswith the Jr-axis. It carrbe easilyshownthat so long as the contrast ofthe secondharmonicremainssmallrelativeto that of the fundamental,then for any phase angle, the peak responseof + cosine and - cosine selective channelsoccursat or very n€ar the peak and trough, resp€ctively, of the fundamental.At theseplaces,the amplitude of the sine component of the second harmonic is zero, and hencedoes not contribute to the peak responseof thesechannels.On the other ity of the sine systemunconfoundedby that of the cosine system.Again, the symmetry is due to the pairing of stimuli in th€ experiment(i.e. 9010 vs 270+ 0). The primary point to be noted is that the channels isolated in this manrer appear to be sumcient to handlethe data of Expe ment l. In other words, if the proceduredescribedhereis succ€ssful in isolating the cosineand sinesystems,then it would seemthat thesetwo systemsare sufficientto account for the thresholddata of Experiment1. This doesnot mean that someother set of channelscenteredabout some other set of phaseangleswould not also accountfor the data. At this point we simply concludethat the four channels are sufficientIt might seemsurprisingthat this complexdiscrimination was even possiblefor the observels.In fact they found the task rath€r easy.Apparcntly,the two i isolated systemsmap onto the two stmtegiesdea scribed earlier. When discdminations are forced along the 90-270(sine)axis,the subjectsadoptedthe edgestrategyand when discriminationswere forced n along the 0 180 (cosine)axis, the subjectsadopted p the bar strat€gy.It appearsthat Experiment2 tapped 3 the natural categodesused by the subjects. Unfortunately,this ph€nomenologydoesnot proo o % ContEstoi Cosin€Component vide adequateevidencethat the four channelsare necessadlyinvolved. Experiment 3 is directed to- Fic. 6. Pr€dictedthresholdsfrom two models of phase wards a more rigorous answerto this question. disc mination. Seetext for details. Phasereve$al discrimination Our modelconsidersonly sineand cosine-selective hand, the cosinecomponentof the secondharmonic, dependingon its sign,will eitherincreaseor decrease channels,so discriminationbetweena pair of stimuli the amplitudeof the peak responseof the + cosine differingonly by a 180deg phaseshift of the second channel while having the opposite effect on the harmonicrequiresa sumcientoutput differencein at - cosinechannel(e.g.increasingthe + cosinecom- leastone ofthe two pairs ofchannels.Therefore,this ponent will increasethe responseof the +cosine model predicts that the results of Expedment 3 the responseofthe -cosine should fall along a rectanglewhose sidesare equi channelwhile decreasing distant and parallel to the i( and l-axis. channel). Altematively,supposetherewerefour channelsbut For th€ sine-selective channels,the role of the sine and cosinecomponentsare reverced.Odd-symmetdc eachselectivelysensitiveto someother phaseangle, receptivefieldsintroducea 90 degphaseshift between say 45, 135,225 and 315deg.With such a set of responsesto fundamental and second harmonic. channels,we shouldfind thresholdsthat fall along a Therefore,the rcspolse of a +sine-selectiv€lwhich rectangleat a different orientation to the axes,as has receptivef,elds like thos€ in Fig. l(c)], to a shown in Fig. 6(a). Experiment3 should be able to stimuluswhosesecondharmonicis in 90 degphaseis diferentiate betweenthesetwo models. the same as the responseof a +cosine-selective channel[e.g. Fig. l(a)] to a stimuluswith a second Results harmonic in 0deg phase. Consequently,the peak The resultsof this expe ment are shown in Figs 7 responseof the sineselectivechannelswill be affected and 8. In Fig. 7 they are plotted on rcctangular by the sine componentof the secondharmonic,but coordinatessimilar to those of Fig. 4, that is, log remain unaffectedby its cosinecomponent. thresholdcontrast of the secondharmonic vs base Therefore,all stimuli which have a given amount phaseangle.As in Experimentl, thresholdcontrast ofcosinecomponentof the secondharmonic,regard- variessystematically with basephaseangl€,having a lessof the amount of sine component,will produce maximumnear 90 degand a minimum in the vicinity a constant differencein the peak outputs of the of 0 deg. In other words, discriminationbetween0 +cosine and -cosine-selectivechannels.Suchstim- and 180deg requiresabout 0.4 log units lesscontrast by a pair of vertical linesthat than between90 and 270.Presumablythis meansthat uli can be represented are equidistantfrom thel-axis, asshownin Fig. 6(b). the cosinesystem,which on our hypothesismediates These same stimuli produce zero differencein the the former discrimination,is more sensitivethan the peak outputsof the sin€-selective chanlels.Similarly, sinesystem,which mediatesthe latter discrimination. all stimuli which contain a given amount sinecomThe error bars in Fig. 7 are roughly the samesize ponent, regardlessof the cosine component, will as thosein Fig. 4, even though they are basedon 8 producea constantdifferencein the peak outputs of rather than 4 blocks of trials. Evidently when only the sineselectivechannels(and zero differencein the one frequency component is varied, the relevant cosine-selective channels).Suchstimuli can be repre- psychometricfunction is shallower,leadingto morc sentedby a pair of horizontal linesequidistantfrom va able thrcsholdestimates. the -Y-axis. In Fig. 8, the resultsof Experim€nt3 are replotted on polar coordinates for easier comparison with theoreticalexpectationsdepictedin Fig. 6. Note that data pointsfor differentbasephaseanglesare plotted in reverseorder in th€ two f,gures:e.g. the poilt plotted on the positive horizontal axis in Fig. 8 correspondsto the point plotted above 0 deg base phaseanglein Fig. 7. Consistentwith the predictions of our four channelmodel,the resultsof Experiment 3 fall along a rectanglewhosesidesare parallel and equidistantfrom the x- and l-axis. Th€ data ar€ inconsistentwith a four channelmodel whosechannelsare selectivelysensitiveto any otherphaseangles. However,theseresultsalsosuggest an evenstronger conclusion:no channelsother than those of our model play any role in theseexperiments.For example, considerthe point in Fig. 8 which representsthe -30 threshold contrast for the 45 vs 225deg discrimio 9 0 a o nation. Supposethat this discriminationwere mediphose ongle {deg) Bose ated by channels selectivelysensitiveto 45 and Fig. ?. Log contrastthresholdof the secondharmonicfor 225deg.In that case,all points along a line perpenl8odeg phase shift discriminationas a function of base joinilg the plotted point to the phaseangle.Contrastofthe fundamentalwasheld constant dicular to the ray at l% for both obs€rvers. Error barsrcpresentf I SE in log origin would be equally discriminablebecauseall thesepoints would producethe sameoutput in such contrast(r = 8). 340 DAVD J. FrELD and JAcoB NAcHi,trAs even-symmetric [or odd-symmetricas postulatedby Stromeyer and Klein (1974)1,whose output is a "neural image" resultingfrom the convolutionof the retinal image with the receptivefield profile. Phase discrimination and phenomenalappearancewould then depend upon application of different computational algorithms at later stagesof the visual E system:for example,comparison of the distances betweena peak and the nearesttrough to the left and to the right of the peak, or comparisonof peak and trough amplitudes.It is not entirelyobviousto what cxt€nt this accountis actuallydiffe.ent from that of the four chann€lmodelwe havebeenadvocating.The consequenc€s of postulating a limited number of computational algorithms are similar to those ol postulatinga limited number of receptivefield types. Finally, it should be stressedthat the evidence favoring a drastic limitation of either kind, comes from a mther modestvarietyofexperiments.Perhaps -a2a ooo o.25 other receptivefield types(o! other algorithms)exist Fig.8. Polarrepresentation of dataof Fig. 7 (Experimentbut werc simply rlot tappedby theseexperiments.A 3).Dashed linesrepresent predictions. theor€tical Errorbars more vigorous search for them is currently in repres€nt tl SEin contrast(, =8). Seetextfor details. progress. Acknottledgement-lh|s research was supported by NEI a systemof channels.Neither observer'sresultsseem traininggrant Ey 02035for res€arch in vision to the to lie along such a line. A similar argumentcan be Unjversityof Pennsylvania and by NSF grant BMS used to disconfirmthe possiblerole in theseexperi- 80-08669 to the second author. ments of any other phase-sensitive channels,other than those postulatedby our four channel model. RIFERENCIS DISCUSSION The psychophysical and phenomenological observationsreportedin this paper are all consistentwith a four channelmodel basedon the earlier work of Shapelyand Tolhurst (1973),Kulikowski and KingSmith (1973),and Tolhurst and Dealy (1975).In ord€r to account for the discdmination between fundamental/secondharmonic complex gratings differing by a l8Odeg phase shift of the second harmonic,it is sufficientto assumethat (a) thereexist four classesof broad-banddetectors.two with even symmetricreceptivef,elds (bdght-bar and dark-bar selectivedetectors) and two with odd symmetric rec€ptiv€ fields (left-edge and right-edge selective detectors),(b) discriminationrequiresthat therebe a sufficient differencebetween the peak outputs of eitherthe two classesof bar-selectivg detectorsor the two classesof edge-selective detecton, (c) phenomenal appeamnceof the barely discriminablepatterns dependson th€ relative activity of thesedetectols. Furthermorc, considerationof the nature of the symmetryof the obtained resultsseemsto rule out alternativefour channelmodels,postulatingreceptive fields of neitherevennor odd symmetry. Th€ r€sultsat hand do not, however,excludeother possibleexplanations.For example,therc could be only one class of broad-band receptivelield, say Atkinson J. and CampbellF. W. (19?4)The efiectof phase on the perceptionofcompound gratings.Vision Res.14, 159-t62. BurIrD. C. (1980)Sensitivityto spatialphase.Visio Re:.20, 391-396. Campbell F. W. and Robson J. G. (1968)Application of Fourier analysisto the visibility of gratings.J. Pr-|,riol. r9, 551 566. Graham N. (19801In Vkual Coding and Adaptability (Edi,ted by Harris C. S.). Lawr€nceErlbaum, Hillsid€, NJ. Graham N. and NachmiasJ. (1971)Detectionof grating patterns containing tuo spatialfrequencies: a compariron of single-channel and multi-channelmod€ls. I/,sro, Rer. lt, 25t-259. Kulikowski J. J. and King-Smith P. E. 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