,/'r'b,i Rer. Vol.24, No.4. pp. 333 340, 1984
Printedin Great Brirain. All righrs reserved
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CopyrishtO 1984Pergamon
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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
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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
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two classesof edge-selective
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