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Human frequency response functions of harmonic 2, 4, 8 and 16 cycle angular frequencyfilters.

MARIA LÚCIA DE BUSTAMANTE SIMAS1

NATANAEL ANTONIO DOS SANTOS2

1LabVis-UFPE - Departamento de Psicologia – Universidade Federal de PernambucoRua Acadêmico Hélio Ramos s/n, 9º Andar, Recife, 50670-901, PE, Brasil

mlbslabv@elogica.com.br and mlbslabvis@npd.ufpe.br

2Programa de Neurociências e Comportamento – Departamento de Psicologia – Universidade de São PauloAv. Professor Mello Moraes, 1721, São Paulo, 05508-900, SP, Brasil

naansant@usp.br

Abstract. We measured narrow-band human frequency response functions for four harmonic angularfrequency filters whose test frequencies were centered at either 2, 4, 8 or 16 cycles using a supra-thresholdsummation method. The four filter response functions of 17 experimental conditions (each) were measuredeight to nine times among five observers (NAS, MMM, ERB, MC and TPL). The results show that, for thearbitrarily selected filter phases, maximum summation effect occurred at test frequency for filters centered at 2and 4 cycles. For both 8 and 16 cycles test frequencies, maximum summation occurred at the lower harmonics.While at 8 cycles, for F8(θ), a relative strong summation effect was observed, at 16 cycles, for F16(θ), almostmaximum inhibition occurred. These results allow us to conclude that there are narrow-band angularfrequency filters operating in human visual system either through summation or inhibition of specificfrequency ranges. Furthermore, as a general result, it appears that addition of higher angular frequencies to lowones disturbs lower angular frequency perception (i.e. 2 and 4 cycles) whereas addition of lower harmonics tohigher ones seems to improve detection of higher angular frequency harmonics (i.e. 8 and 16 cycles).

Keywords: Spatial Frequency, Angular Frequency, Narrow-Band Filters, Polar gratings, Radial Gratings.

Introduction

Our work has focused mainly in characterizing the humanvisual system through its responses to contrast of spatialfrequency targets defined in Polar coordinates. Thus, wehave been calling spatial frequency varying along theradius as radial frequency and its orthogonal component,varying along angles, as angular frequency. Some of ourmeasurements related to contrast sensitivity to radial andangular frequency as well as to angular and radialfrequency filters are reported elsewhere [Simas, 1985;Simas and Dodwell, 1990; Simas, Frutuoso and Vieira,1992; Simas, Frutuoso and Santos, (accepted); Simas,Santos and Thiers, 1997; Simas and Santos, 1997]. Thepresent study focuses on narrow-band angular frequencyfilters similarly to our study of 1992 [Simas, et al. 1992].Since then we have changed our selection of phases for afamily of angular frequency stimuli. The work reportedhere measures harmonic filters at test angular frequenciesof 2, 4, 8 and 16 cycles. First, we are going to outlinesome preliminary aspects taken into consideration andrelevant to the present study.

Angular Frequency Stimuli

Angular frequency stimuli have appeared in the literatureunder a variety of names. Some studies have used termslike radial targets, Polar gratings, windmill stimuli, star-like, among others. For the sake of consistency we havekept our original naming since it was adopted in 1985[Simas, 1985]. Thus, we define angular frequency as aspatial frequency varying sinusoidally (or cosinusoidally,depending on how phase is considered) with angle withina circle being an adimensional stimulus, integer andwhose frequency is independent from the distance of theobserver.

This stimulus was conceived as the orthogonalcomponent based on Equation #4 [from Simas and Santos,1997] derived from Sneddon [Sneddon, 1961]. Suchequation coupling radial and angular components is:

∞exp(i2πρrsinθ) = Σ Jn(2πρr) {cos(nθ) + i sin(nθ)}

n=−∞

Eq#1

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Phase Selection

Our earlier works involving angular frequency [Simas,1985; Simas and Dodwell, 1990; Simas, Frutuoso andVieira, 1992; Simas, Santos and Thiers, 1997] usedarbitrary phases based on the origin of the trigonometricalcircle, i.e. maximum luminance set to the right. All stimuliwere generated having that origin in common.

The present study uses differently selected phasesbased on some known psychological phenomenon. Forinstance, when we observe a circle with shadows such thatthe lighter part is placed at the top and the darker to thebottom we perceive the stimulus as protruding from theplane while the inversion produces a converse effect.Other orientations were selected on the assumption thatangular frequency might be involved in face processingand other related perceptual phenomena.

Thus, we have selected phases based on symmetriesof hemispheres and quadrants. Our axis choice wasvertical versus horizontal. So, n-even angular frequenciespresent symmetric quadrants under these conditions whilen-odd ones present symmetric vertical hemispheres. Wearbitrarily set minimum luminance for all n=even and odd(except for 1 and 6 cycles) at position north (i.e.traditionally 90 degrees). These exceptions for 1 and 6cycles will be discussed elsewhere. They are based on aconfiguration after-effect phenomenon we found.

Figure 1 shows the phases used for the four testangular frequencies in this experiment.

Contrast Sensitivity to Angular Frequencies

Why work with angular frequencies when evidence fromcells at the striate cortex point to orientation selectivity?Figure 2 shows contrast thresholds for angular frequenciesas compared to sine-wave gratings. This result replicatesthat one reported in 1997 [Simas, Santos and Thiers,1997]. We found that the visual system is at least two foldmore sensitive to angular frequencies than to sine-wavegratings in their respective maximum sensitivity ranges.This would not necessarily be expected because somestudies have shown inhibition among sine-wave gratingsat orthogonal or other orientations [e.g. Tyler, 1975;1978].

We have been working with angular frequencies inpsychophysical experiments since 1985 based only on theEq.#1 derived from Sneddon. At that time evidence fromthe work of Hubel and Wiesel [Hubel and Wiesel, 1962;1968] did not lead in this direction. Only recentlyphysiological evidence that favor this type of stimuliconfiguration became available [Gallant, Braun and VanEssen, 1993; Gallant, Connor, Rakshit, Lewis and VanEssen, 1996]. Additional evidence comes from studies ofcells sensitive to expansion/contraction and rotation[Tanaka and Saito, 1989; Tanaka, Fukada and Saito, 1989;Kobatake and Tanaka, 1994] and involve movement.

The fact is that if angular frequency is being used byany visual system it requires integration over wide areasassembling information across hemispheres andquadrants. This implies integration across higher areas of

Figure 2. Contrast thresholds for MTF and aMTF.

Figure 1. Test angular frequencies used (see text).1 10 100

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the visual system. Furthermore, in our view, theprocessing of angular frequencies is coupled to that ofradial frequencies except for n=0 (refer to Eq.#1).

Narrow-band Angular Frequency Filters

Existence of narrow-band spatial frequency filters firstdemonstrated by psychophysical work [e.g. Blakemoreand Campbell, 1969; Graham, 1972] is now wellestablished. This integration seems to occur mainly atstriate and extra-striate cortex. Integration at higher levelsin the visual system are object of much investigationincluding the more recent physiological work mentioned.

We first tried to demonstrate the existence ofnarrow-band angular frequency filters in 1992 [Simas,Frutuoso and Vieira, 1992]. We used the old arbitrarydefinition of phase at background frequencies 1, 2, 3, 4, 6,9, 13, 16, 24, 32, 47, 64 and 96. We measured sevenfilters at test angular frequencies of 2, 4, 9, 13, 16, 24 and47 cycles. We used the same supra-threshold method ofthe present work. This method is based on that of sub-threshold summation [Kulikowski and King-Smith, 1973].Instead of using sub-threshold levels of contrast tomeasure the peak of the function, we used supra-thresholdsummation where the test frequency is summated to abackground frequency of higher contrast. While contrastof the test frequency is varied according to the observerssensitivity, contrast of the background frequency is fixedabove threshold (i.e.supra-threshold). In this manner, if astimulus containing only the background frequency iscompared to a stimulus where the background frequencyis summated to the test frequency the only way ofdifferentiating between the pair will be to detect thepresence of the test frequency in one of them. Our resultsshowed absolute or relative summation effects at the usedtest frequencies surrounded by strong inhibition. Weconcluded for the existence of some selectivity forspecific ranges of angular frequencies.

Other Theoretical Contexts

Literature involving angular frequency stimuli is scatteredand rare. One of the main streams has been based on theproposition by Dodwell [Dodwell, 1983] derived from thework of Hoffman [Hoffman, 1966] who introduced theapplication of Lie Transformation Groups to visualprocesses. For these authors filtering in Polar coordinatesis a by-product of the Lie Transformation Group forNeuropsychology, LTG-NP, model. A variety ofpsychophysical [e.g. Flanagan and Dodwell, 1991;Emerson, Humphrey and Dodwell, 1985; von Grünau,Kwas and Dube, 1994] and physiological studies [e.g.Gallant et al. 1993; Gallant et al., 1996] have been aiming

to obtain evidence in favor of the LTG-NP model. Otherphysiological studies by the group of Tanaka [Tanaka andSaito, 1989; Tanaka et al., 1989; Kobatake and Tanaka,1994] also have shown the existence of cells that couldconform to this model. However, these authors do notaddress to this issue.

Research concerning radial frequency has beenmainly directed towards local filtering of spatialcharacteristics employing Gabor’s or DOG functions aselementary stimuli to probe local processing [e.g. Pollen,Nagler, Daugman, Kroaner and Koenderink, 1984;Graham, Sutter and Venkatesan, 1993; Hess and Hayes,1994; Hess and Wilcox, 1994; Hess and Field, 1995].

Other approaches involving or discussing models ofvisual processing related to Polar stimuli are those byKoenderink and van Door [1990] and the group of Wilson[e.g. Wilson and Wilkinson, 1997; Wilson, Wilkinson andAsaad, in press]. While Koenderink argues for a Wavelet,a Cartesian and a Polar representation of cell’s receptivefeilds, Wilson argues for nonlinear interactions [1997] anduses random dot “Glass patterns” to investigate “non-Cartesian units” [in press].

Our own view of elementary stimuli as defined byEq.#1 is presented in Simas and Santos [1997].

The Present Study

In the experiments reported here we assume that higherareas of the visual system (e.g. V4 and IT) might beprocessing information in terms of coupled radial andangular frequency.

Gallant and colleagues [Gallant et al. 1993; Gallantet al. 1996] showed the existence of groups of cells fordetecting angular frequency in V4. We take this as astrong encouragement to follow the theoretical lineunderlying our psychophysical work.

The present study seek to measure harmonic angularfrequency filters centered at 2, 4, 8 and 16 cycles. Weused the same supra-threshold method and increased thesample to 17 background angular frequencies.

The main objectives are to establish the existence ofnarrow-band angular frequency filters using the newphase selection and observe how integration may be madeas we shift from lower fundamental angular frequencies(e.g. 2 and 4 cycles) to higher harmonics.

We must have in perspective that as angularfrequency is increased, sensitivity increases andinformation becomes more redundant (presentingsymmetric quadrants). So, it is important to establish how

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higher angular frequencies are related to lower ones (thatrequire more integration).

Methods

Subjects

Five 19-30 year old subjects with normal or correctedvision participated of the measurements.

Equipment and Stimulus Material

All images were displayed on a 20-inch monitor SonyBVM-1910 controlled by a 386 IBM-compatible micro-computer through a frame grabber DT-2853. Experimentswere run on-line. Measurements were made using pairs ofstimuli composed of a single background angularfrequency or of the sum of a background-angular-frequency-plus-test-frequency.

Background angular frequencies were either 1, 2, 3,4, 5, 6, 7, 8, 9, 10, 12, 16, 24, 32, 48, 64 or 96 cycles. Thetest angular frequencies, were either 2, 4, 8 or 16 cycles.Figure 1 shows the four used test frequencies at theirarbitrarily selected phases: at top left, 2 cycles, top right,4 cycles, bottom left, 8 cycles and bottom right, 16 cycles.

Procedure

The supra-threshold summation procedure for anexperimental condition consisted of having the observerselecting between the two components of a pair which onecontained the sum background-plus-test-frequency. Onlycontrast of the test frequency was increased or decreasedaccording to a forced-choice method [Wetherill andLevitt, 1965]. Contrast of the background in both imagesof a pair was constant at 42%. The criterion for varyingcontrast of the test frequency was that of three correct

Figure 3. (A) Pairs of stimuli used to measure F2(θ) and (B) Pairs of stimuli used for measuring F4(θ)

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choices to decrease contrast of a unit, and one incorrectchoice to increase it by the same amount. Allmeasurements were made binocularly at 150 cm distance,the mean luminance being 2.0 fL. Maximum andminimum luminance were 2.2 and 1.8 fL, respectively.

The temporal sequence was initiated by a warningsignal immediately followed by a 2 s presentation of thefirst stimulus followed by a 1 s inter-stimulus intervalfollowed by a 2 s presentation of the second stimulus andthe answer of the observer. The order of the stimulus pairwas randomly selected. If the response was a correct one,it was followed by a beep and a 3 s inter-trial intervalwould start. The whole experimental session would varyin length depending on the errors and correct choicesmade by the observer as a total of 10 pairs of peaks andvalleys was necessary to end the session. Generally itlasted about 15-25 min.

Each of the 17 experimental conditions required tomeasure each of the four filters was run at least threetimes in different days by at least three differentobservers. Thus, a total of 8-9 functions were measuredfor each filter yielding a sample of 160-180 values to beaveraged across observers per each of the 17 points. Forfilters F2(θ) and F4(θ) nine measurements were made foreach (NAS:3, ERB:3, TPL:3 and NAS:3, TPL:3 andERB:3, respectively). For filters F8(θ) and F16(θ) eightmeasurements were made for each (NAS:3, ERB:3,MMM:2 and NAS:3, MC:3, ERB:2, respectively).

Figures 3 and 4 illustrate pairs of stimuli for filters attest angular frequencies of 2, 4, 8 and 16 cycles. Figure3.A shows pairs 1 and 1+2 cycles (top), 2 and 2+2 cycles(center), and 3 and 3+2 cycles (bottom) for filter F2(θ)centered at 2 cycles. Figure 3.B shows pairs 3 and 3+4cycles (top), 4 and 4+4 cycles (center), and 5 and 5+4cycles (bottom) for F4(θ) centered at 4 cycles. Figure 4.A

Figure 4. (A) Pairs of stimuli used to measure F8(θ) and (B) Pairs of stimuli used for measuring F16(θ).

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shows pairs 4 and 4+8 cycles (top), 8 and 8+8 cycles(center), and 10 and 10+8 cycles (bottom) for F8(θ)centered at 8 cycles. Figure 4.B shows pairs 4 and 4+16cycles (top), 8 and 8+16 cycles (center), and 16 and16+16 cycles (bottom) for F16(θ) centered at 16 cycles.

Results

Figure 5 shows the frequency response functions for thefour measured filters F2(θ), F4(θ), F8(θ) and F16(θ).Angular frequency of the background stimulus is plottedas a function of the amount of contrast necessary toidentify the presence of the test angular frequency in thesummated pair. We are naming this threshold for the testangular frequency summated to each backgroundfrequency as contrast threshold of test in sum. Legends attop right indicate the test frequency.

Our statistical treatment was to estimate the standarderror of the mean for each distribution of 160-180 valuesmeasured per point and correct for sample size using the t-Student statistic to obtain the 99% confidence level .

In previous studies we have established this estimateas being more rigorous than testing mean throughcomparison for two correlated samples or ANOVAtreatment. In the confidence level method, when two errorbars show a 50% overlap, the t-statistic obtained fortesting difference between mean for correlated samplehave been found to be significant to at least p<.05,sometimes smaller than that. Most of the error bars that donot overlap stand in significance at the level p<.000.ANOVAS tend to show significant effects in all factorsand interactions and do not add much information.

We can classify the resulting measured functions intotwo characteristic responses. For filters F2(θ) and F4(θ),

Figure 5. Contrast threshold of test angular frequencies in presence of each of the 17 background frequencies.

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maximum summation effect occurred at test frequencieswith a secondary maximum at their neighboring higherand lower harmonics, respectively. For filters F8(θ) andF16(θ) this tendency changes. It started to change at F8(θ)where maximum summation effect occurred at 2 cycles(lower harmonic) and a relative strong summation effectoccurred at the 8 cycles test frequency. As for F16(θ), thetendency is definitely changed. Maximum summationoccurs at 8 cycles (neighbor lower harmonic). At the testfrequency of 16 cycles, summation was minimum or, inother words, inhibition was almost maximum. We shouldnote, however, that background frequency contrast was setat 42% for all filters regardless of the absolute thresholdfor each test angular frequency. As we can observe fromFigure 1, contrast thresholds for 8 and 16 cycles areconsiderable lower than for 2 and 4 cycles. Nevertheless,the ratio between maximum summation and maximuminhibition for all filters ranged from 2.19 to 2.47indicating clear filtering effects. F2(θ) shows band-pass at2 and 24-64 cycles. F4(θ) shows band-pass at 4, 2 and 48cycles. These results are less clear for test frequencies at 8and 16 cycles. F8(θ) shows band-pass at 2 cycles, 8 cyclesand 48-64 cycles while F16(θ) shows band-pass at 8cycles, 1-5 cycles and 96 cycles. Furthermore, it shows asignificant increase in sensitivity at its higher harmonic,32 cycles, and strongly inhibits passage at 9-24 cycles.Another observation is that generally the frequency ofpassage is surrounded on both sides by inhibition.

Discussion and Conclusion

The results clearly show filter selectivity for the used testfrequencies. This was not true for F16(θ). All filtersshowed again a main result observed in our work of 1992[Simas, et al, 1992], that is, the test frequencies showsummation surrounded on both sides by strong inhibition.We see a trend for filters F2(θ) and F4(θ) which differsfrom F8(θ) and F16(θ). In the former, summation is verysharp and inhibition by higher frequencies [e.g. 5-12cycles for F2(θ) and F4(θ)] are present. Inhibition at 1 and3 cycles is also clear. As for F8(θ), although summation issharp at 8 cycles, we observe that adding frequencies inthe ranges of 1-4 cycles and 32-96 cycles also increasesensitivity to the test frequency, i.e. 8 cycles. The case ofF16(θ) is quite different, adding frequencies in the range of1-5 cycles as well as its lower harmonic, 8 cycles,increases sensitivity to 16 cycles. On the other hand,frequencies at 9-24 strongly inhibits detection of 16cycles. We should remind that additions were made inphase within the vertical/horizontal axis.

While results for test frequencies at 2, 4 and 8 cyclesconforms to our expectations, the characteristic responsefor F16(θ) is not so easy to interpret. One possibility is thatthe visual system is integrating primarily in a lower rangeof angular frequencies (i.e. 1-8 cycles) and interpolateshigher frequencies. Another possibility is due to the factthat we used 42% contrast for all the backgroundfrequencies. This amount of contrast is 3.4, 5.9, 11 and 15times the absolute threshold for 2, 4, 8 and 16 cycles,respectively. So, we could replicate the experiment for 8and 16 cycles using background frequency contrastsadjusted to be no higher than 5 times their respectiveabsolute thresholds. Finally, we would like to emphasizeour interest in evaluating the involvement of lowerangular frequencies in face processing. We believe thatcells found to be selective for faces [e.g. Bruce, Desimone& Gross, 1981] are responding to coupled angular/radialfrequencies as in Eq.#1.

Note

We thank CNPq (Grant # 52.3791/95-0) andFACEPE (Grant # APQ-0606-7.07/97) for financialsupport of this work.

ReferencesBlakemore, C., & Campbell, F.W. (1969) On the existence ofneurones in the human visual system selectively sensitive to theorientation and size of retinal images. Journal of Physiology,203: 237-260.

Bruce, C.J. Desimone, R. & Gross, C.G. (1981) Visualproperties of neurons in a polysensory area in superior temporalsulcus of the macaque. Journal of Neurophysiology, 46, 369-384.

Dodwell, P. C. (1983) The Lie group transformation model ofvisual perception. Perception & Psychophysics, 34, 1-16.

Emerson, V.F., Humphrey, G.K. & Dodwell, P.C. (1985)Colored aftereffects contingent on patterns generated by Lietransformation groups. Perception & Psychophysics, 37: 155-162.

Flanagan, P. & Dodwell, P.C. (1991) Pattern-specific contrastthreshold elevation. Spatial Vision, 5: 159-187.

Gallant, J.L., Braun, J. & Van Essen, D.C. (1993) Selectivityfor polar, hyperbolic, and Cartesian gratings in macaque visualcortex. Science, 259, 100-103.

Gallant, J.L., Connor, C.E., Rakshit, S., Lewis, J.W. & VanEssen, D.C. (1996) Neural responses to Polar, Hyperbolic, andCartesian gratings in area V4 of the macaque monkey. Journalof Neurophysiology, 76, 2718-2739.

M. L. B. SIMAS E N. A. SANTOS

Anais do XI SIBGRAPI, outubro de 1998

8

Graham, N., (1972) Spatial frequency channels in the humanvisual system: effects of luminance and pattern drift rate. VisionResearch, 12: 53-68.

Graham, N., Sutter, A. & Venkatesan, C. (1993) Spatial-frequency- and orientation-selectivity of simple and complexchannels in region segregation. Vision Research, 33, 1893-1911.

Hess, R.F. & Hayes, A. (1994) The coding of spatial positionby human visual system: Effects of spatial scale and retinaleccentricity. Vision Research, 34, 625-643.

Hess, R.F. & Wilcox, L.M. (1994) Linear and non-linearfiltering in stereopsis. Vision Research, 34, 2431-2438.

Hess, R.F. & Field, D.J. (1995) Contour integration acrossdepth. Vision Research, 35, 1699-1711.

Hoffman, W.C. (1966) The Lie algebra of visual perception.Journal of Mathematical Psychology, 3, 65-98. Errata, ibid., 4(1967), 348-440.

Hubel, D.H. & Wiesel, T.N. (1962) Receptives fields andfunctional architecture in cat’s visual cortex. Journal ofPhysiology, 160: 106-154.

Hubel, D.H. & Wiesel, T.N. (1968) Receptives fields andfunctional architecture of monkey sriate cortex. Journal ofPhysiology, 195: 215-243.

Kobatake, E. & Tanaka, K. (1994) Neuronal selectivities tocomplex object features in the ventral visual pathway ofmacaque cerebral cortex. Journal of Neurophysiology, 71: 856-867.

Koenderink, J.J. & van Doorn, A.J. (1990) Receptive fieldfamilies. Biological Cybernetics, 63:291-297.

Kulikowski, J.J. & King-Smith, P.E. (1973) Spatial arrangementof line, edge and grating detectors revealed by sub-thresholdsummation. Vision Research, 13: 1455-1478.

Pollen. D.A., Nagler, M., Daugman, J.G., Kroaner, R. &Koenderink, J.J. (1984) Use of Gabor elementary functions toprobe receptive field substructure of posterior inferotemporalneurons in the owl monkey. Vision Research, 24, 233-242.

Simas, M. L. de B. (1985) Linearity and domain invariance inthe visual system. PhD thesis dissertation. Queen's University atKingston, Ontario, Canada. University Microfilms International.Ann Arbor: Michigan, 1985. "Publication n. 86-17940".

Simas, M. L. de B. and Dodwell, P. C. (1990) Angularfrequency filtering: a basis for pattern decomposition. SpatialVision, 5, 59-74.

Simas, M. L. de B., Frutuoso, J. T., & Vieira, F. M. (1992)Inhibitory side bands in multiple angular frequency filters in thehuman visual system. Brazilian Journal of Medical andBiological Research, 25: 919-923.

Simas, M.L. de B., Frutuoso, J.T. & Santos, N.A. dos (accepted)Jo targets and sinewave gratings seem to be processed by distinctcontrast mechanisms as revealed by suprathreshold summation.

Simas, M.L. de B., Santos, N.A. dos & Thiers, F.A. (1997)Contrast sensitivity to angular frequency stimuli is higher thanthat for sinewave gratings in respective middle range. BrazilianJournal of Medical and Biological Research, 30: 633-636.

Simas, ML de B. & Santos, NA (1997) Human visual contrastdetection of radial frequency stimuli defined by Bessel profilesj0, j1, j2, j4, j8 and j16 and its relation to angular frequency.Proceedings of the II Workshop on Cybernetic Vision, IEEEComputer Science, 219-224.

Sneddon, I.N. (1961) Special functions of mathematical physicsand chemistry, New York; Interscience Publishers, p.105.

Tanaka, K. & Saito, H-A. (1989) Analysis of motion of thevisual field by direction, expansion/contraction, and rotationcells clustered in the dorsal part of the medial superior temporalarea of the macaque monkey. Journal of Neurophysiology, 62:626-641.

Tanaka, K., Fukada, Y. & Saito, H-A. (1989) Underlyingmechanisms of the response specificity of expansion/contractionand rotation cells in the dorsal part of the medial superiortemporal area of the macaque monkey. Journal ofNeurophysiology, 62: 642-656.

Tyler, C.W. (1975) Spatial frequency filters in cat visual cortex?Vision Research, 15: 303-304.

Tyler, C.W. (1975) Selectivity for spatial frequency and barwidth in cat visual cortex. Vision Research, 18: 121-122

Von Grünau, M. , Kwas, M. & Dube, S. (1994) The interactionof depth parameters in motion integration with polar plaids.Spatial Vision, 8: 119-140.

Wetherill, G. B. & Levitt, H. (1965) Sequential estimation ofpoints on a psychometric function. The British Journal ofMathematical and Statistical Psychology, 18, 1-10.

Wilson, H.R. & Wilkinson, F. (1997) Evolving concepts ofspatial channels in vision: from independence to nonlinearinteractions. Perception, 26: 939-960.

Wilson, H.R., Wilkinson, F. & Asaad, W. (in press) Evidencefor non-Cartesian units in human form vision.