Stereopsis from contrast envelopes
ABSTRACT We report two experiments concerning the site of the principal nonlinearity in second-order stereopsis. The first exploits the asymmetry in perceiving transparency with second-order stimuli found by Langley et al. (1998) (Proceedings of the Royal Society of London B, 265, 1837–1845) i.e. the product of a positive-valued contrast envelope and a mean-zero carrier grating can be seen transparently only when the disparities are consistent with the envelope appearing in front of the carrier. We measured the energy at the envelope frequencies that must be added in order to negate this asymmetry. We report that this amplitude can be predicted from the envelope sidebands and not from the magnitude of compressive pre-cortical nonlinearities measured by other researchers. In the second experiment, contrast threshold elevations were measured for the discrimination of envelope disparities following adaptation to sinusoidal gratings. It is reported that perception of the envelope’s depth was affected most when the adapting grating was similar (in orientation and frequency) to the carrier, rather than to the contrast envelope. These results suggest that the principal nonlinearity in second-order stereopsis is cortical, occurring after orientation- and frequency-selective linear filtering.
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ABSTRACT: We recorded the initial vertical vergence eye movements elicited in monkeys at short latency ( approximately 70 ms) when the two eyes see one-dimensional (1D) horizontal grating patterns that are identical except for a phase difference (disparity) of one-quarter wavelength. With gratings composed of single sine waves, responses were always compensatory, showing Gaussian dependence on log spatial frequency (on average: peak = 0.75 cycles/deg; SD = 0.74; r(2) = 0.980) and monotonic dependence on log contrast with a gradual saturation well described by the Naka-Rushton equation (on average: n = 0.89; C(50) = 4.1%; r(2) = 0.978). With gratings composed of two sine waves whose spatial frequencies were in the ratio 3:5 and whose disparities were of opposite sign (the 3f5f stimulus), responses were determined by the disparities and contrasts of the two sine-wave components rather than the disparity of the features, consistent with early spatial filtering of the monocular inputs before their binocular combination and mediation by detectors sensitive to disparity energy. In addition, responses to the 3f5f stimulus showed a nonlinear dependence on the relative contrasts of the two sine waves. Thus on average, when the contrast of one sine wave was 2.3 times greater than that of the other, the one with the lower contrast was largely ineffective as though suppressed, and responses were determined almost entirely by the sine wave of higher contrast: Winner-Take-All. These findings are very similar to those published previously on the vertical vergence responses of humans, indicating that the monkey provides a good animal model for studying these disparity vergence responses.Journal of Neurophysiology 10/2008; 100(5):2907-18. · 3.30 Impact Factor
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ABSTRACT: There is a long history of research into depth percepts from very large disparities, beyond the fusion limit. Such diplopic stimuli have repeatedly been shown to provide reliable depth percepts. A number of researchers have pointed to differences between the processing of small and large disparities, arguing that they are subserved by distinct neural mechanisms. Other studies have pointed to a dichotomy between the processing of 1st- and 2nd-order stimuli. Here we review literature on the full range of disparity processing to determine how well different proposed dichotomies map onto one another, and to identify unresolved issues.Vision research 01/2009; · 2.29 Impact Factor
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ABSTRACT: In this work we propose a first principles dynamic optimization model for an intensive energetically integrated process, a natural gas processing plant, within a simultaneous dynamic optimization framework. We have developed rigorous models, including thermodynamics with a cubic equation of state, for separation tanks, distillation columns, turboexpanders and cryogenic countercurrent heat exchangers with partial condensation. The resulting partial differential algebraic equation system is transformed into an ordinary differential algebraic equation one (DAE) by applying the method of Lines for the spatial coordinate. The high integration between process units as well as path constraints have been efficiently handled by a simultaneous dynamic optimization approach in which the DAE is transformed into a large nonlinear programming problem through orthogonal collocation over finite elements in time and solved with an interior point algorithm. In the case study, we maximize ethane recovery under a ramp change in natural gas feed flowrate. The model provides temporal and spatial profiles of controlled and manipulated variables that are in good agreement with plant data.01/2010;
Wer ep orttwoexperimentsc oncerningthesiteofthe principalnonlinearity in
second?orderstere opsis?The?rst exploitstheasymmetry inperc eivingtr ansp arency
with sec ond?or derstimulifound byL angley etal????????i?e?? the product ofa
p ositive?valuedc ontr astenvelope andame an?zeroc arriergratingc anbe se entrans?
parently onlywhen thedisparities areconsistent withthe envelopeappe aringin
fr ontofthecarrier?Wemeasuredtheenergy atthe envelopefrequencies thatmust
be addedin orderto ne gatethis asymmetry?Werep ortthatthis amplitudec an
bepredictedfromthe envelopesideb andsandnot from the magnitudec ompr essive
pr e?cortic alnonlinearitiesme asuredby otherr ese archers?In thesec ondexperiment?
c ontr ast threshold elevations were measured forthediscrimination ofenvelope dis?
p arities followingadaptation tosinusoidal gratings?It isr eported thatperc eption
oftheenvelope?sdepthwasa?ectedmost whentheadapting gratingwas similar
?inorientation andfre quency?to thec arrier?rather than to thecontr astenvelope?
Theser esultssuggest thattheprincipalnonlinearityinsecond?order stereopsisis
cortical?occurringafterorientation?and frequency?sele ctivelinear ?ltering?
KeyW ords? Second?orderStereopsis?MultiplicativeTransparency
Portionsofthisresearchwerepresentedat ECVP????and????andHibbard?sPh?D Thesis?
Department ofPsyc hology? UniversityCollege London? London?U?K?
Department ofComputing andInformationScience?Queen?sUniv ersity? Kingston?Canada?
ular band?pass signals ?e?g??Cormack?Stevenson andSchor?????? DeAngelis?Ohzawa
andF reeman?????? Mallot?ArndtandBultho?? ?????Fleet?Wagner andHeeger? ??????
But there areclassesof signals?often callednon?Fourieror second?order?? thepercep?
tion ofwhichis notw ellcapturedbythese mo dels?Examples includetextureboundaries
?Frisby andMayhew???????motionboundaries?Halpern???????andcontrast env elopes
?Liu?Schor andRamachandran??????Sato andNishida? ?????????? HessandWilco x?
?????WilcoxandHess?????? ????? ?????FleetandLangley?????b? Hibbard?Langley
andFleet?????? LinandWilson? ??????
Thereare several plausiblemodelsfor second?order visualprocessing? allofwhichinvolve
animp ortantnonlinearityat somestage?Thereare single?channelmo delsthatinvolvean
early? pre?corticalnonlinearity ?e?g??Burton???????Suchmo delshavereceived attention
in the context ofsecond?ordermotion?e?g?? Bro wn??????Scott?Samuel andGeorgeson?
?????? Accordingto this mo del?thenonlinearityin troducesa distortion productinto the
visual signalat thefrequenciesof the contrastenv elope?Derrington and Badcock? ??????
Thisdistortionproductwould thenbe processedbya conven tional ?rst?ordermodel of
binocular matc hingto deliv erthedisparityinformation presentin the second?ordersignal?
There isincreasingsupp ort?how ev er? foratw o?c hannel model?Thetw o?c hannel models
used to explainstereopsisresem blethoseforwarded invisual motion?with aseparate
c hannelforpro cessing the second?order signal?Chubband Sperling? ?????T urano and
Pan tle??????Victor and Conte? ?????Wilson?F errera?andY o? ?????Zanker?????? Fleet
andLangley?????a? Langley? ??????????? These modelsdi?erin severalways?including
boththesite andthe mathematicalform ofthenonlinearity?Manymodelssupposea
corticalnonlinearity?sothatthevisual signalisprocessedbylinear?lterstuned tospatial
nonlinearityisimplied?p erhapsowing to thecompressivenonlinearity of neuronsfound
in theLGN ?Sclar etal?? ????? Scott?Samueland Georgeson? ??????Similarissuesexist
concerningthe perceptionofmonocularcon trastenvelop es ?Burton?????? Henninget al?
?????Derringtonand Badcock??????Langleyetal??????? Mareschal andBak er? ??????
andin thecon text of second?order motion?Chubband Sp erling??????Wilsonet al??????
Wilcox andHess ??????posited thatthenonlinearityoccursbeforethebinocular com?
bination ofmonocular signals? They alsoshow edthatsecond?orderstereopsis couldbe
di?cultwhen thecarrier orien tationsin contrast?modulated inputswhere di?erent? sug?
gesting thatorien tation?speci?c processing precedes thenonlinearity? Incon trast?Langley
etal? ???????seealsoLiu etal?? ????? havesho wnthatbino cular depthperception from
con trastenv elopes isp ossiblewhen carriergratings di?er signi?can tlyin spatial frequency?
Thereasonsforthis discrepancycouldre?ect di?erences inthe paradigmsandstim uli
This paperaddresses thesiteof thenonlinearitywithtwo experimen ts?The?rst exploits
theresults ofLangley etal??????? who examineddisparitythresholds for sums??rst?
order?and products?second?order?of binocular gratingslike those usedhere?seeFig? ???
Theyfoundthat?whenview edbinocularly ?the sumoftwo ??dgratingsof di?erent spatial
frequencyand orientation maybeseen transparently?Eithergrating canbeperceiv edin
frontof orb ehindtheother?dependingon the disparitiesof theresp ectivegratings?When
thesametwogratingsweremultiplied togethertransparencyalsooccurred? but therewas
an asymmetry? Thecon trastenvelopewasonlyperceiv edina separate depthplanewhen
infront of thecarrier? itw as nev erseenas theunderlaying surface?
Toexplain thedepthasymmetryLangleyetal????????seealsoKersten? ?????appliedthe
thetransparentsurface?Ifoneassumes that these constraintsin?uencebino culardepth
p erception?then theasymmetrycanbeexplained?
Metelli?sconstrain tsalso predictthat the product of thetwopositive?v aluedsignals used
in ourstudy maybep erceived symmetricallyindepth? eithersignalmaybeseenin?frontof
orbehindthe other? Thisisb ecauseb othsignalssatisfyMetelli?s mono cular constraintson
transparencysothatamono cularcue couldnot be usedby the visualsystemtoin?uence
binoculardepthp erception?see alsoKersten???????Thiscase?consistent withthe results
of Langleyetal???????? isexploitedinthe ?rstexperiment here?In particular?onecan
addpow er atthe env elopefrequencies toa second?ordersignal tocreatea stim ulusthat
is equal toapro ductofpositive?valued gratings?andisthereforeperceiv edsymmetrical ly
transparen t?Here?wemeasured theamplitudeofadditive signalatenv elope frequencies
thatw as requiredtoov erridethedepth asymmetry?Wethen comparedthese thresholds
tothose predicted fromtwomo dels?namely?asinglechannel model withacompressive
nonlinearit yas measuredinother studies ?Brown? ?????Scott?Sam uelandGeorgeson?
??????anda modelthatdepends ontheenergy ofthe second?ordersignal?
We report thatthis amplitudemaybe predictedfrom thesumof thecontrast env elope?s
ratherthan thethat expectedfrom the magnitudeofpre?corticalnonlinearities
teststim ulus? Ifthe site ofadaptationw asafterthenonlinearity?thenonewouldexpect
Thesidebandsare thefrequencies on eithersideof thecarrier frequencyatwhic hthereisnonzero
power? Thesesidebands gro winamplitude withthedepthofcontrast modulation?
quencyand orientationoftheadaptinggratingwerecloseto thecarrierratherthan tothe
env elope?Moreov er? theadaptationwasorien tation andfrequencysp eci?c?From these
data?weconcludethatthe site ofthe nonlinearityo ccursin thecortex?after orientation
andfrequency selective ?ltering?
Three subjectsw ereused for each experiment? Twosubjectsw ere authors?The thirdsub?
ject didnotknowthe purpose of theexperiments? All subjectshad normal orcorrected?
Binocularimageswere presentedonaSONYmonitorwitharefreshrateof??Hz and
??? greyscales?Eight?bit quantization? althoughcrude?was thoughtsu?cientb ecause
con trastthresholds for thediscriminationofenvelope disparityin our exp erimentswere
t ypically closeto???Theluminance ofthe monitorw as linearisedby takingluminance
measuremen tswitha photometer?towhicha logarithmiccurvew as ?ttedand alinear
lookup tablegenerated?The residual errorfrom the?ttedcurvewasnomorethan????
oftheluminance atany one ofthesamplep oin ts?Themean luminancew as????cd?m
Experimen tsw ere carriedout inadarkened room?The only visibleilluminationoriginated
The basicsecond?order stimuliwerecontrast?modulatedsinusoidal gratings? Someexam?
ples are showninFig??A? The envelop e?
???mE?x?y ???w asanapproximation toa
square?wave gratingformedbysummingthe?rst andthird harmonics? Here?m isthe
depth of modulation ?b etw een?and ??? andE?x?y?isgivenby?
where? isthefundamen talfrequency ofthe env elope? and? isc hosensothatE oscillates
Given the envelope anda carrier?C?x?y ?? themathematicalform oftheleft andright
contrast?mo dulatedtest stimuli is?
where? isthemeanluminance?adenotesthecontrastofthecarrierprior to thecontrast
modulation?andd is thep ositional disparity?Thestimuliwerevisibleonlywithina
circularwindowassho wninFigure?? For Experimen t??the surroundsofthe circular
windo wwere blac kas inFigure??InExperimen t??the surrounds were equaltothe
meanluminance?For allthese experiments? C?x? y?was ahorizontalsin usoidalgrating?
Its spatialfrequency waseither????or??? cycles per degree?cpd??
Figure ??Examplesofthebinocularstimuli? ?A?A multiplic ativecombinationofa
vertic allyorientedcontr astenvelopeandhorizontalc arriergr ating? ?B?Each imageshows
anadditivec ombinationoftwogr atingslikethosein ?A??Cr oss?eyed fusion yieldsa
transpar encyinfr ont?left?c enterpair?? orbehind?c enter?rightp air?thehorizontal grating?
Subjectswereseated withtheir heads stabilizedinac hinrest in front ofthe Wheatstone
stereoscop e?They resp ondedusingacomputer mousein forced?choicediscrimination
tasks?In Experiment? sub jectsw ere asked torespondas quic kly asthey could?but
werenot constrainedbythe viewingtimeotherwise? InExperiment ?? theadaptation
exp eriment? thetest imagewaspresen tedfor ???msec followingthe adaptinggrating?
receptorequation ?Scott?Samuel andGeorgeson???????Asdiscussed inAppendixA?also
see Burton??????Henninget al???????Brown?????Scott?Samueland Georgeson???????
acompressive nonlinearityappliedtoacon trast?modulated signal intro ducesadistortion
productatthefrequency ofthe contrastenv elope?Thedistortion product is???
of phasewith the envelop e?and its amplitudeat theenvelopefrequency increases asa
functionof mo dulationdepthand thesquareof imagecon trast?
An alternativemo delp ositsthat thestrength of theenvelope?sdisparity signaldepends
isa measureofthe energyinthesecond?order signal? Equation????oftheapp endixshows
thatthismodelpredictsthatthe amplitudeoftheenvelopesignalincreases linearlyas
afunctionof modulation depthand con trast?These predictions aredi?erent fromthe
compressivenonlinearity model because ofthedi?eren tlinearco e?cientsthatgo vernthe
e?ectof modulationdepth?and because ofthe linear versusquadratic dep endenceon
Todecideb etween thetwomodels? weexploitedthedepthasymmetryfoundforcon trast
en velopes ?FleetandLangley ?????b?Hibbard etal??????? Hibbard??????Langleyet al??
?????? Wemeasuredthecontrastof anadditiv econtribution of E?x? y?that wasrequired
to override thedepthasymmetry ?so that E?x? y?could beseentransparen tlyat adi?erent
???and????butwithanadditive contributionof E?x?y??
???mE ?x?d???y??C?x? y?
power?Thesesidebandsgro win amplitudewiththedepthofcontrastmodulation?
for the thresholdofb? the contrast of thecarrierwas ?xed at????The spatial frequencies
ofE?x?y? andC?x?y?w ere????and??? cpd resp ectively? Notethatwhileadistortion
pro duct intro ducespow er that is ???
outof phasewith the envelop e?the additive signal
here isin?phasewith the envelop e?
Oneach trial? subjectsrep ortedwhetherthecom binedenv elopeandluminancesignal
?referred tob elowastheh ybridenvelop e?wasseenb ehind the horizontalgrating? When
bw assu?ciently large?subjects reported thatthehorizontal carrierw asp erceiv edinthe
depthplaneof the ?xationsp ot?Wurgerand Landy ??????Langley etal?? ??????withthe
h ybridenv elope appearingtransparen tlybehind it?On the otherhand?whenbw astoo
small?sub jectsrep orted a ?at?coheren t?surface?This canveri?ed bycross?ey edfusion
ofFig? ?A?Thedisparit y oftheenvelop ewasuncrossed and?xed at??min utes? Atthis
disparity?with b???allsub jectsreported a?at?coherent?surface?
Within asinglesession?for a?xed valueof m?? thecontrastof the additivesignal?b?
w ascon trolledfromtrial totrial inanadaptive fashionusing APE?Watt andAndrews?
??????Psychometricfunctionswere?tted tothedata?Themagnitudeofbatwhic hthe
abilitytoobserve thevertical structure ata di?erentdepththan thehorizontalstructure?
Proportion Transparent Responses
Figure ???A?Apsychometricfunction fromsubjectJB showstheprobability ofp er?
ceivingtransparencyasafunctionofthecontr ast?b? oftheadded signal?Theenvelope
modulation depthwas????Thethresholdrepresentsthe valueofbrequiredbysubjects
toseetransparency on???ofthetrialssuchthatthe hybridenvelope wasrep ortedto
liebehindthe carrier?Thepsychometricfunctionwas generated usingAPE? Itshouldbe
note dthatduring each session?thenumb eroftrialsat each contrast levelwouldhave been
di?erentbec ause APEis anadaptivemetho d??B? Transp arency thresholdsar eshown as
afunctionofmodulationdepth? Theerr or bars representthe standar derrorof subjects?
???thr esholdsoverthr eedi?erentsessions?
becamemore likely ?Figure?Bshowsthecontrast ofthe additiv esignal that was required
to perceivethe h ybridenvelopebehindthecarrieron???ofthetrials?averagedoverall
threesubjects? Noticethatthis contrastthresholdincreaseswithmo dulationdepth?For
eac hsubject?smeasurements?yieldedslopesof???? ??????KL?????? ??????JB??and
??????????PB?withinterceptsof ????????????????????and???? ?????respectively ?
ofeach subject?smeasurements?w as foundtobe??????????
Eachmeasurementisrep ortedherewithits???con?denceinterv altakenfromatwo?tailedstudent?s
explainssomespeci?cpredictions ofthesetwomo delsthatare usedbelow?
The?rstproblemwiththeearly nonlinearity model concernsthe expectedamplitudeof
sho wninFig? ??withm???? anda?????the contrastofthe env elope?sdistortion
productwas estimatedat??????However?a ????cpdluminancegrating alone atthis
contrast isinsu?cien ttoyield adepth p ercept? Bycomparison?Fig? ?B sho ws thatthe
meandisparit ythresholds whenthe modulationdepthwas zero?whichis a simpleadditive
transparency conditionasshown in Fig??B? was ???? ??????Therefore?theamplitude
ofthedistortionproducts introduced bytheNak a?Rushtonnonlinearit yare belo wthe
con trastthresholdrequired toproduce areliabledisparit ysignalforthestimuliused in
Equation ???? intheappendixalsoshowsthat?accordingtothis formofnonlinearity? the
subjects was ???? ??????Thisvaluedepartssigni?cantlythepredicted valueof??????
Thereforeonemigh texpectthatthereshouldbeavalue ofbforwhichtheycombinede?
structivelyandcancel?In thiscaseonewouldexpectnoresultant disparitycueandaloss
in depthsensation?We foundnocircumstancesinwhich addedluminanceinformation
n ulledthe depthsignalinthe contrastenv elope? Thisw as alsothe casewhen weadded
Gaussian functionsmaynot bedetectedbya?rst?orderprocess?
Insteadoflooking toapre?corticalnonlinearitytoexplain thedata inFig??B?onemight
consideracorticalnonlinearity thatextracts the modulating envelope afterband?pass
?ltering?Withsuchamodel?one canshowthattheamplitude ofthe modulating envelope
isalinear function ofthe amplitude ofthecarrier sidebands?Asdiscussedin theapp endix?
wthatthesidebandamplitudes willv ary linearlywith thecontrast aandwith
the depthof modulation m?In particular?with a????in theexp erimen there? it predicts
thatthe sideband amplitudesshouldincreaselinearlyasa function ofm withaslop eof
??????This ism uch closertothe averageslop eof????foundfromFig? ?Bthantheslop e
predicted bythe earlynonlinearit ymo delab ove?
This view isalsoconsisten twith thepredictions obtainedfromMetelli?s??????mono cular
constraintsontransparency? namely ?that theproduct oftwopositive?valuedfunctions
maybep erceiv edsymmetricallyinfrontof? orbehindeachother?With ourstim uli it is
easyto showthat?when b?
? the stimulusinEquation???isequivalent toa product
con trasta?isv erycloseto thatinEquation ???? that describes theamplitudeofthe
signal?s sideband frequencycomp onen ts?afterthesignal hasbeen compressedbyaNaka?
thatthe depthasymmetryfound forcontrast envelop esoccurssolelybecause ofdi?erent
additiveversusmultiplicative com bination rulesthatwere used toconstruct thestimuli?
Itp oints toour h ypothesisthat binocularasymmetriesre?ectaprop erty of second?order
processing?consistentwithatw o?channel model?
Inoursecond experimen t?subjectswere?rst adaptedtoahigh contrast sinusoidalgrating?
and thenaskedto reporttherelativedepth oftheenv elope inacon trast?mo dulatedtest
stimulus? Thepremisebehind theexp erimentwas that? ifthe siteof adaptationwasafter
the nonlinearity?thenthe en velop esignal would bepresent?asa?rst?ordersignal??and
onew ouldthereforeexp ectthatthemost e?ectiveadaptingfrequencies wouldbecloseto
theen velop e frequency?If thesite oftheadaptation was beforethenonlinearit y?thenthe
en velop esignal remainsimplicitin the sidebandfrequenciesnearthe carrier?Thereforewe
wouldexpectthatadaptationto frequencies nearthe carrierwoulda?ect theperceived
depthoftheenvelopesigni?cantly? In additiontothesetwofactors?iftheadaptation
wasstronglyorientation? andfrequency?speci?c?then thisimplies thatito ccursinvisual
cortex?Inthiscaseonemigh tpredictthatcontrastthresholds?as afunctionofthe
adaptinggrating?sfrequency? wouldsho wthesamefrequencyandorientationdependen t
Toexploretheseideas? weusedteststimulithat werethesameasthosede?nedinEqua?
tions???and????Thedepthofmodulation mwas?xedatunit y?whilethecontrast a
was variedasanindependen tvariable?In theseexperimen ts?thespatialfrequency ofthe
ask edtorep ortwhetherthecontrastenvelopewas infrontorb ehinda?xationspot?This
w asimmediately followedbyatop?upadaptationperiodof?seconds?followedbyanother
teststimulus oneac hsubsequent trial?Thecon trastofthe teststim ulus?ainEquations
???and????wasv ariedusingthe method ofconstant stim uli?
Baseline con trastthresholdsweremeasuredb eforeeachsessionby repeating thesame
taskwithoutanadaptinggrating? Thesedataw ereobtained sothatwe couldcompute
threshold elev ationsafteradaptation asam ultipleof thebaselinethreshold?
Alogistic function?ranging from ???to ??w as?ttedto the data tak enfrom eachsubject
in eachsession? From theresulting psychometric curves? ????correctdisparit ythresholds
were measured?Eac hmeasurementw asdividedbythe ????correctdisparitythresholds
tak enfrom thebaseline tasktoobtain the thresholdelev ations? Eachsession consisted
oftest imagespresented at ??di?eren t contrast levels? ?times each?Eac hsession was
repeatedthree times? yielding?? trials ateac h con trastlev el?
Three sets of sessionsw ere run?
?Inthe ?rst set ofsessions thefrequency of theadaptinggratingwas equal to the
gratingand thecarriergrating?This allow edus toexaminetheimportanceof
orien tationwhenadapting frequencieswereclosetothe carrier? Weusedanglesof
cies?????and???cpd? wereusedtosho wspatialfrequencytuning?Thefrequencies
Relative Orientation Between Test
Carrier and Adapting Grating (deg)
0 204060 80
Spatial Frequency of Adapting Grating
Relative Orientation Between Test
Carrier and Adapting Grating (deg)
Figure ?? ?A?Thresholdelevationsare shownasa functionofthe anglebetweenthe
wasp aralleltotheadapting gr ating?The twocarrier fre quenciesareindicated on the
horizontal axis??C? Thr esholdelevations asafunctionofthe anglebetwe enthe c arrier
and adapting grating?The frequenciesofthe adapting grating and theenvelop e were
e qual? Err orb arsrepr esentthestandar d err ortaken calculate dfr omeach subject?s mean
of theadapting gratingsdi?eredfromthe carrierbyfactors of? ???? ?????????and??
We also usedanadaptingfrequency thatw asequal tothefundamen talfrequency
ofthe env elopeandone thatwas ?octavelo w er thantheen v elopefrequency ?
? Inthethird set?thefrequency oftheadaptinggrating wasequal tothefundamental
frequency oftheenvelop e?Thenwevariedtheanglebetweentheadaptinggrating
and the carrier?using angles of?? ??and ??deg? Because thecarrieranden velop e
are perp endicular?whentheorientationbet weentheadaptinggratingandcarrier