Two-dimensional approach to psychophysical orientation tuning

Vol. 7, No. 1/January 1990/J. Opt. Soc. Am. A 141

Two-dimensional approach to psychophysical orientationtuning

James L. Dannemiller

Department of Psychology, University of Wisconsin-Madison, Madison, Wisconsin 53706

James N. Ver Hoeve

Department of Ophthalmology, University of Wisconsin-Madison, Madison, Wisconsin 53706

Received March 6, 1989; accepted September 1, 1989

Orientation bandwidths of psychophysical channels in the human visual system were inferred from contrastthresholds for a special class of polar-separable, two-dimensional patterns. Detection thresholds for these patternsconformed to a model with linear filtering by orientation-selective channels followed by probability summationacross these channels. The number of channels (6-8 pairs) and their half-bandwidths at half-sensitivity (approxi-mately 170) did not differ at 4 and 10 cycles/degree. The results extend the many one-dimensional, linear systemmodels of vision to two dimensions.

INTRODUCTIONThere is considerable support for the view that the humanvisual system is composed of psychophysical channels selec-tively responsive to both the orientation and the spatialfrequency of image features.' These channels may repre-sent collections of two-dimensional (2D) filters. An impor-tant characteristic of such a channel, or bank of spatiallylocalized 2D filters, is the range of orientations over whichthe individual filters are responsive. Several psychophysi-cal paradigms have been used to infer the angular selectivityof these channels. In general, these procedures rely on thestrategy of altering the sensitivity of a proposed channel bystimulating nearby orientations. Adaptation studies yieldestimates (half-bandwidths at half-sensitivity) between 6.50and 10°.2-5 Subthreshold summation studies yield esti-mates between 150 and 24° when corrected for probabilitysummation 6 (cf. the uncorrected value of 3° obtained byKulikowski7). Masking studies yield estimates from 60 to310.6,812 The differences in these estimates among studiesmay be due in part to the fact that bandwidth estimates varywith the metric chosen to index the threshold elevationeffect in both adaptation and masking paradigms.'3 Thesepsychophysical estimates of orientation bandwidth comparefavorably with the range of bandwidths found in orienta-tion-tuned cells in the primate visual cortex.'4 There areaspects of these psychophysical techniques, however, thatmake it difficult to determine whether these measures re-flect the properties of a single channel.

One interpretive difficulty with techniques such as adap-tation and masking is that a single test grating could con-ceivably produce responses in several channels with overlap-ping tuning functions. The use of an off-channel detectionstrategy may result in detection of the test stimulus bydifferent channels before and after adaptation or in thepresence and absence of the masking stimulus.'0 Blake andHolopigian'0 used spatial noise oriented symmetrically froma test in order to combat this strategy and found, as predict-

ed, wider orientation bandwidths (23°-31°) than those ob-tained by using asymmetric masking. However, to avoid theoff-channel criticism completely it would be desirable toestimate orientation bandwidths with anisotropic stimulithat contain power modulated continuously around 3600.Explicitly 2D stimuli of this type would permit measure-ments of orientation selectivity through the use of tradition-al contrast detection thresholds.

Recent advances in the functional neuroanatomy of thevisual system also suggest the need for an explicitly 2Dapproach to orientation selectivity. Neurophysiologicalstudies clearly show that the receptive fields of neurons invisual cortex have finite, 2D response properties.'5 ,1 6 Daug-man'2 "17 has illustrated the necessity of placing models ofspatial vision within the constraints of the anisotropic, 2Dnature of the underlying neural substrate. Mostafavi andSakrison18 used 2D, filtered spatial noise to estimate theangular selectivity of a channel in human vision and foundbandwidth estimates between 100 and 14°. The broadbandspatial frequency content of 2D noise, however, makes infer-ences about orientation tuning at a specific spatial frequen-cy difficult.

For all the reasons outlined above, it would be informativeto use radially harmonically pure images in conjunction with2D models to derive estimates of the tuning of channels at asingle spatial frequency. Detection thresholds for one classof harmonically pure, 2D stimuli were obtained first by Kel-ly'9' 20 and later by Kelly and Magnuski." These studiesmeasured 2D contrast sensitivity functions, using circularlysymmetric, radial Bessel functions of the first kind, order 0.These functions consist of a single spatial frequency withequal amplitude and phase at all orientations. For an iso-tropic visual system, this transfer function could in principlebe used to predict the contrast threshold for any circularlysymmetric, complex 2D pattern. However, the constraint ofcircular symmetry precludes the use of anisotropic stimuli,which are necessary for determining orientation tuning.

0740-3232/90/010141-11$02.00 © 1990 Optical Society of America

J. L. Dannemiller and J. N. Ver Hoeve

142 J. Opt. Soc. Am. A/Vol. 7, No. 1/January 1990

In this paper we note that the isotropic pJby Kelly' 9 is a singular case of a more gEperiodically anisotropic, 2D spatial functioterns are polar separable and contain a s:spatial frequency, with the amplitude at Iquency varying sinusoidally as a function ofthey are anisotropic and radially harmonic,isotropic pattern used by Kelly is a singufamily of patterns because it contains no arDaugman 22 recently introduced a class ofselective frequency-independent patterns.noted that the latter patterns contain powerfrequencies at a given orientation, whereas iBessel patterns described below have all the:trated at a single spatial frequency at any giWe used contrast detection thresholds for reterns in conjunction with a multiple-channtrast detection to derive estimates of orientand the number of channels contributing topredictions of the multiple-channel modewith those of a power-detection model. 23

METHODS AND PROCEDURES

Stimulus Patterns and Display CharacterisContrast thresholds were collected for 2D pspatial luminance distribution in polar form

L(r, 0) = L0 [1 + C cos(n0)Jn(f 0 r)]

L(r, 0) = Lo

where

L is luminance,r is the radial distance in degrees from t]

pattern,0 is the angle from the horizontal in the viLo is the mean luminance of the pattern,C is an amplitude scalar,Jn is a Bessel function of the first kind, off is a spatial frequency scalar, andr is the pattern radius in degrees.

Examples of these patterns are shownisotropic Bessel functions used by Kelly are:this class of pattern when n = 0. It is showithat this family of spatial functions possesseand useful properties. First, these 2D spatiessentially all their significant power confspatial frequency when more than a few rpresent. Second, the order of the Bessel fitrols the angular periodic variation in both t]spatial frequency domains. The spatial pacorresponding spectra possess n-fold rotatiThe angular period of the cosinusoidal amrin the 2D spatial frequency domain is 27r/n.tion of Fig. 1 shows that for orders n > 0 thcomposed of single spatial frequency comp,two phases-0° and 180°-with respect to Ipattern. The trade-off between the numbetion bands and the angular width of each be

attern used first patterns useful for testing detection models with angularneral family of bandwidth and number of channels as parameters.ns. These pat- Contrast thresholds were collected for orders n = 0, 2, 4, 6,ingle, dominant 8, 12, 24. Thresholds were collected with fo set to produce;hat spatial fre- patterns at 4 and 10 cycles per degree (cpd). Field diameterorientation (i.e., varied inversely, with spatial frequency being 40 at 4 cpd andally pure). The 1.6° at 10 cpd. This ensured an equal number of radialdlar case of this cycles at both spatial frequencies. Viewing distance was setigular variation. at 72.5 and 181.25 cm at 4 and 10 cpd, respectively. In this2D orientation- way we ensured that the spatial transfer function of the

It should be display did not differentially affect the results. When n > 0,at many spatial these patterns possess 2D spectra with 2n maxima and mini-

the higher-order ma. Hardware limitations prevented us from randomizingir power concen- the absolute orientation of these patterns in the visual fieldyven orientation. across trials. The patterns were presented so that one pairidial Bessel pat- of maxima/minima at orthogonal orientations always oc-el model of con- curred in the primary meridia of the visual field (e.g., 12ation selectivity o'clock and 3 o'clock).detection. The The Bessel functions required by Eq. (1) were calculated1I are compared with a numerical integration routine and stored as lookup

tables. The actual spatial pattern for any condition wascalculated according to Eqs. (1) and (2) with a Harris 800computer and downloaded into a Chromatics 780 graphicsstation with 1024 X 1024 pixel spatial resolution. Each ofthe three color guns in the 60-Hz, noninterlaced monitor was

atterns having a controlled by an independent 8-bit digital-to-analog con-atterinsd havig a verter. An additional 8-bit color lookup table was used ton defined by ensure that the entire 8-bit tone scale was used as efficiently

r < ro, (1) as possible. At the mean luminance of 7 cd/M2 , the mini-mum available modulation depth was 0.0189. All threshold

r > ro, (2) estimates in all conditions exceeded this value.Only the red and green phosphors were activated (in

phase), producing yellow luminance contrast patterns. Therelationship between input voltage and output luminance,measured with a Spectra-Prichard photometer, was deter-

he center of the mined separately for each gun and was gamma corrected toremove nonlinearities. Calibrations were checked with an

sual field, automated routine before each session.The patterns were presented as 100-msec rectangular

pulses. Between trials the large uniform field remainedorder n, constant at 7 cd/M2 . Two small fixation marks were ar-

ranged diametrically on the circumference of the patternfield. The pattern appeared at their implied midpoint.

Contrast was defined as (Lmax - Lmean)/Lmean. This defi-in Fig. 1. The nition of contrast was used since the isotropic pattern (n = 0)singular cases of contains asymmetric variation near the mean luminance.i in Appendix As several unique Proceduresal patterns have A two-temporal-alternative forced-choice procedure wasined to a single used to measure contrast thresholds. The two 100-msecradial cycles are display periods were signaled by auditory tones. Auditoryinction (n) con- feedback on the correctness of the forced choice was provid-he space and the ed to the observer after each trial. A testing run consisted oftterns and their the measurement of two contrast thresholds for the sameional symmetry. order n pattern at the same spatial frequency. These two)litude variation thresholds were measured using two randomly interleavedFinally, inspec- two-down/one-up staircases with eight reversals each.ese patterns are These two staircases were always started from different ini-onents with just tial contrast levels. The step size was set at 3.0 dB for thethe center of the first two reversals and at 1.5 dB for the last six reversals.or (n) of orienta- Only the last six reversals from each staircase were averagedand makes these to obtain a threshold estimate. Each data point shown



U,0.5

-JCLF0.0

-0.5

:LU

0

IL-J

0.5 F

0.0 F

-0.5

1.0

LU0

iM

-J0.

0.51-

0.0 -

-1.0 -0.5 0.0 0.5 1.0DEGREES

Fig. 1. Left-hand column, examples of radial, Bessel functions of the first kind: top, Jo; middle, J 4 ; bottom, J8. Middle column, horizontalcross-section luminance profile through each pattern. Right-hand column, the 2D Fourier transforms of the spatial patterns shown on the left.When these photos are viewed as polar plots, the height shows the amplitude of a component of spatial frequency r (radius) at orientation 0.Negative amplitudes indicate components with phases opposite (1800 phase shift) those of the positive amplitude components. Excludingslight artifacts introduced by truncating these patterns at a fixed radius, these transforms show that all the patterns on the left have a single,identical frequency at all orientations (excluding null orientations). Jo is isotropic with only positive, cosine phase components of equalamplitude at all orientations. The other two patterns are anisotropic with the order (n) of the Bessel function controlling the number and dis-tribution of amplitude cycles. The peak spatial contrast necessary to produce a given peak amplitude in the 2D spatial frequency domaindecreases as n increases.

below represents the average of at least three staircase runs.Presentation of the seven orders at each of two spatial fre-quencies was randomly distributed across the required test-ing sessions. Viewing was monocular with the preferred eye.Distance was secured with a head-and-chin rest.

ObserversThe two authors served as observers in all the conditions.One of the authors, JVH, is emmetropic, and the other, JLD,is a corrected-to-normal myope. Partial data on a third,naive, observer showed the same pattern of results. Onlythe full data from the two experienced psychophysical ob-servers are presented below.

Detection ModelsThese 2D patterns contain power at many different orienta-tions. We consider two detection models. The multiple-channel model computes the responses of several channelswith differing center orientations to a pattern and combinesthese responses, using probability summation to determine

threshold. This model is referred to as the spatial contrastmodel. The alternative model assumes that the stimulus isfirst filtered through isotropic mechanisms. Threshold oc-curs in this model when the spatial power in the filteredoutput reaches some criterion level. This model is identicalto the one used by Jamar and Koenderink23 with one-dimen-sional patterns and by Greenwood24 and Mitchell2 5 with 2Dpatterns. This model is referred to as the spatial powermodel. Thse models are described in detail in Appendix B.

RESULTS

The symbols in Figs. 2 and 3 show the observed contrastthresholds for both observers at both spatial frequencies.The curves represent the best fit of the spatial contrastmodel shown in Eq. (B16) of Appendix B to the data. Thepredictions are connected with lines for display purposes,although it is understood that the order parameter, n, cantake-on only discrete, integer values. The parameters esti-mated from the data are shown in Table 1. The x2 is also

wvV~Jvw

W\AAfJv�JWVVVV



Angulor Bondwidth (Deg)00 90.0 45.0 30.0 22.5 18.0 15.0

.- )

0L

c 100U

0O4 8 12 16 20 24J Order

Fig. 2. Contrast thresholds for observer JLD at 10 cpd (circles) and4 cpd (squares). The curves show the predictions from the best fitof Eq. (B16) to each data set. Error bars are 1 standard error of themean.

Angulor Bondwidth (Deg)oo 90.0 45.0 30.0 22.5 18.0 15.0

U)0Lc 100UT

0 4 12 16 20 24J Order

Fig. 3. Same as Fig. 2 but for observer JVH.

Table 1. Estimated Model Parameters from Eq. (B16)

foObserver (cpd) 0.5 0' k To x

2

JLD 10 15.000 7.50 0.28 0.326

JVH 10 17.740 6.00 0.30 0.205JLD 4 16.770 7.00 0.07 0.082JVH 4 17.460 6.00 0.08 0.206

shown for each data set. The model in Eq. (B16) could notbe rejected as an adequate fit to any of the observed data setsbased on the x2 value. Threshold contrasts at a given spa-tial frequency vary over a range of approximately 0.2 log unitas the order parameter, n, is varied. This is identical to the0.2-log-unit range of threshold contrasts observed by Mosta-favi and Sakrison' 8 (see Fig. 8 of Ref. 18) as the angularbandwidth of 2D noise was varied.

First note the estimates of half-bandwidth at half-sensi-tivity. The estimates at both spatial frequencies are quitesimilar. The average half-bandwidth at half-sensitivity es-timate is 16.740. This value is quite similar to those report-ed previously for cases in which other paradigms with one-dimensional stimuli were used. Channels with medium ori-entation bandwidth that contribute to the detection of one-dimensional patterns apparently also contribute to thedetection of patterns with significant 2D variation. Theangular selectivity values are similar to those reported byDeValois et al.14 for simple cells in macaque visual cortex.Similarly, Daugman 26 found in a survey of orientation-tun-ing studies of simple cells in cat visual cortex that the meanhalf-width at half-response was 17.50.

Next, note the estimates of the number of channels. Thisparameter represents the number of channels contributingto the detection of the isotropic pattern. The estimatesrange from six to eight channels. Regan and Price27 sug-gested recently that a smaller number of channels, perhapsas few as four at a given spatial frequency, contribute todetection and discrimination at any location in the visualfield. Our estimates are slightly higher than those of Reganand Price. The estimate of six to eight channels refers to thenumber of channels with the same phase preference contrib-uting to the detection of the isotropic pattern. The modelused to derive this estimate assumed paired channels withphase preferences differing by 1800 of phase angle at eachorientation. Doubling the values shown in Table 1 yields anestimate of 12 to 16 even symmetric channels covering 1800of orientation. Of course, these channels would transferonly the even portion of the phase spectrum of an image; anadditional set of odd symmetric channels in quadrature withthe even set would be necessary to transfer the phase spec-trum fully. This would at least double the estimate, yield-ing 24 to 32 channels, all having the same optimal spatialfrequency covering the portion of the visual field occupiedby our input pattern.

Figures 4 and 5 show the threshold data expressed as rmscontrast for both observers. Recall that the spatial powermodel predicts that thresholds should be constant as the nparameter is varied. The results shown in Figs. 4 and 5violate the predictions of the spatial power model describedabove. The slope of the least-squares regression linethrough each set of data was tested against an expectedvalue of 0 as predicted by the spatial power model. Forobserver JLD the slope of log rms contrast threshold versuspattern order (n) at 4 cpd differed significantly from 0 [t(5)= 2.82, p < 0.05], as did the slope at 10 cpd [t(5) = 4.04, p <0.01]. For observer JVH the slope at 4 cpd differed signifi-cantly from 0 [t(5) = 2.94, p < 0.05], as did the slope at 10 cpd[t(5) = 3.52, p < 0.02]. These results are consistent withthose of Greenwood,24 who also found that a single-channel,spatial power model could not account for the detection of2D noise images. The results, however, are inconsistent

\y1


J. L. Dannemiller and J. N. Ver Hoeve Vol. 7, No. 1/January 1990/J. Opt. Soc. Am. A 145

-1. 0

U')

&-1.5.DI-

CC-2. 0.

J

-2.5 56 10 1S 20

PATTERN OROER (N)25

Fig. 4. The logarithm of rms contrast at threshold for observeJLD at 10 cpd (circles) and 4 cpd (squares). The best-fitting linethrough the data are also shown. The spatial power model predicta slope of zero in both cases; both slopes deviate significantly fronzero.

-1. 0

-1. 5

-2. 0

-2. 5

-5 0 5 10 15 20PATTERN ORDER (N)

Fig. 5. Same as Fig. 4 but for observer JVH. Bothsignificantly from zero.

25

slopes differ

with those of Jamar and Koenderink,2 3 who found that sucha model could account for detection of one-dimensionalnoise images. It is not clear why Greenwood 2 4 and Jamarand Koenderink23 reached different conclusions regardingthe detectability of noise patterns, but one obvious possibili-ty is that the dimensionality of the stimulus influenced thedetection of the visual pattern. Both the present study andthat of Greenwood 2 4 used 2D images, whereas Jamar andKoenderink 2 3 used one-dimensional images. Support forthis interpretation comes from Mitchell,2 5 who found, unlikeJamar and Koenderink,2 ' that the bandwidth of the stimu-lus did affect the detection of 2D noise patterns. Thepresent results are compatible with the idea that 2D imagesare detected after being filtered through orientation-selec-tive mechanisms that are also phase selective (also see Ap-pendix B).

Finally, it is interesting to note that only when channelspossess medium orientation bandwidth does the model pre-dict relatively equal contrast thresholds from n = 2 to n = 24at a given spatial frequency. The observed data show thisrelative invariance of spatial contrast threshold across theseorders despite large differences in the spatial distribution of

contrast and the number of spatial maxima/minima withinthe pattern. Essentially the only characteristic that thesepatterns have in common is that for fixed fo they are allcomposed of just one spatial frequency in many differentorientations. It should be emphasized that equal detect-ability in the spatial domain is in contrast to peak amplitudeAn at threshold in the 2D spatial frequency domain wherethreshold amplitude increases by a factor of approximately 2as n increases from 2 to 24. These results are consistentwith other subthreshold summation studies that show thatthe contrast threshold for a one-dimensional grating is rela-tively unaffected by the addition of gratings of the samespatial frequency at orientations sufficiently different from

)o the first.2 8'2 9 Our results also show, however, that a singlechannel sums inputs in neighboring orientations and that

zr this within-channel summation across neighboring orienta-zs tions is phase selective.ts

DISCUSSION

The detection of patterns with significant 2D spatial varia-tion conforms to a simple model consisting of an initial linearfiltering of the pattern by medium-bandwidth, phase-selec-tive channels followed by probability summation over thesechannels. The results also indicate the usefulness of thisfamily of patterns for studying the properties of 2D channelsin human vision.

The data from this study are consistent with those ofprevious studies that have applied Fourier models to explainthe detection of 2D patterns. Kelly and Magnuski2l haveshown that the detection of isotropic patterns identical tothe ones used in the present study depends on the maximumamplitude of these patterns in the 2D spatial frequencydomain. Kelly29 shows similar results. Kelly30 also showsthat the detection of checkerboards is compatible with a 2Dspatial frequency model.

A significant aspect of these results is the support for theassumption that orientation-selective channels are alsophase sensitive. Simulations with the model in Eq. (B16)show that phase-insensitive integration within medium-bandwidth orientation channels leads to the prediction ofrapid increases in the spatial contrast necessary for detec-tion as the order parameter n increases. This resulf is rela-tively easy to understand. At higher n orders with phase-insensitive integration, both 00 phase and 1800 phase com-ponents with equal peak amplitudes fall within the sameorientation channel. These opposite-signed componentswould cancel one another in a linear, phase-insensitive chan-nel, producing little net response. In contrast, the data areconsistent with a model containing pairs of phase-selectivechannels at each orientation. These channels preserve theeven phase information in the signal and reject the signalcomponents opposite their preferred phases.

The simple model presented here ignores the possibilitythat the outputs of neighboring orientation channels mayinhibit one another. Blakemore and Tobin3l and DeValoiset al.14 present neurophysiological evidence from cat, andBlakemore et al.

3 2 present human psychophysical evidenceconsistent with flanking orientation inhibition. Both ofthese studies used high-contrast patterns in demonstratingthis inhibition. The magnitude of this effect may be dimin-ished at the threshold level contrasts used in the present

0

a * *EU

i 0I: 5

0 a

*" I I I I1 I I .1 ._m E_ HI

5 3.


study. We know of no subthreshold summation study of

orientation selectivity that shows that the presence of asubthreshold grating in one orientation decreases the de-

tectability of a grating at a neighboring orientation.Daugman, 26 DeValois et al.,

33 and Webster and DeValois16

have suggested that the early visual filtering of spatial pat-terns is probably accomplished by units that are tuned si-multaneously for spatial frequency, orientation, and phase.The Gabor model captures this simultaneous tuning alongthese dimensions as well as implying that such units are also

spatially localized.3 3'3 4 The patterns used in this study werenarrow band for spatial frequency. Our results, therefore,cannot tell us anything about the spatial frequency tuning ofthe orientation channels. However, at a fixed spatial fre-quency our results suggest that the orientation dimension ispaved with roughly six to eight pairs of even symmetric

channels having half-bandwidths of approximately 17°.Daugman17'2 6 has pointed out that 2D channels are not polarseparable, which means that orientation selectivity will de-pend greatly on spatial frequency. We have been able to

avoid this complication by using novel 2D patterns that are

extremely narrow band in terms of spatial frequency (radialspectral variable), thus allowing us to probe orientation se-lectivity at a given spatial frequency.

Finally, we may ask why channels with narrower selectiv-

ity for orientation are not used to encode the image. Such

narrowly tuned filters could have a contrast sensitivity ad-vantage over more broadly tuned filters for detecting orient-ed image features such as edges and lines. For example,

compare the responses of an isotropic, center-surroundmechanism (e.g., circularly symmetric difference-of-Gauss-ian models) and an even symmetric Gabor filter with a long

bar whose width corresponds to the width of the centralexcitatory portions of their respective receptive fields. TheGabor unit clearly has an advantage in detecting optimallyoriented features. However, this gain in sensitivity wouldbe purchased at some cost because more such units would

then be necessary at each location in the field to pave the

orientation dimension smoothly and avoid possible anisotro-py. Narrowly tuned filters would also seem to be more

advantageous for orientation discrimination. It is probablythe case, however, that orientation-discrimination thresh-olds of the order of 0.5-1° could be achieved by using more

broadly tuned filters with a neural comparison between theoutputs this network of orientation filters.3' 2 7'3 5 Such anarrangement has a clear analog in color vision. Wavelength

discriminations of the order of 1-2 nm are still possible with

only three broadly tuned photopigments. Subsequent post-receptoral interactions between the outputs of photorecep-tors containing these photopigments probably permit finechromatic discriminations. The 170 half-bandwidths ob-served in human vision may then represent a compromise

between the demands of an adequate signal-to-noise ratiofor the transfer of information about oriented features in theimage and an adequate difference signal for the discrimina-tion of a change in feature orientation.

APPENDIX A

It is desired to find the 2D spatial functions whose 2D Fouri-

er transforms are given by the following equations:

F(p, 6) = sin(n0)6(p - po) for n odd. (Alb)

This function contains a single spatial frequency (po),which is the same at all orientations 0. The amplitude of

this spatial frequency varies with the cosine or the sine of 0.The parameter n controls the angular period between peakamplitudes in the 2D spatial frequency domain.

The inverse 2D transform of a polar function, F(p, 6), isgiven as

27, af(r, O) = J J F(p, 6)expLjrp cos(o - 0)]pdpdO.

L o

Therefore the inverse 2D Fourier transform of Eq. (Ala) is

2rf(r, /) = PoL cos(nO)expLirpo cos(, - 0)]dO,

since the function in Eq. (Ala) depends only on 0 because of

the behavior of (p - po) under integration.By substitution,

cos(nO) = 1/2(e'n° + eino),

f(r, O) = 2 {J en0 expUrpo cos(k - 6)1dO

2,,+ J e-jnO expUrpo cos(o - 0)IdO .

Let g(6) = expLjrpo(o - 0)]. The function g(O) is periodic

in 0.Expand it in a Fourier series:

g(0) = > cmeimo,m

(A2)

where27

cm = 2 J g(O)e-imsdO.

Substituting into Eq. (A2),

2r [ e" cmelmodO + J e" mjnZ cmelm O2 mf(r, ) = Po cm (L ejnOejmodO + e-jnoejmod6)]

f(r, 0) = P [ 27rcm(6mn + bm-d

2'7rpof(r, 0) = 2 (Cn + C-d)2

Now find Cn:

1 2,Cf= 2 j exptjrp 0 cos(,o - 0)]exp[-j(+n)0]d0,

27r1 2rHCl~fl = - cos[rpo cos~o - 6) ±1 n6]d6.

(A3)

F(p, 0) = cos(n0)6(p - po) for n even,


(Ala) Let O' = 0- 0:


l 2rc = 2r exptj[rp0 cos0' 1 n(ji- 6')]jdO,

C'n = exp(±jn F) L2 7| exp(jrpo cos 6' L nO')dOl-

By the definition of the Bessel function of the first kind,order n,

C~n= exp(±jn0)J~n(rP 0)-

Substituting into Eq. (A3),

f(r, 0) = P [e-noJn(rp0) + ein0J-n(rP0)1,2

if n is even

f(r, 0) = po cos(nO)Jn(rpo); (A4a)

if n is odd

f(r, q) = -po sin(n0)Jn(rp 0 ). (A4b)

Equations (A4a) and (A4b) show the spatial modulationfunctions whose 2D Fourier transforms were given in Eqs.(Ala) and (Alb). For n > 0 these functions are periodicallyanisotropic in the spatial domain and radially harmonicallypure for a field of infinite extent.

APPENDIX B

Spatial Contrast Model

Response of a Single Two-Dimensional ChannelThe effect of a 2D selective channel with linear filteringcharacteristics on an input pattern is given by 2D convolu-tion of the channel's spatial impulse response function withthe spatial luminance distribution of the pattern. This op-eration is also defined in the 2D spatial frequency domain.It is then necessary to multiply the Fourier transform of theimpulse response by the Fourier transform of the spatialpattern. For the model detailed in this subsection, all com-putations were carried out in the 2D spatial frequency do-main. The response of an orientation channel was comput-ed by integrating the product of the pattern's input spec-trum with the channel's impulse response spectrum. Thismethod of computing a channel's response assumes that thechannel integrates spatially over an area at least as large asthe input pattern. The diameters of the patterns used inthis study were 40 and 1.60 at 4 and 10 cpd, respectively.This model differs from other models, which assume proba-bility summation across localized, spatial detectors.3 6 Herewe assume probability summation across channels definedin the 2D spatial frequency domain.

For analytical convenience we assumed that the orienta-tion-tuning function of a single 2D channel at the radialspatial frequency of the pattern was a triangular function oforientation. Since these channels were meant to correspondto real weighting functions in the space domain, they haveHermite symmetry, i.e., each channel defined in the 2Dspatial frequency domain actually consists of a pair of trian-gular tuning functions symmetric with respect to the origin.This same orientation-tuning function was used by Thomasand Gille.37 Campbell et al.3 8 have demonstrated this typeof orientation tuning in simple cells in cat visual cortex.

The estimates of orientation bandwidths derived belowwould differ by at most a scalar if other monotonically de-creasing tuning functions were used. 37

Mathematically, the orientation-sensitivity function ofchannel i at a single spatial frequency is given by

(Bi)

with Si being the peak sensitivity of the channel at theoptimal orientation (assumed to be 00 for analytical conve-nience), 0 being the angular deviation from this optimalorientation, and 0' being the one-sided angular deviation atwhich the channel's sensitivity falls to 0. The half-band-width at half-sensitivity is then simply 0.5 6'. The sensitiv-ity of the channel is taken as 0 for 0 > 6'.

The response of the channel to the various patterns de-scribed above is given by multiplying Eqs. (B3) below by the2D spectra of these patterns and integrating the productover the polar 2D spatial frequency plane. Here we makethe simplifying assumption that all the power of these spa-tial patterns is concentrated within a vanishingly narrowannular ring centered at the spatial frequency determinedby fo. Discrete, 2D fast Fourier transforms supported thisassumption for the pattern diameters used in this study.The ring has an annular width of 26 for all orders. Thisallows us to neglect the tuning of the orientation channel inthe radial (spatial frequency) direction. The response ofchannel i to the order n pattern is then given by

Rin = 47rf0 , SP(l - 1/0')An cos(nO)dO,-o

(B2)

where An cos(nO) represents the 2D spatial frequency spec-trum of the order n pattern. This integration is performedwith the spatial pattern oriented in the visual field so as tostimulate maximally the channel's optimal orientation.

Equation (B2) neglects the phase specificity of the chan-nel. Here we make the explicit assumption that these chan-nels are phase selective. Pollen and Ronner15 have suggest-ed that channels occur in pairs with differences of 1800 intheir preferred phases. Spitzer and Hochstein39 have shownthat simple cells in cat visual cortex are phase selective.Owing to their low-to-zero maintained responses these unitsrespond as half-wave rectifiers to an input signal. Channelscomposed of many pairs of such units with opposite phasepreferences (on cente'r, off center) would then transfer fullythe even portion of the phase and amplitude of an inputsignal. DeValois 4 0 and Legge4l have shown that patternadaptation in the human visual system is phase selective,and Caelli and Bevan4 2 have shown that 2D textures appearto be processed through phase-selective mechanisms. Weformulated a phase-selective detection model after review-ing these data.

Figure 6 shows a schematic version of paired, phase-selec-tive orientation channels. The sinusoidal transform of ahigher n order pattern is also shown. The channel labeledR+ with a triangular sensitivity profile weights and inte-grates only the positive phase components of the input fall-ing within its bandwidth. The channel labeled R- similarlyweights and integrates only the negative phase componentsof the input falling within its bandwidth. There are severalways to represent this phase-selective integration. We usethe formulation


Fj(0) = Si 1 - 101O/ ),


0I

I¢=XL

Fig. 6. Schematic version of paired, phase-selective channels. An-other Hermite-symmetric pair is not shown but is necessary to makethe detectors in each channel correspond to real spatial weightingfunctions. The angular selectivity profile of channel, labeled R+,weights and integrates only the positive phase components of. thesinusoidal amplitude spectrum falling within its bandwidth. Thechannel labeled R- weights and integrates only the opposite, nega-tive phase components falling within its bandwidth. The sinusoi-dal amplitude spectrum of a higher-order radial Bessel function isalso shown. Values below the horizontal axis indicate componentswith phase values of 7r; values above the horizontal axis indicatecomponents with phase values of 0.

Rin = 47rf0 6 Sj(1 - 101/0')An{j [cos(nO) + Icos(n0)I]}do,

(B3a)

Rin = 4rf 0 J Si(1 - 101/0')An{ [cos(nO) - Icos(n6)I]}do.

(B3b)

Analytic solutions for Eqs. (B3a) and (B3b) were used inderiving the estimates below. We next turn to the issue ofhow the responses of this pair of phase-selective orientationchannels are combined with the responses of channels atdifferent orientations to determine threshold.

We wish to be clear about the sense in which we are usingphase here. These even order patterns have associatedphase and amplitude spectra. The spectra of these patternswhen n > 2 contain only two phase values, 0 and 7r. For n =0, the spectrum contains only one phase value, 0 if the centerof the pattern is bright (above the mean luminance) and irfor the complementary pattern with a dark center. For n =0, the phase of the component at every orientation is thesame with respect to the center of the pattern. For n = 2,the phase flips discontinuously from 0 to 7r at null orienta-tions and remains fixed at one of these two values betweennull orientations. This is why in Figs. 1 and 6 we haveplotted the signal spectra above and below a plane (Fig. 1) ora line (Fig. 6). All spatial frequencies between the two nullorientations have the same phase values. Similarly, thetriangular orientation-selective profile in Fig. 6 above the

horizontal line is meant to indicate a channel coniposed of

primarily on-center mechanisms, whereas the profile belowthe line is meant to indicate a channel composed of off-center mechanisms.

Combining the Responses of Different Pairs of Phase-Selective Orientation Channels across OrientationsWe assume that the pattern is detected when the response ofat least one orientation channel with either phase preferencereaches threshold. This is equivalent to assuming that theoutputs of these n channels are passed through an OR gate.We are interested in predicting relative contrast thresholdsat a fixed spatial frequency as the n parameter in Eq. (Bi) isvaried. To accomplish this we will develop the detectionmodel to predict thresholds for all the patterns relative tothreshold for the isotropic (n = 0) pattern. This patterncontains equal amplitude at all orientations, so it can conve-niently serve as a reference pattern.

First, consider the ensemble of responses by different ori-entation channels to the isotropic n = 0 pattern. The Fouri-er transform of this spatial pattern contains a single spatialfrequency of equal amplitude and positive phase at all orien-tations. The response of a single channel is therefore givenby evaluating Eq. (B3a) with n = 0. This yields

Rio+ = 4rf 0t5SiA 00', (B4)

with terms defined as before.The probability that channel i will detect the n = 0 pattern

is assumed to be related to the channel's response by theequation

pio = 1 - exp(-IRio+ll). (B5)

Equation (B5) therefore relates the probability of detectionto pattern amplitude AO in the 2D spatial frequency domainsince Rin+ is linearly related to amplitude Ao as shown in Eq.(B3a). The relationship in Eq. (B5) was used previously43' 44

for contrast detection. The ,3 parameter in Eq. (B5) is relat-ed to the steepness of the psychometric function slope.45-47

Next, consider the probability that the isotropic stimuluswill be detected in a two-alternative forced-choice paradigmby at least one of k orientation channels. This probability isgiven by

k

po = 1 - 0.5 II (1 - Pi).i=1

Combining Eqs. (B5) and (B6) yieldsk

pO = 1 - 0.5 1J exp(-WRiolI'),i=1

which in turn yields

P( = 1 - 0.)5 ex( -71 IRi0+If).

(B6)

(B7)

(B8)

For a two-down/one-up staircase, threshold is defined asthe amplitude necessary to achieve po = 0.71. Therefore thethreshold amplitude AO can be found by substituting 0.71 forpo in Eq. (B8) and noting that the responses from all kchannels are identical when n = 0. This leads to the conclu-sion that threshold will occur when the response from atleast one channel equals the right-hand-side quantity in Eq.(B9):

Rio' = (0.54/k)W/. (B9)

We can now combine Eqs. (B4) and (B9) to solve for thresh-old amplitude, which is given by



A - (0.54/k)hI ( lO°O = to50 (B10)

Next, by a similar formulation we can determine the prob-ability that an anisotropic pattern with n > 0 will be detect-ed by at least one orientation channel. Here we introducetwo other assumptions. First, for patterns with n > 0, weassume that the patterns are detected by means of responsesin the maximally active orientation channels. This meansthat the number of maximally active orientation channelsfor a pattern is determined by n, the number of maxima/minima in the 2D spatial frequency domain. There areactually 2n maxima/minima in the 2D spatial frequencydomain. However, for even values of n, the components at 0and 6 + 7r stimulate the same, even symmetric channel. Theresponses in these n active channels will be identical, i.e.,1Ri+1 = IRj+1I and IRjl = IRj-1. Regan and Beverly 3 have alsosuggested that detection occurs by means of responses in themaximally stimulated channels. Second, we assume that ifk channels contribute to the detection of the isotropic pat-tern, then no more than k channels can contribute to thedetection of patterns with n > 0. Practically, this meansthat there may never be more than k channels contributingto the detection of any pattern at a fixed spatial frequency.The justification for this assumption is that the maximumnumber of available channels should be stimulated by theisotropic pattern, so that it would be impossible to stimulatemore than this number of channels, k, with any other pat-tern.

With these assumptions in mind we can now determinethe amplitude, An, necessary to detect a pattern with order n> 0. Since these patterns may elicit responses in a pair ofphase-selective channels at each of n orientations for largervalues of n, we must also consider the combination of thesephase-sensitive responses given in Eqs. (B3a) and (B3b).This leads to the following:

n = 1- 0.5 exp[- (IRi, Il + IRinIP)] (B11)

with n limited by k as required by the second assumptionnoted above. Threshold occurs when the responses from atleast one pair of opposite-phased channels at an orientationconform to Eq. (B12):

Equation (B14) shows how relative sensitivities in the 2Dspatial frequency domain should vary as a function of theorder parameter n. Since the observed data are contrastthresholds in the spatial domain, Eq. (B14) should be trans-formed to make space-domain predictions. This is accom-plished by knowing the relationship between spatial con-trast and maximum amplitude, An, for each order, n. Ofcourse, with n and fo fixed, this relationship is linear; that is,if the spatial contrast is increased by a factor of x, then thepeak amplitude in the 2D spatial frequency domain alsoincreases by a factor of x. However, the relationship be-tween peak spatial contrast and peak amplitude variesacross n orders. As shown schematically in Fig. 1, for a fixedpeak amplitude in the 2D spatial frequency domain thespatial contrast corresponding to that amplitude decreasesas n increases. This relationship is shown in the secondcolumn of Table 2. For example, Table 1 shows that thepeak spatial contrast for n = 24 is approximately 23% of thecontrast for n = 0 when these two patterns have equal peakamplitudes in the 2D spatial frequency domain. This rela-tionship can be used to transform the predictions in Eq.(B14) into the space domain.

One other factor must be taken into consideration whenmaking this transformation. Equations (1) and (2) showthat the stimulus pattern was truncated at ro. This trunca-tion is equivalent to convolving the 2D spatial frequencyspectrum of the pattern by the Fourier transform of a 0/1cylinder function of spatial radius ro. This decreases thepeak amplitudes at the higher n orders. The correctedtransformation factors are shown in the third column ofTable 1.

Representing these frequency-to-space-domain transfor-mation factors by Gn yields

Tn SO

To Sn(B15)

where Tn represents the spatial contrast threshold for theorder n pattern and To is the spatial contrast threshold forthe n = 0 pattern. Finally, substituting from Eq. (B14) andrearranging Eq. (B15) yields

Tn~~ GnTOn= G/ 01(n)1'/#F[(I Qin+i'+IQiQ.l If(B16)

An[IRin+l' + W~in-1#P]" = 0.54.

Solving for An yields the following:

(0.54/n) 1 /1

47rf06Si(QinQ1# + IQin I )"'

(B12)

(B13)

Qin+ and Qin- are normalized single-channel responses de-fined by Eqs. (B3a) and (B3b) to the order n pattern withinput peak amplitude An = 1.00. The normalization factoris 47rfobSi.

With sensitivity (in the 2D spatial frequency domain) foran order n pattern defined as Sn = 1/An, the relative sensitiv-ity Sn/So is given by taking a ratio of Eqs. (B10) and (B13):

Sn = (n)1/' [r(Qin+Il + IQin') 113](B14)

6' represents twice the half-bandwidth at half-sensitivity inradians.

Equation (B16) shows the predicted spatial contrastthresholds as n is varied. This equation has four free pa-rameters, 6', k, fi, and To. On a plot of log To versus n, the To

Table 2. Ratios of Peak Spatial Contrasts across nOrder for Constant Peak Amplitude in the 2-D Spatial

Frequency Domain

GnOrder n Infinite Field Truncated Field

0 1.000 1.0002 0.489 0.4864 0.400 0.4126 0.355 0.3738 0.325 0.348

12 0.286 0.34024 0.230 0.274



Angular Bandwidth (Deg)00 90.0 45.0 30.0 22.5 18.0 15.0

0d11

I.

I-llJ

LC0:

0 4 8 12 16 20 24PATTERN ORDER (N)

Fig. 7. Predictions of the model shown in Eq. (B16) with k = 6 and0 = 3.00. The 6' parameter (full bandwidth at self-sensitivity) isindicated by its one-sided values. Also shown is the limiting predic-tion (6) for channels with infinitesimally narrow tuning. The pat-tern order (n) shown on the abscissa refers to the order of the Besselfunction in Eq. (Bi). The angular bandwidth refers to the angularperiod of the pattern.

parameter simply shifts the curve in the vertical direction.The value of 3 was set to 3.14 for observer JLD and 2.31 forobserver JVH.4 5 This left three free parameters in the mod-el used to fit the observed data. To understand the behaviorof the model with respect to variations in the assumed angu-lar selectivity of orientation channels, we have generated anillustrative example using Eq. (B16) with various values of0'. The number of channels, k, was set to 6, and ,B was set to3.0. The value of To was arbitrarily chosen as 1.0 withoutloss of generality. Figure 7 shows the effect of varying thebandwidth (full bandwidth at half-sensitivity) parameter, 6',from 40 to 640. Also shown as the lowest curve in Fig. 3 are

the limiting predictions for channels with tuning functionsthat are essentially delta functions of orientation-sharplytuned channels. This curve simply represents the numbersin the last column of Table 1. The effects of this bandwidthparameter are straightforward when considered in the 2Dspatial frequency domain. For a narrowly tuned channelwith a given center orientation, only the spectral compo-nents immediately neighboring this center orientation con-tribute to the response of the channel. In the extreme caseof a channel whose tuning function is a delta function of

orientation, the responses of this channel to all orders (n)would be equal when the spatial contrasts were in the ratiosshown in the last column of Table 1. As the bandwidth of achannel becomes broader, however, the responses of thischannel to the anisotropic patterns fall below the response tothe isotropic pattern. This occurs because the isotropicpattern contains equal amplitude at all orientations, where-as the amplitudes of the anisotropic patterns fall off sinusoi-dally around the peak orientations. This causes a rise in the

spatial contrast thresholds for the anisotropic patterns rela-tive to threshold for the isotropic pattern.

It is also interesting to note that the nonmonotonicity ofthe predictions shown in Fig. 3 for a given bandwidth is duein part to the assumption that orientation channels arephase selective. As the order parameter n is first increased,spectral components with phases opposite the preferredphase of a channel begin to fall within the channel's orienta-tion bandwidth. These components do not contribute tothe response of this phase-selective channel (they do, howev-er, contribute to the response of a channel at a nearby orien-tation with the opposite phase preference). As n is in-creased further, additional spectral components with theappropriate phases again fall within the channel's band-width, contributing positively to the channel's response.This is a form of aliasing or undersampling the orientationdimension. Regan and Price27 recently presented evidencefor undersampling of the orientation dimension by orienta-tion-selective mechanisms. Thus increases in the order pa-rameter n affect the response of a channel in a manner thatdepends on the bandwidth and the phase selectivity of thechannel. This leads to the nonmonotonicities in the predic-tions shown in Fig. 7. Mostafavi and Sakrison18 also shownonmonotonic predictions from a model with angularly se-lective channels when these channels are stimulated withpatterns having differing orientation bandwidths.

Finally, it should be pointed out that the logic used togenerate the model in Eq. (B16) also could be used to predictthresholds for patterns composed of the addition of variousone-dimensional, single-spatial-frequency gratings in differ-ent orientations. The model predicts that the responses togratings within a single channel will linearly summate,whereas gratings in different channels will contribute todetection by means of probability summation. In thissense, the model in Eq. (B16) is derived from a more generalone that applies to the detection of 2D, radially harmonicallypure patterns with discrete or continuous spectra.

For the spatial contrast model, the data for each observerat each spatial frequency were fitted by using Eq. (B16).The Qin+ and Qint terms in Eq. (B16) were evaluated byusing an analytic solution to the normalized versions of theintegrations shown in Eqs. (B3a) and (B3b). Values for thethree free parameters in Eq. (B16) (3 was held fixed) wereestimated by using a least-squares-error criterion for the fitbetween observed and predicted data. STEPIT4 8 was used toperform the required error minimization.

Spatial Power ModelThe spatial power model assumes that the pattern is firstfiltered through isotropic mechanisms. Threshold occurswhenever the spatial power in the filtered image reaches acriterion level. The spatial power of the stimulus is definedas the rms value of

Lo-L(r, O)

from Eq. (1). This model makes the straightforward predic-tion that thresholds plotted in terms of rms contrast shouldbe constant as the order parameter n is varied. This modelcan easily be tested by fitting rms contrast thresholds with astraight line and testing the slope against an expected valueof 0.0.



ACKNOWLEDGMENTS

We thank B. E. A. Saleh for many helpful discussions re-garding properties of 2D Bessel functions and J. Robson fordiscussion of psychophysical results. Computing time wassupported by National Institutes of Health core grantHD03352 to the Waisman Center.

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Two-dimensional approach to psychophysical orientation tuning