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Detectability of amplitude- and frequency-modulation of suprathreshold sine-wave gratings
0042.6989/82/030407-10803.OWO Pergamon PressLtd
VisionRet Vol. 22. pp. 407 10 416. 1982 Printed in Great Britain
DETECTABILITY OF AMPLITUDEAND FREQUENCY-MODULATION OF SUPRATHRESHOLD SINE-WAVE GRATINGS J. H. T.
JAMAR,
J. C.
CAMPAGNE
and J. J.
KOENDERINK
Department of Medical and Physiological Physics, State University of Utrecht, Princetonplein 5, 3584 CC Utrecht, The Netherlands (Receioed 19 February
1981; in revisedform
15 June
1981)
Abstract-The detectability of amplitude- and frequency-modulation of sine-wave carrier gratings was measured for a number of carrier frequencies, modulation frequencies and carrier contrasts. The results show that for low modulation frequencies (spatial frequency components close together; separation less than 0.5 x carrier frequency) modulation thresholds are determined by both the amplitude and phase spectrum of the stimulus; for high modulation frequencies (frequency components more widely separated) they are determined by the amplitude spectrum only. The dependence of modulation thresholds on carrier contrast was found to be very similar, irrespective of the modulation frequency, carrier frequency and kind of modulation selected, and was in agreement with known results of contrast-difference threshold measurements and of masking experiments.
INTRODUCTION From many psychophysical experiments evidence has been obtained to support the hypothesis that contrast perception is mediated at any retinal location by several, functionally distinct mechanisms rather than by one mechanism only. These mechanisms are commonly regarded as contrast-sensitive units which have sensitivities that are confined to limited ranges of spatial frequency and to limited regions in space. Ever since the hypothesis of independent mechanisms selectively sensitive to limited ranges of spatial frequency (“spatial frequency channels”) was postulated (Campbell and Robson, 1968) a great deal of attention has been paid to the frequency-domain properties of the contrast-detecting mechanisms. Recently the space-domain aspects of contrast detection have begun to receive more attention (MacLeod and Rosenfeld, 1974; Koenderink and van Doorn. 1978; Wilson and Bergen, 1979). The space-domain and frequency-domain properties of contrast-detecting units are difficult to derive from experimental results because little is known about the interactions among the units, both in space and in the frequency domain. For instance, the results of Sachs et al. (1971) and Kulikowski and King-Smith (1973) led to estimates of l/4 to l/2 octave for the bandwidth of the spatial frequency channels. However, Stromeyer and Klein (1975). Bergen et al. (1979) and Henning et al. (1981) showed that bandwidth estimates are very dependent on the assumptions that are made about how the output of the individual contrast-sensitive units situated at different retinal locations and/or tuned to different ranges of spatial frequency is processed. They arrived at bandwidth estimates of about 1 octave (Stromeyer and Klein) and 1.75 octave (Bergen et al.). A number of interaction types have been
proposed, such as probability summation, spatial pooling (Stromeyer and Klein, 1975) inhibition and disinhibition between spatial frequency channels (Tolburst, 1972; Tolhurst and Barfield, 1978) and “phaseinhibition” between spatially adjacent bar and edge mechanisms (Klein and Stromeyer, 1980). Bodis-Wollner et al. (1973) reported results on the detectability of contrast differences, which they generated by spatially modulating the contrast of a sinewave grating. They found that their results could be predicted from the observer’s contrast sensitivity at the spatial frequency of one of the “sideband frequencies” (the frequency components introduced by the modulation of the contrast). We replicated their results and expanded them to a number of carrier frequencies and carrier contrasts. We also used frequency-modulated sine-wave gratings having approximately the same amplitude- (but not phase-) spectra as the corresponding contrast-modulated gratings. By varying the spatial frequency of the (sinusoidal) modulation function (“modulation frequency”) we investigated how the spatial interactions between contrast-detecting units depend on distance. By comparing our results for contrast and frequency modulation we obtained a bandwidth estimate that is independent of previous bandwidth estimates. METHODS Apparatus
Stationary vertical gratings were generated on the face of a cathode-ray tube by a method similar to that described by Campbell and Green (1965). We used a large screen monitor (HP 1317A; white P-4 phosphor). An analogue multiplier was used to modulate the amplitude of the Z-axis signal (and thus to modulate the amplitude of the contrast on the screen). A
408
J. H. T. JAMARet al.
frequency-modulated Z-axis signal could be produced by means of the frequency-control input of the Z-axis function generator. The relationship between the voltage applied to this input and the output frequency was checked and found to be linear. The amplitudes of Z-axis signal and modulation signal were controlled by a subject-operated microprocessor system. Using the method of Carter and Henning (1971) we measured the harmonic distortion of an (unmodulated) sine-wave grating and the intermodulation distortion caused by amplitude modulation of a sinewave grating. For contrasts up to 30’4 the amplitudes of the higher harmonics were at least 1.75 log units below the first harmonic. All results reported here were measured at contrasts below 30%, except for three points on each of the curves of Fig. 6. Intermodulation distortion (a frequency component at the modulation frequency) depended on both contrast and on modulation percentage. We checked that this component was well below its threshold contrast for all amplitude-modulation thresholds reported here. The target subtended 4 x 4 deg2; mean retinal illuminance was 20 td and the surround was dark. The target was viewed monocularly through an artificial pupil (2.8 mm dial. There was a small black dot in the centre of the screen to aid fixation and accommodation
It is useful to define the modulationindex m’, which is related to the modulation percentage m according to: (3a)
(3b) In the appendix we derive that AM and FM gratings with equal, small modulation indexes have approximately equal amplitude spectra. There are spatial frequency components at f, and fc + fm with amplitudes equal to c and m’c/2 respectively. However, the phase relationships between the spectral components are different in AM and FM gratings. According to (2). +m& is also the maximum phase shift (in radians of phase angle) of points in the frequency modulated grating with respect to the unmodulated grating. The maximum shift in position is thus + mkM/2nfc(degrees of visual angle). Experimental
procedure
A sine-wave grating, the contrast of which was suprathreshold, was presented to the subject. The subject determined his AM or FM modulation threshold (i.e. the threshold value of the modulation percentage Stimuli m) by the method of limits. The criterion was to detect The stimuli used in these experiments were ampliany irregularity in the grating. Thresholds were detertude-modulated (AM) and frequency-modulated (FM) mined for a number of settings offcrf, and c. Within sine-wave gratings. This means that either the conone session, lasting l-2 hr. only one or two carrier trast or the spatial frequency varies across the grating. frequencies were used. Both the AM and the FM The modulation was always sinusoidal. The luminthreshold for the same setting of fc, f, and c were ance is given as a function of position by the formulas always determined close together in time. The (1) and (2) for AM and FM gratings respectively (a measurements were carried out by two experienced detailed explanation of these formulas is given in the observers, J.C. and J.J. aged 26 and 24 yr respectively. Both observers had corrected-to-normal acuity appendix) : (-2D/C = - 1.5 axis 167 and -0.75 D respectively). All results presented are averages of at least three L(x) = Lo’ 1 + c 1 + $+- sin 27rf,x sin 2nfcx threshold determinations. For all the data points 0 > I I i shown the standard error was about 0.05 log unit. The (1) dependence of modulation thresholds on modulation frequency was determined for carrier frequencies of 0.5, 1, 2, 4, 8, 16 and 32 c/deg. Between one and four x L(x) = LO’ 1 + c.sin Z&-Y - __m .Los27r~ mI 100”” b contrast levels were used at each carrier frequency. L ( We chose contrast levels 0.3, 0.9, 1.5 and 2.1 log units above the threshold contrast for the unmodulated In these formulas .x = horizontal position. where grating in order to have comparable contrast perceptions or “subjective contrasts” at different carrier frex = 0 represents the centre of the stimulus, c = averquencies, i.e. in order to correct for the differences in age contrast, f, = carrier frequency and f, = modulation frequency. L = average luminance of the the contrast sensitivity for different carrier frequenscreen; m = modulation percentage (0 < m I lOO’$$. cies. For this reason we began the experiments by A modulation percentage of for instance 15% determining the threshold contrast as a function of spatial frequency. means for AM that the extreme values of the contrast differ by 15% from the average contrast c. For FM this means that local spatial frequency differs by a RESULTS maximum of 15% from the average fe (see the appenThe dependence of modulation thresholds on dix for a definition of local frequency). Figures la and modulation frequency for carrier frequencies of 1, 4 lb show examples of such gratings.
Fig. la. Example of a frequency-modulated frequency of this stimulus was 8 c/deg and Fig. lb. Example
of an amplitude-modulated
sine-wave grating. In the experimental the modulation frequency was 1 c/deg 4 x 4deg’). sine-wave 409
grating.
Same frequencies
set-up the carrier (target subtended as in Fig. la.
411
Detectability of AM and FM of sine-wave gratings
w
mIxI
10 -
t . 0.1
IP
I
.
1
1-
1
f,,, Icpd) Fig. 2. Threshold values of amplitude modulation percentage (circles) and frequency modulation percentage (squares) plotted against modulation frequency for carrier frequencies of 1, 4 and 16 c/deg (from left to right). Carrier frequencies are indicated by dashed lines. These results were obtained with mean contrast set 0.9 log unit above the threshold contrast for the unmodulated carrier grating. Observer J.C.
I
m(r
4 .
4
1
10
f, kpdl
Fig. 3. Same results for observer J.J. than AM thresholds and the dependence on modulation frequency is rather weak; for f_ 7 O.Sfc the dependence is stronger and AM and FM thresholds become comparable. It appeared from the observers’ reports that in the latter case (particularly for carrier frequencies above 2 c/deg) they were actually performing a contrast threshold determination for the sideband spatial frequency fc - f, or fc -t f,, whereas at low modulation frequencies they had the impression that they were detecting an irregularity, which, when modulation was raised slightly above threshold, could be seen to consist of a hi%ence in contrast or spatial frequency between stimulus positions. In Figs 4 and 5 threshold curves for different carrier frequencies are plotted together. Here, contrast is *One might think that there would have been a com- 0.9log unit suprathreshold, too. In Fig. 4 (AM mon asymptote if a different measure of modulation thresholds) the low-&-ends of the curves fall around strength had been used. From equation (2) it can be seen a common asymptotic curve: in this range AM that in an FM grating some cycles are shifted to the right thresholds are almost independent of Carrie; freor to the left with respect to the unmodulated grating. Thus we might have plotted threshold values of this shift quency when contrast is kept 0.9log unit supra[expressed in degrees of visual angle or in degrees of phase threshold. The same holds for the results obtained at angle of the carrier frequency (cf: Wcstheimer, 197811 the other contrasts levels (0.3, 1.5 and 2.1 log unit against modulation frequency. However, neither of these possibilities yields a common asymptotic FM curve, nor is suprathreshold) and those obtained with subject J.J. There is no common asymptote for the FM threshold such a curve obtained if we plot threshold values of m’ against modulation frequency. curves (Fig. 5)+.
and 16cfdeg is shown in Figs 2 and 3 for observers J.C. and J.J. respectively. These. results were obtained for the contrast level 0.9 log unit above the threshold for the unmodulated grating. It should be noted that retinal inhomogeneity within our 4 deg target field might have a non-negligible effect on the results when we are working with high carrier frequencies and low modulation frequencies. Figures 2 and 3 show that this may have a serious effect only in the case of the highest fe (16 c/deg) and the lowest jm (0.25 c/deg). Generally two ranges of modulation frequency can be distinguished: for low modulation frequencies (fm -z OSj;) FM thresholds are considerably lower
412
J. H. T.
JAMAR V[ d
m I%)
10 -
f,
(CPd)
Fig. 4. Amplitude
modulation thresholds for 6 carrier frequencies (filled circles: 0.5 c/deg: empty circles: 1 cideg: filled triangles: Zc/deg: empty triangles: 4c/deg; filled squares: 8 c/deg: empty squares: 16 c/de@. Carrier contrast 0.9 log unit above threshold contrast. Observer J.C.
The effect that changing mean contrast c has on frequency modulation thresholds is shown in Fig. 6. The dependence for AM thresholds was very similar. It is obvious from the figure that the dependence on mean contrast is also very similar for all modulation frequencies. There is a tendency for the slope of the curves to decrease with increase in contrast and to approach zero for the highest contrasts.
can be predicted using the hypothesis of independent spatial frequency channels (Campbell and Robson. 1968): modulation is detected as soon as one of the sideband frequencies reaches an amplitude equal to its threshold contrast when measured in isolation. This means that the predicted modulation thresholds are determined merely by the amplitude spectrum of the stimulus. Bodis-Wollner et al. (1973) found that their AM threshold data could be fairly well predicted in this way. (They used .fl = 18 c/deg andf, from 1.8 to 16.2 c/deg; carrier contrast “well above threshold”.) However, Henning et ul. (1975) presented results where the measured AM threshold was a factor of three below the predicted one (,/, = 9.5 c/deg. fm = 1.9c;deg. carrier contrast 6.3”“). In Fig. 7 the predictions are compared with the measured modulation thresholds. The results of Fig. 2 have been re-plotted. the modulation thresholds now being expressed as the modulation index ~1’. Because AM and FM stimuli with equal modulation index tti have (approximately) equal sideband amplitudes. (approximately) equal values of m’ are predicted for AM and FM (difference <0.014 log unit for m’ < 0.5). Comparing predictions and AM thresholds of Fig. 7, we see that our results he somewhere between those of Bodis-Wollner ef al. (1973) and those of Henning er (11.(1975). However. there is no real inconsistency here. We find that the dependence of AM thresholds on carrier contrast is such that the results of Henning er ul. are replicated at lOOr
r
_
DISCUSSION
When we modulate the amplitude or the spatial frequency of a sine-wave grating, we introduce sideband spatial frequency components into the stimulus. One should bear in mind that what we call here a spatial frequency component represents in fact a spatial frequency band having a width of about l/4 c/deg because of the limited width of the target field. We investigated to what extent modulation thresholds
10
I
ml%) 10 1
100 10
m (%I
10
I
Fig. 6. Dependence of FM thresholds on carrier contrast for carrier frequencies of 1. 4 and 16c/deg. Observer J.C. Upper set of curves: h = 1 c/deg and ,‘, = 0.25 c/deg 1”
1
f,
(cpd)
Fig. 5. Frequency modulation thresholds. Same carrier frequencies, contrast and observer as in Fig. 4.
(circles). and 0.8 c/deg (squares). Middle set: fc = 4 c/deg and& = 0.25 c/deg. (circles) and 3.2 c/deg (squares). Lower set: fc = 16 c/deg and fn = 0.25 c/deg (circles), 4 c/deg (triangles) and’12.8 c/deg (squares). The dashed lines indicate the threshold contrast for each of the carrier frequencies,
413
Detectability of AM and FM of sine-wave gratings
m’ t 1 t
:b--._ --__ - ---__,
0.1
I
.
I
1
I
1
1
,
10
f, (cpd) Fig. 7. Same data as in Fig. 2. Here thresholds are expressed as the modulation index m’ instead of modulation percentage m. Circles are AM results, squares are FM results. The dashed curves are predictions based on the assumption of independent detection of a sideband spatial frequency (see text).
(roughly at 24 times threshold contrast) and those of Bodis-Wollner et al. at somewhat higher contrasts (roughly 10 times threshold contrast). For even higher contrasts measured thresholds are up to a factor of 4 above the predicted ones. Figure 7 shows also (a) that the prediction of equal threshold values of m’ for AM and FM holds only for a limited range of modulation frequencies (fm > 0.5fC) and (b) that even in this range the predictions differ from the measured thresholds. low contrasts
A way of interpreting (a) is suggested by Zwicker and Feldtkeller (1967) who performed hearing experiments analogous to our visual experiments: they measured modulation thresholds for amplitude and frequency modulation of tones. Figure 8 is a re-plot of their Fig. 69.3. Note the striking similarity to the curves in Fig. 7. They found that the modulation frequency at which the AM curve and the FM curve join is equal to half the width of a “critical band” centred on the carrier frequency. They stated that from this
Fig. 8. Amplitude and frequency modulation thresholds for a I kHz carrier tone as a function of modulation frequency, measured by Zwicker and Feldtkeller (1967) for three distinct sound pressure levels (re-plot).
J. H. T.
JAMAR
i 1
contrast Fig. 9. Filled and empty symbols: ratio of sideband amplitude at modulation threshold to threshold amplitude (threshold contrast) for sideband frequency when presented alone (log units) plotted against carrier contrast. These results all relate to a sideband frequency 1.8 times the carrier frequency. Circles: f< = 0.5 c/deg: triangles: fc = 1 c/deg; squares: f, = Zc/deg; inverted triangles: A = 4c/deg: filled and empty symbols for observers J.C. and J.J. respectively. Because this figure relates to the “high&-range”, where t&, and m& do not differ significantly. we present averages of AM and FM results. Symbols with centre dot: results re-plotted from Fig. 2 of Tolhurst and Barfield (1978). Increase in sensitivity to a test spatial frequency of 8.5 c/deg when a masking stimulus of 4.25 cideg is presented simultaneously, plotted against contrast masking stimulus. Data for two observers.
of the
modulation frequency upwards sideband frequencies differ sufficiently to stimulate distinct “critical bands” (in this case relative phases of frequency components would not be important. yielding equal threshold values of m’ for AM and FM). If the same argument applies to vision, then the bandwidth of visual mechanisms can be estimated. From Fig. 7 it is seen that the joining point is at f, = (0.65 k 0.1) /‘f,. yielding a bandwidth of (1.3 + 0.2)j; or about 2 octaves. However, in our experiments one of the two sidebands generally remained subthreshold because of the slope of the contrast sensitivity curve. Thus it seems that for the visual case we would have to suppose that from the joining point upwards. one of the sidebands and the carrier frequency stimulate different channels. In that case we would find a bandwidth of about 1 octave, which is in agreement with the results of Stromeyer and Klein (1975). The fact that the predictions differ from the measurements even for high modulation frequencies (b) can be related to the results of the masking experiments of Tolhurst and Bartfield (1978). In Fig. 9 we compare some of their results with some of our own results. In this figure we interpret the difference between measurements and predictions as an increase or decrease in contrast sensitivity to the sideband frequency, caused by the presence of the carrier. For all the data shown in the figure the ratio of sideband (or test) frequency to carrier (or mask) fre-
or u/
quency is about 2. The results of Tolhurst and Barfield re-plotted in Fig. 9 were obtained only at a masking frequency of 4.25 c/deg. Although their psychophysical method was rather different from ours (they used a forced choice technique) there is reasonable agreement between their results and those we obtained at 4 c/deg. However, there is little indication of increased sensitivity to the sideband frequency for carrier frequencies below 2 or above 4cpd. In our results the sensitivity increase might result from the contribution of both sidebands to detection. because when the carrier frequency is 2 or 4 c/deg (i.e. near the maximum of the contrast sensitivity curve) there is only a small difference in the contrast sensitivity to the sideband spatial frequencies. However. the fact that Tolhurst and Barfield found the same effect suggests that there is some other, unknown cause for the sensitivity increase. We shall now consider the data for low modulation frequencies. Henning rt al. (1975) presented data showing that an AM grating and a grating at its modulation frequency interact. This suggests that AM might be detected by the spatial frequency channel at the modulation frequency. However, we do not believe this to be the case, because if it was, one would expect AM thresholds to depend on b in the same way as contrast thresholds depend on frequency. We found a much weaker dependence. Also, the observers’ reports (they reported seeing an “irregularity” in the grating) seem to indicate that the observers did not use the spatial frequency channel at the modulation frequency for detecting AM. The fact that the curves for AM and FM in Fig. 7 are far apart at low modulation frequencies means that for low J;. modulation thresholds are not determined by the amplitude spectrum of the stimulus only: the phase relationships have to be taken into account. This can be done in the most straightforward way by considering the space-domain structure of the stimuli. The task of the visual system is thus to compare the contrast or the spatial frequency between patches of grating which have a width of one halfcycle of the modulation frequency and whose centres are separated by the same amount. With increase in fm modulation thresholds might either decrease (because of the decrease in the comparison distance) or increase (because of the decrease in the number of carrier-cycles per half-cycle of modulation). The results show that the first effect is the more important and leads to a decrease in modulation thresholds particularly AM thresholds) with increasingf,. This holds up to a modulation frequency of about 0.5 XJ$, where only one cycle of carrier is available per half-cycle of modulation. This suggests that the spatial extent of contrast detecting units does not exceed one cycle. Part of the difference in the behaviour of AM and FM thresholds for low modulation frequencies can perhaps be understood in the following way. Consider a collection of contrast detecting units of similar size (and thus having similar spatial frequency tuning).
415
Detectability of AM and FM of sine-wave gratings
The (amplitude or frequency) modulation of a sinewave stimulus will cause the stimulation of the units of that collection to be a function of retinal position. Modulation detection is possible if the visual system has some “detector” for the stimulation differences. For FM there is another clue for detection: the fact that stimulation is distributed over a broader collection of unit sizes than in the case of an unmodulated grating. This might account for the differences between the AM and FM thresholds. Figure 6 shows how modulation thresholds depend on carrier contrast. Contrast-difference thresholds depend on contrast in rather the same way (e.g. Kohayakawa,
1972; Legge, 1981). This is not surpris-
ing as far as AM detection for low modulation frequencies is concerned, because here the observer in fact detects contrast difference. Figure 9 shows (indirectly, because increase in sensitivity has been plotted instead of the threshold values themselves) that the dependence of modulation thresholds on carrier contrast (curves indicated by filled and empty symbols) resembles the dependence of threshold contrast on the contrast of a masking grating presented along with the test grating (symbols with centre dot). This is not surprising for high modulation frequencies, because in this case we were in fact performing masking experiments. However, it is remarkable that the dependence is similar for all modulation and carrier frequencies and for both kinds of modulation, in spite of the fact that the subjects probably used various ways of detecting the modulation in various situations. This point is particularly salient when formulated in terms of spatial frequency channels: when modulation frequency is low, modulation detection has to be performed within one channel, as all frequency components are very close together. For high modulation frequencies modulation is detected by separate detection of a sideband spatial frequency, i.e. by a different channel. It is remarkable that in both cases thresholds are affected similarly by the carrier contrast.
Carter B. E. and Henning G. B. The detection of gratings in narrow-band visual noise. J. physiol., Land. 219. 355-365.
Goldman S. (1948) Frequency analysis, modulation and noise. Republication 1967. Dover publications, New York. Henning G. B., Hertz B. G. and Broadbent D. E. (1975) Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency. Vision Res. IS, 887-897. Henning G. B.. Hertz B. G. and Hinton J. L. (1981) Effects of different hypothetical detection mechanisms on the shape of spatial-frequency filters inferred from masking experiments: I. noise masks. J. opt. Sot. Am. 71, 574-581. Klein S. and Stromeyer C. F. III (1980) On inhibition between spatial frequency channels: adaptation to complex gratings. Vision Res. 20, 459-466. Koenderink J. J. and van Doorn A. J. (1978) Visual detection of spatial contrast; influence of location in the visual field, target extent and illuminance level. Biol. Cybernc:. .30, 157-167. Kohayakawa Y. (1972) Contrast-difference thresholds with sinusoidal gratings. J. opt. Sot. Am. 62, 584-587. Kulilowski J. J. and King-Smith P. E. (1973) Spatial arrangement of line, edge and grating detectors revealed by subthreshold summation. Vision Res. 13, 1455-1478. Legge G. E. (1981) A power law for contrast discrimination. Vision Res. 21, 457-467. MacLeod I. D. G. and Rosenfeld A. (1974) The visibility of gratings: spatial frequency channels or bar-deteciing units? Vision Res. 14, 909-915. Sachs M. B., Nachmias J. and Robson J. G. (1971) Spatial frequency channels in human vision. J. opt. Sec. Am. 61. 1176-1186. Stromeyer C. F. II1 and Klein S. (1975) Evidence against narrow-band spatial frequency channels in human vision: the detectability of frequency modulated gratings. Vision Res. 15, 899-910.
Tolhurst D. J. (1972) Adaptation to square-wave gratings: inhibition between spatial frequency channels in the human visual system. J. Physiol., Lond. 226, 231-248. Tolhurst D. J. and Barfield L. P. (1978) Interactions between spatial frequency channels. I/ision Rex 18. 951-958.
Westheimer G. (1978) Spatial phase sensitivity for sinusoidal grating targets. Vision Res. 18, 1073-1074. Wilson H. R. and Bergen J. R. (1979) A four mechanism model for threshold spatial vision. Vision Rex 19, 19-32. Zwicker E. and Feldtkeller R. (1967) Dus Ohr u/s Nuchrichrenempjuenger. S. Hirzel, Stuttgart.
Acknowledgements-The authors are indebted to the Netherlands Organization for the Advancement of Pure Research (Z.W.O.) for giving financial support for part of this investigation. We are grateful to C. Noorlander and W. A. van de Grind for making critical remarks on the manuscript.
For an unmodulated sine-wave grating the luminance as a function of position x is given by:
REFERENCES
L(X) = I-0’ I1 + cO.sin ZrrjI:u!
Bergen J. R., Wilson H. R. and Cowan J. D. (1979) Further evidence for four mechanisms mediating vision at threshold: sensitivity to complex gratings and aperiodic stimuli. J. opt. Sot. Am. 69. 1580-1587. Bodis-Wollner I., Diamond S. P., Orlofsky A. and Levinson J. (1973) Detection of spatial changes of the contrast of a grating pattern. J. opt.-Sot. Am. 6, 1296 (A). Campbell F. W. and Green D. G. (1965) Optical and retinal factors affecting visual resolution. J. Physiol., Lond. 181, 576593.
Campbell F. W. and Robson J. G. (1968) Application of Fourier analysis to the visibility of gratings. J. Physiol., Lond. 197, 551-566.
APPENDIX
IAl)
where: LO =
mean luminance
CO =
comast
/ =
=
LL,,,
-
L,,,,,J!(L,,,,,, + L ,“,“I
spatial frequency.
In our stimuli x = 0 always represents the centre of the stimulus. When we replace co by a sinusoidal function of x (the value of that function can then be called local contrast). we get an amplitude-modulated sine-wave grating: m -sin loo”,,
2nf+,x) sin 2nL.y\ (AZ)
416
J. H. T.
JAMAR
where:
ef (I/.
For FM: c = average contrast m = modulation percentage.
L(x) = LO. I I + (” [J,(X) sin 27rfc_\-
We now have to distinguish between modulation frequency (fm) and carrier frequency (fc). If we want to produce a frequency-modulated sine-wave. i.e. to make local frequency a function of position, we need a definition of local frequency, The usual definition is (Goldman, 1948):
- J,(X)COS27r(f, +.fm,,.x- J,(X)cos2n(~;
- f,).Y
- J,(X)sin2n(fc
+ 2&).x - JL(X)sin2n(f;
- 2/,).x
+ J,(X)cos2n(f;
+ 3,/,J.Y+ J,(X)cos2n(j,
- 31,)s
+ Jq(X)sin 2rr(/; + 4f,J.x + J,(X) sin 2n( 1; - 4f,)u (A%
- . ..]I (A3)
fL(X) = & f:’ which is applicable to signals of the form:
(A41
S(x) = sin 0(x).
where X = m/100”,, x fc/fm and J, indicate Bessel functions of the first kind and of order n. When X is less than unity:
Note that this definition gives the correct value off for unmodulated sine-waves. We want fL(x) to be a sinusoidal function of position according to
J,(X) - I J,(X) 10.5x J,(X) 10
fL(x) = j;.
I + i&
0
(
‘sin 2rrf,x)
(A5)
So again we can call m the modulation percentage. Combining (A3) and (A5) and integrating we obtain: O(x) = 27rQ - $&
f- .cos27&.x + oo.
(n > I)
In this case it follows that AM and FM gratings have approximately equal amplitude spectra when their modulation percentages tnAMand rrrFMare such that:
(A6)
” L If we choose (I, = 0, our FM gratings are given by: L(x) = La.
i
I + c,sin
i
2nL.x - i&.L
II .I,
.cos2rrS,x
tAl0)
11 (A7)
The frequency spectra of AM and FM gratings [(A2) and (A7) respectively] are given by the Fourier expansions: For AM:
-
~
2 loo”,,
I - 2 &
(Al I)
cos2x(f, + f”)Z
0
cosZn(.f, - fnt)x
Ii
0x3)
Thus AM and FM gratings which have equal modulation index M’ will have approximately equal amplitude spectra. provided m’ is less than I.
VisionRet Vol. 22. pp. 407 10 416. 1982 Printed in Great Britain
DETECTABILITY OF AMPLITUDEAND FREQUENCY-MODULATION OF SUPRATHRESHOLD SINE-WAVE GRATINGS J. H. T.
JAMAR,
J. C.
CAMPAGNE
and J. J.
KOENDERINK
Department of Medical and Physiological Physics, State University of Utrecht, Princetonplein 5, 3584 CC Utrecht, The Netherlands (Receioed 19 February
1981; in revisedform
15 June
1981)
Abstract-The detectability of amplitude- and frequency-modulation of sine-wave carrier gratings was measured for a number of carrier frequencies, modulation frequencies and carrier contrasts. The results show that for low modulation frequencies (spatial frequency components close together; separation less than 0.5 x carrier frequency) modulation thresholds are determined by both the amplitude and phase spectrum of the stimulus; for high modulation frequencies (frequency components more widely separated) they are determined by the amplitude spectrum only. The dependence of modulation thresholds on carrier contrast was found to be very similar, irrespective of the modulation frequency, carrier frequency and kind of modulation selected, and was in agreement with known results of contrast-difference threshold measurements and of masking experiments.
INTRODUCTION From many psychophysical experiments evidence has been obtained to support the hypothesis that contrast perception is mediated at any retinal location by several, functionally distinct mechanisms rather than by one mechanism only. These mechanisms are commonly regarded as contrast-sensitive units which have sensitivities that are confined to limited ranges of spatial frequency and to limited regions in space. Ever since the hypothesis of independent mechanisms selectively sensitive to limited ranges of spatial frequency (“spatial frequency channels”) was postulated (Campbell and Robson, 1968) a great deal of attention has been paid to the frequency-domain properties of the contrast-detecting mechanisms. Recently the space-domain aspects of contrast detection have begun to receive more attention (MacLeod and Rosenfeld, 1974; Koenderink and van Doorn. 1978; Wilson and Bergen, 1979). The space-domain and frequency-domain properties of contrast-detecting units are difficult to derive from experimental results because little is known about the interactions among the units, both in space and in the frequency domain. For instance, the results of Sachs et al. (1971) and Kulikowski and King-Smith (1973) led to estimates of l/4 to l/2 octave for the bandwidth of the spatial frequency channels. However, Stromeyer and Klein (1975). Bergen et al. (1979) and Henning et al. (1981) showed that bandwidth estimates are very dependent on the assumptions that are made about how the output of the individual contrast-sensitive units situated at different retinal locations and/or tuned to different ranges of spatial frequency is processed. They arrived at bandwidth estimates of about 1 octave (Stromeyer and Klein) and 1.75 octave (Bergen et al.). A number of interaction types have been
proposed, such as probability summation, spatial pooling (Stromeyer and Klein, 1975) inhibition and disinhibition between spatial frequency channels (Tolburst, 1972; Tolhurst and Barfield, 1978) and “phaseinhibition” between spatially adjacent bar and edge mechanisms (Klein and Stromeyer, 1980). Bodis-Wollner et al. (1973) reported results on the detectability of contrast differences, which they generated by spatially modulating the contrast of a sinewave grating. They found that their results could be predicted from the observer’s contrast sensitivity at the spatial frequency of one of the “sideband frequencies” (the frequency components introduced by the modulation of the contrast). We replicated their results and expanded them to a number of carrier frequencies and carrier contrasts. We also used frequency-modulated sine-wave gratings having approximately the same amplitude- (but not phase-) spectra as the corresponding contrast-modulated gratings. By varying the spatial frequency of the (sinusoidal) modulation function (“modulation frequency”) we investigated how the spatial interactions between contrast-detecting units depend on distance. By comparing our results for contrast and frequency modulation we obtained a bandwidth estimate that is independent of previous bandwidth estimates. METHODS Apparatus
Stationary vertical gratings were generated on the face of a cathode-ray tube by a method similar to that described by Campbell and Green (1965). We used a large screen monitor (HP 1317A; white P-4 phosphor). An analogue multiplier was used to modulate the amplitude of the Z-axis signal (and thus to modulate the amplitude of the contrast on the screen). A
408
J. H. T. JAMARet al.
frequency-modulated Z-axis signal could be produced by means of the frequency-control input of the Z-axis function generator. The relationship between the voltage applied to this input and the output frequency was checked and found to be linear. The amplitudes of Z-axis signal and modulation signal were controlled by a subject-operated microprocessor system. Using the method of Carter and Henning (1971) we measured the harmonic distortion of an (unmodulated) sine-wave grating and the intermodulation distortion caused by amplitude modulation of a sinewave grating. For contrasts up to 30’4 the amplitudes of the higher harmonics were at least 1.75 log units below the first harmonic. All results reported here were measured at contrasts below 30%, except for three points on each of the curves of Fig. 6. Intermodulation distortion (a frequency component at the modulation frequency) depended on both contrast and on modulation percentage. We checked that this component was well below its threshold contrast for all amplitude-modulation thresholds reported here. The target subtended 4 x 4 deg2; mean retinal illuminance was 20 td and the surround was dark. The target was viewed monocularly through an artificial pupil (2.8 mm dial. There was a small black dot in the centre of the screen to aid fixation and accommodation
It is useful to define the modulationindex m’, which is related to the modulation percentage m according to: (3a)
(3b) In the appendix we derive that AM and FM gratings with equal, small modulation indexes have approximately equal amplitude spectra. There are spatial frequency components at f, and fc + fm with amplitudes equal to c and m’c/2 respectively. However, the phase relationships between the spectral components are different in AM and FM gratings. According to (2). +m& is also the maximum phase shift (in radians of phase angle) of points in the frequency modulated grating with respect to the unmodulated grating. The maximum shift in position is thus + mkM/2nfc(degrees of visual angle). Experimental
procedure
A sine-wave grating, the contrast of which was suprathreshold, was presented to the subject. The subject determined his AM or FM modulation threshold (i.e. the threshold value of the modulation percentage Stimuli m) by the method of limits. The criterion was to detect The stimuli used in these experiments were ampliany irregularity in the grating. Thresholds were detertude-modulated (AM) and frequency-modulated (FM) mined for a number of settings offcrf, and c. Within sine-wave gratings. This means that either the conone session, lasting l-2 hr. only one or two carrier trast or the spatial frequency varies across the grating. frequencies were used. Both the AM and the FM The modulation was always sinusoidal. The luminthreshold for the same setting of fc, f, and c were ance is given as a function of position by the formulas always determined close together in time. The (1) and (2) for AM and FM gratings respectively (a measurements were carried out by two experienced detailed explanation of these formulas is given in the observers, J.C. and J.J. aged 26 and 24 yr respectively. Both observers had corrected-to-normal acuity appendix) : (-2D/C = - 1.5 axis 167 and -0.75 D respectively). All results presented are averages of at least three L(x) = Lo’ 1 + c 1 + $+- sin 27rf,x sin 2nfcx threshold determinations. For all the data points 0 > I I i shown the standard error was about 0.05 log unit. The (1) dependence of modulation thresholds on modulation frequency was determined for carrier frequencies of 0.5, 1, 2, 4, 8, 16 and 32 c/deg. Between one and four x L(x) = LO’ 1 + c.sin Z&-Y - __m .Los27r~ mI 100”” b contrast levels were used at each carrier frequency. L ( We chose contrast levels 0.3, 0.9, 1.5 and 2.1 log units above the threshold contrast for the unmodulated In these formulas .x = horizontal position. where grating in order to have comparable contrast perceptions or “subjective contrasts” at different carrier frex = 0 represents the centre of the stimulus, c = averquencies, i.e. in order to correct for the differences in age contrast, f, = carrier frequency and f, = modulation frequency. L = average luminance of the the contrast sensitivity for different carrier frequenscreen; m = modulation percentage (0 < m I lOO’$$. cies. For this reason we began the experiments by A modulation percentage of for instance 15% determining the threshold contrast as a function of spatial frequency. means for AM that the extreme values of the contrast differ by 15% from the average contrast c. For FM this means that local spatial frequency differs by a RESULTS maximum of 15% from the average fe (see the appenThe dependence of modulation thresholds on dix for a definition of local frequency). Figures la and modulation frequency for carrier frequencies of 1, 4 lb show examples of such gratings.
Fig. la. Example of a frequency-modulated frequency of this stimulus was 8 c/deg and Fig. lb. Example
of an amplitude-modulated
sine-wave grating. In the experimental the modulation frequency was 1 c/deg 4 x 4deg’). sine-wave 409
grating.
Same frequencies
set-up the carrier (target subtended as in Fig. la.
411
Detectability of AM and FM of sine-wave gratings
w
mIxI
10 -
t . 0.1
IP
I
.
1
1-
1
f,,, Icpd) Fig. 2. Threshold values of amplitude modulation percentage (circles) and frequency modulation percentage (squares) plotted against modulation frequency for carrier frequencies of 1, 4 and 16 c/deg (from left to right). Carrier frequencies are indicated by dashed lines. These results were obtained with mean contrast set 0.9 log unit above the threshold contrast for the unmodulated carrier grating. Observer J.C.
I
m(r
4 .
4
1
10
f, kpdl
Fig. 3. Same results for observer J.J. than AM thresholds and the dependence on modulation frequency is rather weak; for f_ 7 O.Sfc the dependence is stronger and AM and FM thresholds become comparable. It appeared from the observers’ reports that in the latter case (particularly for carrier frequencies above 2 c/deg) they were actually performing a contrast threshold determination for the sideband spatial frequency fc - f, or fc -t f,, whereas at low modulation frequencies they had the impression that they were detecting an irregularity, which, when modulation was raised slightly above threshold, could be seen to consist of a hi%ence in contrast or spatial frequency between stimulus positions. In Figs 4 and 5 threshold curves for different carrier frequencies are plotted together. Here, contrast is *One might think that there would have been a com- 0.9log unit suprathreshold, too. In Fig. 4 (AM mon asymptote if a different measure of modulation thresholds) the low-&-ends of the curves fall around strength had been used. From equation (2) it can be seen a common asymptotic curve: in this range AM that in an FM grating some cycles are shifted to the right thresholds are almost independent of Carrie; freor to the left with respect to the unmodulated grating. Thus we might have plotted threshold values of this shift quency when contrast is kept 0.9log unit supra[expressed in degrees of visual angle or in degrees of phase threshold. The same holds for the results obtained at angle of the carrier frequency (cf: Wcstheimer, 197811 the other contrasts levels (0.3, 1.5 and 2.1 log unit against modulation frequency. However, neither of these possibilities yields a common asymptotic FM curve, nor is suprathreshold) and those obtained with subject J.J. There is no common asymptote for the FM threshold such a curve obtained if we plot threshold values of m’ against modulation frequency. curves (Fig. 5)+.
and 16cfdeg is shown in Figs 2 and 3 for observers J.C. and J.J. respectively. These. results were obtained for the contrast level 0.9 log unit above the threshold for the unmodulated grating. It should be noted that retinal inhomogeneity within our 4 deg target field might have a non-negligible effect on the results when we are working with high carrier frequencies and low modulation frequencies. Figures 2 and 3 show that this may have a serious effect only in the case of the highest fe (16 c/deg) and the lowest jm (0.25 c/deg). Generally two ranges of modulation frequency can be distinguished: for low modulation frequencies (fm -z OSj;) FM thresholds are considerably lower
412
J. H. T.
JAMAR V[ d
m I%)
10 -
f,
(CPd)
Fig. 4. Amplitude
modulation thresholds for 6 carrier frequencies (filled circles: 0.5 c/deg: empty circles: 1 cideg: filled triangles: Zc/deg: empty triangles: 4c/deg; filled squares: 8 c/deg: empty squares: 16 c/de@. Carrier contrast 0.9 log unit above threshold contrast. Observer J.C.
The effect that changing mean contrast c has on frequency modulation thresholds is shown in Fig. 6. The dependence for AM thresholds was very similar. It is obvious from the figure that the dependence on mean contrast is also very similar for all modulation frequencies. There is a tendency for the slope of the curves to decrease with increase in contrast and to approach zero for the highest contrasts.
can be predicted using the hypothesis of independent spatial frequency channels (Campbell and Robson. 1968): modulation is detected as soon as one of the sideband frequencies reaches an amplitude equal to its threshold contrast when measured in isolation. This means that the predicted modulation thresholds are determined merely by the amplitude spectrum of the stimulus. Bodis-Wollner et al. (1973) found that their AM threshold data could be fairly well predicted in this way. (They used .fl = 18 c/deg andf, from 1.8 to 16.2 c/deg; carrier contrast “well above threshold”.) However, Henning et ul. (1975) presented results where the measured AM threshold was a factor of three below the predicted one (,/, = 9.5 c/deg. fm = 1.9c;deg. carrier contrast 6.3”“). In Fig. 7 the predictions are compared with the measured modulation thresholds. The results of Fig. 2 have been re-plotted. the modulation thresholds now being expressed as the modulation index ~1’. Because AM and FM stimuli with equal modulation index tti have (approximately) equal sideband amplitudes. (approximately) equal values of m’ are predicted for AM and FM (difference <0.014 log unit for m’ < 0.5). Comparing predictions and AM thresholds of Fig. 7, we see that our results he somewhere between those of Bodis-Wollner ef al. (1973) and those of Henning er (11.(1975). However. there is no real inconsistency here. We find that the dependence of AM thresholds on carrier contrast is such that the results of Henning er ul. are replicated at lOOr
r
_
DISCUSSION
When we modulate the amplitude or the spatial frequency of a sine-wave grating, we introduce sideband spatial frequency components into the stimulus. One should bear in mind that what we call here a spatial frequency component represents in fact a spatial frequency band having a width of about l/4 c/deg because of the limited width of the target field. We investigated to what extent modulation thresholds
10
I
ml%) 10 1
100 10
m (%I
10
I
Fig. 6. Dependence of FM thresholds on carrier contrast for carrier frequencies of 1. 4 and 16c/deg. Observer J.C. Upper set of curves: h = 1 c/deg and ,‘, = 0.25 c/deg 1”
1
f,
(cpd)
Fig. 5. Frequency modulation thresholds. Same carrier frequencies, contrast and observer as in Fig. 4.
(circles). and 0.8 c/deg (squares). Middle set: fc = 4 c/deg and& = 0.25 c/deg. (circles) and 3.2 c/deg (squares). Lower set: fc = 16 c/deg and fn = 0.25 c/deg (circles), 4 c/deg (triangles) and’12.8 c/deg (squares). The dashed lines indicate the threshold contrast for each of the carrier frequencies,
413
Detectability of AM and FM of sine-wave gratings
m’ t 1 t
:b--._ --__ - ---__,
0.1
I
.
I
1
I
1
1
,
10
f, (cpd) Fig. 7. Same data as in Fig. 2. Here thresholds are expressed as the modulation index m’ instead of modulation percentage m. Circles are AM results, squares are FM results. The dashed curves are predictions based on the assumption of independent detection of a sideband spatial frequency (see text).
(roughly at 24 times threshold contrast) and those of Bodis-Wollner et al. at somewhat higher contrasts (roughly 10 times threshold contrast). For even higher contrasts measured thresholds are up to a factor of 4 above the predicted ones. Figure 7 shows also (a) that the prediction of equal threshold values of m’ for AM and FM holds only for a limited range of modulation frequencies (fm > 0.5fC) and (b) that even in this range the predictions differ from the measured thresholds. low contrasts
A way of interpreting (a) is suggested by Zwicker and Feldtkeller (1967) who performed hearing experiments analogous to our visual experiments: they measured modulation thresholds for amplitude and frequency modulation of tones. Figure 8 is a re-plot of their Fig. 69.3. Note the striking similarity to the curves in Fig. 7. They found that the modulation frequency at which the AM curve and the FM curve join is equal to half the width of a “critical band” centred on the carrier frequency. They stated that from this
Fig. 8. Amplitude and frequency modulation thresholds for a I kHz carrier tone as a function of modulation frequency, measured by Zwicker and Feldtkeller (1967) for three distinct sound pressure levels (re-plot).
J. H. T.
JAMAR
i 1
contrast Fig. 9. Filled and empty symbols: ratio of sideband amplitude at modulation threshold to threshold amplitude (threshold contrast) for sideband frequency when presented alone (log units) plotted against carrier contrast. These results all relate to a sideband frequency 1.8 times the carrier frequency. Circles: f< = 0.5 c/deg: triangles: fc = 1 c/deg; squares: f, = Zc/deg; inverted triangles: A = 4c/deg: filled and empty symbols for observers J.C. and J.J. respectively. Because this figure relates to the “high&-range”, where t&, and m& do not differ significantly. we present averages of AM and FM results. Symbols with centre dot: results re-plotted from Fig. 2 of Tolhurst and Barfield (1978). Increase in sensitivity to a test spatial frequency of 8.5 c/deg when a masking stimulus of 4.25 cideg is presented simultaneously, plotted against contrast masking stimulus. Data for two observers.
of the
modulation frequency upwards sideband frequencies differ sufficiently to stimulate distinct “critical bands” (in this case relative phases of frequency components would not be important. yielding equal threshold values of m’ for AM and FM). If the same argument applies to vision, then the bandwidth of visual mechanisms can be estimated. From Fig. 7 it is seen that the joining point is at f, = (0.65 k 0.1) /‘f,. yielding a bandwidth of (1.3 + 0.2)j; or about 2 octaves. However, in our experiments one of the two sidebands generally remained subthreshold because of the slope of the contrast sensitivity curve. Thus it seems that for the visual case we would have to suppose that from the joining point upwards. one of the sidebands and the carrier frequency stimulate different channels. In that case we would find a bandwidth of about 1 octave, which is in agreement with the results of Stromeyer and Klein (1975). The fact that the predictions differ from the measurements even for high modulation frequencies (b) can be related to the results of the masking experiments of Tolhurst and Bartfield (1978). In Fig. 9 we compare some of their results with some of our own results. In this figure we interpret the difference between measurements and predictions as an increase or decrease in contrast sensitivity to the sideband frequency, caused by the presence of the carrier. For all the data shown in the figure the ratio of sideband (or test) frequency to carrier (or mask) fre-
or u/
quency is about 2. The results of Tolhurst and Barfield re-plotted in Fig. 9 were obtained only at a masking frequency of 4.25 c/deg. Although their psychophysical method was rather different from ours (they used a forced choice technique) there is reasonable agreement between their results and those we obtained at 4 c/deg. However, there is little indication of increased sensitivity to the sideband frequency for carrier frequencies below 2 or above 4cpd. In our results the sensitivity increase might result from the contribution of both sidebands to detection. because when the carrier frequency is 2 or 4 c/deg (i.e. near the maximum of the contrast sensitivity curve) there is only a small difference in the contrast sensitivity to the sideband spatial frequencies. However. the fact that Tolhurst and Barfield found the same effect suggests that there is some other, unknown cause for the sensitivity increase. We shall now consider the data for low modulation frequencies. Henning rt al. (1975) presented data showing that an AM grating and a grating at its modulation frequency interact. This suggests that AM might be detected by the spatial frequency channel at the modulation frequency. However, we do not believe this to be the case, because if it was, one would expect AM thresholds to depend on b in the same way as contrast thresholds depend on frequency. We found a much weaker dependence. Also, the observers’ reports (they reported seeing an “irregularity” in the grating) seem to indicate that the observers did not use the spatial frequency channel at the modulation frequency for detecting AM. The fact that the curves for AM and FM in Fig. 7 are far apart at low modulation frequencies means that for low J;. modulation thresholds are not determined by the amplitude spectrum of the stimulus only: the phase relationships have to be taken into account. This can be done in the most straightforward way by considering the space-domain structure of the stimuli. The task of the visual system is thus to compare the contrast or the spatial frequency between patches of grating which have a width of one halfcycle of the modulation frequency and whose centres are separated by the same amount. With increase in fm modulation thresholds might either decrease (because of the decrease in the comparison distance) or increase (because of the decrease in the number of carrier-cycles per half-cycle of modulation). The results show that the first effect is the more important and leads to a decrease in modulation thresholds particularly AM thresholds) with increasingf,. This holds up to a modulation frequency of about 0.5 XJ$, where only one cycle of carrier is available per half-cycle of modulation. This suggests that the spatial extent of contrast detecting units does not exceed one cycle. Part of the difference in the behaviour of AM and FM thresholds for low modulation frequencies can perhaps be understood in the following way. Consider a collection of contrast detecting units of similar size (and thus having similar spatial frequency tuning).
415
Detectability of AM and FM of sine-wave gratings
The (amplitude or frequency) modulation of a sinewave stimulus will cause the stimulation of the units of that collection to be a function of retinal position. Modulation detection is possible if the visual system has some “detector” for the stimulation differences. For FM there is another clue for detection: the fact that stimulation is distributed over a broader collection of unit sizes than in the case of an unmodulated grating. This might account for the differences between the AM and FM thresholds. Figure 6 shows how modulation thresholds depend on carrier contrast. Contrast-difference thresholds depend on contrast in rather the same way (e.g. Kohayakawa,
1972; Legge, 1981). This is not surpris-
ing as far as AM detection for low modulation frequencies is concerned, because here the observer in fact detects contrast difference. Figure 9 shows (indirectly, because increase in sensitivity has been plotted instead of the threshold values themselves) that the dependence of modulation thresholds on carrier contrast (curves indicated by filled and empty symbols) resembles the dependence of threshold contrast on the contrast of a masking grating presented along with the test grating (symbols with centre dot). This is not surprising for high modulation frequencies, because in this case we were in fact performing masking experiments. However, it is remarkable that the dependence is similar for all modulation and carrier frequencies and for both kinds of modulation, in spite of the fact that the subjects probably used various ways of detecting the modulation in various situations. This point is particularly salient when formulated in terms of spatial frequency channels: when modulation frequency is low, modulation detection has to be performed within one channel, as all frequency components are very close together. For high modulation frequencies modulation is detected by separate detection of a sideband spatial frequency, i.e. by a different channel. It is remarkable that in both cases thresholds are affected similarly by the carrier contrast.
Carter B. E. and Henning G. B. The detection of gratings in narrow-band visual noise. J. physiol., Land. 219. 355-365.
Goldman S. (1948) Frequency analysis, modulation and noise. Republication 1967. Dover publications, New York. Henning G. B., Hertz B. G. and Broadbent D. E. (1975) Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency. Vision Res. IS, 887-897. Henning G. B.. Hertz B. G. and Hinton J. L. (1981) Effects of different hypothetical detection mechanisms on the shape of spatial-frequency filters inferred from masking experiments: I. noise masks. J. opt. Sot. Am. 71, 574-581. Klein S. and Stromeyer C. F. III (1980) On inhibition between spatial frequency channels: adaptation to complex gratings. Vision Res. 20, 459-466. Koenderink J. J. and van Doorn A. J. (1978) Visual detection of spatial contrast; influence of location in the visual field, target extent and illuminance level. Biol. Cybernc:. .30, 157-167. Kohayakawa Y. (1972) Contrast-difference thresholds with sinusoidal gratings. J. opt. Sot. Am. 62, 584-587. Kulilowski J. J. and King-Smith P. E. (1973) Spatial arrangement of line, edge and grating detectors revealed by subthreshold summation. Vision Res. 13, 1455-1478. Legge G. E. (1981) A power law for contrast discrimination. Vision Res. 21, 457-467. MacLeod I. D. G. and Rosenfeld A. (1974) The visibility of gratings: spatial frequency channels or bar-deteciing units? Vision Res. 14, 909-915. Sachs M. B., Nachmias J. and Robson J. G. (1971) Spatial frequency channels in human vision. J. opt. Sec. Am. 61. 1176-1186. Stromeyer C. F. II1 and Klein S. (1975) Evidence against narrow-band spatial frequency channels in human vision: the detectability of frequency modulated gratings. Vision Res. 15, 899-910.
Tolhurst D. J. (1972) Adaptation to square-wave gratings: inhibition between spatial frequency channels in the human visual system. J. Physiol., Lond. 226, 231-248. Tolhurst D. J. and Barfield L. P. (1978) Interactions between spatial frequency channels. I/ision Rex 18. 951-958.
Westheimer G. (1978) Spatial phase sensitivity for sinusoidal grating targets. Vision Res. 18, 1073-1074. Wilson H. R. and Bergen J. R. (1979) A four mechanism model for threshold spatial vision. Vision Rex 19, 19-32. Zwicker E. and Feldtkeller R. (1967) Dus Ohr u/s Nuchrichrenempjuenger. S. Hirzel, Stuttgart.
Acknowledgements-The authors are indebted to the Netherlands Organization for the Advancement of Pure Research (Z.W.O.) for giving financial support for part of this investigation. We are grateful to C. Noorlander and W. A. van de Grind for making critical remarks on the manuscript.
For an unmodulated sine-wave grating the luminance as a function of position x is given by:
REFERENCES
L(X) = I-0’ I1 + cO.sin ZrrjI:u!
Bergen J. R., Wilson H. R. and Cowan J. D. (1979) Further evidence for four mechanisms mediating vision at threshold: sensitivity to complex gratings and aperiodic stimuli. J. opt. Sot. Am. 69. 1580-1587. Bodis-Wollner I., Diamond S. P., Orlofsky A. and Levinson J. (1973) Detection of spatial changes of the contrast of a grating pattern. J. opt.-Sot. Am. 6, 1296 (A). Campbell F. W. and Green D. G. (1965) Optical and retinal factors affecting visual resolution. J. Physiol., Lond. 181, 576593.
Campbell F. W. and Robson J. G. (1968) Application of Fourier analysis to the visibility of gratings. J. Physiol., Lond. 197, 551-566.
APPENDIX
IAl)
where: LO =
mean luminance
CO =
comast
/ =
=
LL,,,
-
L,,,,,J!(L,,,,,, + L ,“,“I
spatial frequency.
In our stimuli x = 0 always represents the centre of the stimulus. When we replace co by a sinusoidal function of x (the value of that function can then be called local contrast). we get an amplitude-modulated sine-wave grating: m -sin loo”,,
2nf+,x) sin 2nL.y\ (AZ)
416
J. H. T.
JAMAR
where:
ef (I/.
For FM: c = average contrast m = modulation percentage.
L(x) = LO. I I + (” [J,(X) sin 27rfc_\-
We now have to distinguish between modulation frequency (fm) and carrier frequency (fc). If we want to produce a frequency-modulated sine-wave. i.e. to make local frequency a function of position, we need a definition of local frequency, The usual definition is (Goldman, 1948):
- J,(X)COS27r(f, +.fm,,.x- J,(X)cos2n(~;
- f,).Y
- J,(X)sin2n(fc
+ 2&).x - JL(X)sin2n(f;
- 2/,).x
+ J,(X)cos2n(f;
+ 3,/,J.Y+ J,(X)cos2n(j,
- 31,)s
+ Jq(X)sin 2rr(/; + 4f,J.x + J,(X) sin 2n( 1; - 4f,)u (A%
- . ..]I (A3)
fL(X) = & f:’ which is applicable to signals of the form:
(A41
S(x) = sin 0(x).
where X = m/100”,, x fc/fm and J, indicate Bessel functions of the first kind and of order n. When X is less than unity:
Note that this definition gives the correct value off for unmodulated sine-waves. We want fL(x) to be a sinusoidal function of position according to
J,(X) - I J,(X) 10.5x J,(X) 10
fL(x) = j;.
I + i&
0
(
‘sin 2rrf,x)
(A5)
So again we can call m the modulation percentage. Combining (A3) and (A5) and integrating we obtain: O(x) = 27rQ - $&
f- .cos27&.x + oo.
(n > I)
In this case it follows that AM and FM gratings have approximately equal amplitude spectra when their modulation percentages tnAMand rrrFMare such that:
(A6)
” L If we choose (I, = 0, our FM gratings are given by: L(x) = La.
i
I + c,sin
i
2nL.x - i&.L
II .I,
.cos2rrS,x
tAl0)
11 (A7)
The frequency spectra of AM and FM gratings [(A2) and (A7) respectively] are given by the Fourier expansions: For AM:
-
~
2 loo”,,
I - 2 &
(Al I)
cos2x(f, + f”)Z
0
cosZn(.f, - fnt)x
Ii
0x3)
Thus AM and FM gratings which have equal modulation index M’ will have approximately equal amplitude spectra. provided m’ is less than I.