V~ston Rex Vol. IS. pp. R99-910. Pergamon Press 1975. Prmted in Great Britam

EVIDENCE AGAINST NARROW-BAND SPATIAL FREQUENCY CHANNELS IN HUMAN VISION: THE DETECTABILITY OF FREQUENCY MODULATED GRATINGS C. F. STROMEYER, III and S. KLEIN’ Department of Psychology, Stanford University, Stanford, California 94305, U.S.A. (Received 12 February 1974; in revised form 7 October 1974) Abstract-Previous studies have suggested that there are narrow-band spatial frequency channels in human vision. In these studies, the detection of one sinusoidal grating is facilitated by a second subthreshold grating of neighboring spatial frequency. However, the superposition of two gratings produce regions where the contrasts subtract-where the gratings are out of phase. These regions of low contrast might reduce the detectability of the compound grating and thus produce the illusion of narrowly tuned mechanisms. The present experiment showed that a simple sinusoidal grating and a frequency modulated sinusoidal grating of identical contrast were about equally detectable. Roth gratings were centered at 4-5 c/deg near the peak of the subjects’ contrast sensitivity function. The frequency components making up the modulated grating are arranged so that the contrast is constant across the grating. In addition, the frequency components are spaced well apart in spatial frequency so that narrow-band mechanisms cannot effectively sum the components. It is shown that narrow-band mechanisms cannot readily account for the results, but medium-band mechanisms with a one octave bandwidth may be compatible with data on the visibility of compound gratings.

INTRODUCI’ION

It is well known that the ear can “hear out” the spectral components of a complex sound. Recently it has been suggested that the human visual system contains analogous mechanisms that are tuned to rather limited ranges of spatial frequencies (Campbell and Robson, 1968; Sachs, Nachmias and Robson, 1971). These studies typically employ sinusoidal gratings-patterns of stripes in which the luminance along one direction varies sinusoidally. Spatial frequency is the number of stripes per degree of visual angle. Evidence for narrow-band spatial frequency mechanisms is provided by studies in which detection of a sinusoidal grating is facilitated by a second, nearthreshold grating of variable spatial frequency (Sachs et al., 1971; Kulikowski and King-Smith, 1973; Lange, Sigel and Stecher, 1973). The test grating is facilitated only by gratings of rather similar frequency. The bandwidth of a channel tuned to a given frequency may be expressed in terms of the degree to which the second grating, of different frequency, facilitates detection of the test pattern. The bandwidth of the channel centered near 14 c/deg seems indeed sharp: sensitivity falls to a value about 33 per cent of the maximal sensitivity at a distance 20 per cent either side of the peak frequency (Sachs et al., 1971; Lange et al., 1973). Quick (1973) found that the sensitivity of channels near 5 c/deg were of comparable width, with sensitivity -decreasing horn 30 to 60 per cent (depending on observer) only 1.0 c/deg .from the peak frequency. The bandwidth of channels near 4-5 c/deg may be even narrower according to measurements by Kulikowski and KingSmith (1973) and King-Smith and Kulikowski (1975). [Other data suggest that channel bandwidth may be

broader, on a logarithmic scale, at lower frequencies. Sachs et al. (1971) showed that channels below 2-8 c/deg were broader than the channel measured at 14 c/&g. Quick (1973) concludes, on the basis of measurements near 5,7,14 and 21 c/deg, that channel bandwidth on an octave scale may decrease linearly with increasing frequency.] Evidence for narrow-band channels, whose bandwidths are independent of spatial frequency, can be extracted from a study by Hoekstra, van der Goat van den Brink and Bilsen (1974) who measured the threshold of a sinusoidal grating as a function of spatial frequency (1 c/deg to 7 @leg), luminance, and most importantly, the number of cycles. The threshold decreased as the number of cycles increased until a saturation point was reached. As luminance increased from 2 to 165 cd/m2, the number of cycles for saturation increased from 3 to 8. The results can be interpreted (Appendix C) as evidence for the existence of mechanisms whose bandwidths are even narrower than those observed by Kulikowski and King-Smith (1973) for grating detection. There is an alternative explanation for these findings which does not require the existence of narrow-band mechanisms. Whenever gratings of different spatial frequencies are added together, there will be certain limited regions where the contrasts add completely; at many other regions, the patterns will tend to cancel each other. The contrasts of the two gratings will add completely only where the two patterns are exactly in phase, and the contrasts will subtract where the patterns are out of phase.. The patterns are not of a constant contrast but are modulated in amplitude and slightly in frequencyhence, quasi amplitude wwdw bed

Colleges, Claremont, CA 91711. Please requests to this address.

send

gratings.

The facilitation of one grating by another is probably not dependent only on the peak contrast value of the compound grating. Possibly, gratings are detected

I Present address: Joint Science Department. Claremont reprint 899

900

C. F. STROMEYER III and S. KLEIS

at threshold by mechanisms of medium bandwidth (with a one octave bandwidth). which have been found by so many investigators. Since these mechanisms respond to rather local regions of the grating. their response will be weak at the regions of reduced contrast. Detection may be based on some intermediate contrast value and not on the peak contrast value (see Discussion). Thus, when the spatial frequency of the facilitating grating is made different from the test pattern, facilitation may drop off rapidly, and this sharp drop-off may give the illusion of narrow tuning. This paper shows that gratings near the peak of the subjects’ contrast sensitivity function (the function relating contrast sensitivity to spatial frequency) are not detected by narrow-band mechanisms. The test pattern is a frequency modulated (FM) sinusoidal grating, which was chosen because it produces a poor response in narrow-band mechanisms and a good response in medium-band bar and edge mechanisms. The spatialfiequenc!: or bar width, varies across the grating, but the contrast is entirely constant across the grating, unlike the quasi amplitude modulated patterns in previous summation studies. The detectability of this FM grating was compared to the detectability of simple sinusoidal gratings of identical contrast. The hypothetical receptive field (line spread function) of the narrow-band mechanisms resembles a grating of reguhrIy spaced, alternating excitatory and inhibitory bands. For example, Kulikowski and King-Smith (1973) require for their data a hypothetical receptive field with eleven rather significant excitatory bands that are regularly spaced. An Fh4 grating with bars of varying width will poorly match the shape of these regular receptive fields and will produce a weak response. Edge or bar mechanisms, however, will respond to local regions of the grating, and thus irregularities extending over a larger period will have little effect on the response. The experiment may also be visualized in frequency space. The frequency modulated grating has a broad spectrum. A narrow-band mechanism would summate poorly a few spectral components, whereas a medium-band mechanism could encompass many components. METHODS Apparatus

A vertical grating was generated on a CRT (white P-4 phosphor) following the method of Campbell and Green (1965). The grating was a simple sinusoidal grating or sinusoidally frequency modulated sinusoidal grating. The display field was 6” dia with a black surround, viewed from 81 cm. A fine horizontal and diagonal cross hair was placed across the field to aid focussing. A Wavetek Model 112 oscillator served as a frequency. modulator, which provided the Z axis signal. The sine-wave output of this oscillator was sinusoidally frequency modulated by a second sine-wave generator that fed into the Wavetek oscillator. The Wavetek oscillator was calibrated by reading the output frequency with a digital frequency meter for various values of d.c. voltage fed into the oscillator. The relationship between input voltage and output frequency was linear for the range of frequencies in the present experiment. When a sine-wave was fed into the Wavetek oscillator, the peak voltages of the sine-wave input produced a particular maximal deviation of the instantaneous frequency of the output. The range of frequency modulation, AL is said to be the maximal deviation of the instantaneous frequency away from the

mean frequency. This value was known from the d.c calibration and will be used to determme the spectrum of the FM gratings. Contrast of the gratings is conventionally defined

Grating modulation voltage was set with a vacuum tube voltmeter, and frequency was set with a digital frequent! meter. A second oscilloscope was used to monitor the 2 axis signal. Srimull The luminance distribution across a grating may be represented by the general expression L(x) = Lo[l + v(x)]

0)

where v(x) is an arbitrary function of space and &, is the mean spatial luminance. The sinusoidal grating in the present experiment is represented by L(x) = &,[I - AcosZnjx]

(3

wherejis spatial frequency and A is the modulation ampletude. The sinusoidally frequency modulated sinusoidal grating may be represented by 4~) = I+[ 1 - Acos(2lrfi- + m sin 2r$,s)]

(3,

where fm is the modulating spatial frequency and III is the modulation index (see Black 1953). The modulation index is defined as the range of frequency modulation (i.e. the maximal deviation of instantaneous frequency) divided by the modulating frequency, that is m = A,.

(4)

(Measurement of Afis described in Apparatus.) The frequency spectrum of the FM grating of equation (3) is given by the Fourier expansion v(x) = - A{J&n) [cos Zxfx] + J,(m) [Cos 2n(f + 1,)x - cos 2x(f - fm)x] + J*(m) [cos 27r(j + 2f,)x t cos 2n(f - 2f,)r] + J,(m)

[COS 27r(f +

3f,)x - cos 2n(,f - -1S,)x] (5) The spectrum consists of a sinusoidal grating at frequency + . ..)

jplus components spaced symmetrically either side of j at intervals equal to the modulating frequencyf,. The amplitude of each component is given by J.(m), which are Bessel functions of the first kind and order n. The modulation index thus specifies the amplitude of each component. The FM gratings were centered at either 4.0 or 4.5 c/deg and were modulated by a grating of l/3 these frequencies. i.e. f/f,, = 3. The modulation index m was 1.434. Figure 2 shows the theoretical spectra of these gratings. and Fig. 1 shows the actual gratings. Note that the maximal amplitude of the spectral components is 0.55. A simple sinusoidal grating of equiualent contrast has a much larger spectral amplitude of l@). These test patterns were chosen for two reasons. Mediumband mechanisms can effectively sum the three equal-amplitude central components at different phase positions across the gratings, whereas narrow-band mechanisms poorly sum the components. If narrow-band channels are used to detect gratings, then these FM gratings should be less detectable than a simple sinusoidal grating of the same contrast. (These notions are presented in a quantitative form in the Discussion.) Secondly, the FM gratings are centered near the peak of the photopic contrast sensitivity function as measured with sine-wave targets (van Nes and Bouman, 1967; Campbell and Robson, 1968; Sachs er crl.. 1971). It has been

,_,

-

_ ___ _

.” _ -

- _ ^

-

_ .

_ _.

_ _

--

_

-

Fig.1. The test patterns: a sinusoidal grating and a sinusoidally frequency modulated sinusoidal grating. Both patterns have the same central frequency f and the same contrast. The modulated pattern has a modulation index= 1.434 and the ratio of the central frequency to the modulating frequencyf/f, is 3. The patterns shown here are several hundred times threshold contrast.

The detectability of frequency modulated gratings

90 I

Fig. 2. Theoretical spectra of the sinusoidally frequency modulated sinusoidal gratings used in these experiments. The central frequency components of the gratings, /; was either 4.0 c/deg (top spectrum) or 45 c/deg (bottom spectrum). The modulation index m is 1.434, and the ratio of the central frequency to the modulating frequency f/if, is 3. The amplitude of each spectral component is indicated at the end of each line. Positive and negative lines indicate positive and negative cosine components respectively. The sinusoidal gratings of equivalent contrast were centered at 4.0 or 45 c/deg with an amplitude of 1.00. assumed explicitly by Campbell and Robson (1968) and implicitly by Sachs et al. (1971) that the overall contrast sensitivity function represents the envelope of the contrast sensitivity functions of all the independent channels. By placing the simple and FM gratings at the peak of the contrast sensitivity function, we make the central components at 40 or 45 c/deg more visible than the sideband components: Thus the FM grating should not be more detectable than the simple grating merely because one of the sideband components affects a more sensitive channel than does the simple grating. This is the reason why the experiment was done with gratings centered near 4-5 c/deg. The experiment would be more difficult to do with higher frequency gratings. for example at 14 c/deg, for at these frequencies the contrast sensitivity function changes rapidly. Thus the amplitude of each spectral component would have to be adjusted to compensate for the individual’s contrast sensitivity function. Studies on the subthreshold summation of two gratings with similar frequencies (a quasi amplitude modulated pattern) do indeed suggest that there are narrow-band channels near 5 c/dea. Kulikowski and King-Smith 11973) found very little failitation between a gratiig of 5.O‘c/deg and a grating of 3.0 or 4.0 c/deg, and King-Smith and Kulikowski (1975) found very little facilitation between gratings of 3.6, 4.0 and 44 c/deg. Quick (1973) observed that the facilitation effect fell off sharply to about half strength one half cycle away from 5 c/deg. Sachs et al. (1971) and Pantle (1973) found no facilitafion between a grating near 5 c/deg and a second grating one octave lower. And Abadi and Kulikowski (1973) found facilitation between a 5.0 c/deg sinusoidal grating and only those harmonics of a square-wave grating of either 1.0, 1.67 and 5.0 c/deg that were very close in- frequency to the 5.0 c/deg sinusoid. Hoekstra cr a!.‘~ (1974) study on the effect of field size on grating detection’can also be interpreted as evidence for very narrow-band channels within the range l-7 c/deg (see Appendix C). Thus given the evidence for narrowband channels near 5 c/de& it is of interest to see whether these same channels can be demonstrated with FM as opposed to quasi AM gratings.

Procedure

The main experiments compared the detectability of a simple sinusoidal grating and an FM grating of equivalent contrast. Additional experiments were also done to demonstrate that the sinusoidal and modulated gratings were indeed near the peak of the subjects’ contrast sensitivity function. Each run consisted of 75 trials where two types of patterns and blanks were randomly intermixed, each event occurring with l/3 probability. The subject’s task was to rate the visibility of stimuli on a 1 to 5 scale where 1 means blank and 5 means definitely a pattern (Egan, Shulman and Greenberg, 1959). The subject was informed after each trial what the stimulus had been. A pair of patterns was used within a single run. This helped eliminate variability due to sensitivity fluctuations over time. The pairs of patterns were of two types: (1) A simple sinusoidal grating and a frequency modulated grating, of identical contrasts, both centered at the same spatial frequency. (2) Two sinusoidal gratings of identical contrasts and different spatial frequencies. These were used to measure the subjects’ contrast sensitivity function. The stimuli were presented in a repeating sequence: (1) 4 sec. blank field. (2) 4 sec. tone plus blank field. During this period. S pressed a button which, on 2!3 of the trials, presented the test pattern for 750 msec. Each of the two types of test patterns and blanks occurred with l/3 probability. The onset and offset of the pattern produced no visible electrical noise. The subject practiced at the beginning of each run until he felt he was ready to start. The CRT was viewed with the left eye which was carefully refracted. The experiments were run under two luminance levels. Initially the scope luminance was set to 5.0 cd/m2 and no artificial pupil was used. The natural pupil is approx 5.5 mm at this luminance level (Flamant, 1948). and thus retinal illuminance was 120 td. The experiments were then repeated with the scope luminance increased to 75 cd/m2 (produced by adding d.c. voltage to the Z axis). Subject CFS used an artificial pupil

C. F. STROMEYER III and S.

902

KLEIX

of approx 2.3 mm dia (retinal illuminance 310 td). and subject SK used a 3.0 mm pupil (530 td). The purpose of this second condition was to better assure that the contrast sensitivit! function peaked near 4-5c/deg-at the center frequency of the frequency modulated patterns. The size of the artificial pupil was chosen for each subject because it in fact made the peak of the contrast sensitivity function fall near the desired frequency.

with the value of r = @25 found b> Nachmias and Kocher for detecting flashed spots oflight. All d’ values in this paper were calculated with z = 0.25. For all experiments. each of the two types ofsignalsused within a run was compared independently to the blanks. since we were interested in the detectability of that particular signal relative to the blank trials.

Data treatrncrtt

Figure 3 shows the detectability d’ of simple sinusoidal gratings. indicated by open circles. compared to the frequency modulated gratings. closed circles. The bars are -I_1.0 S.E. For these data. the gratings were centered at 40cjdeg; mean luminance was 5.0 cdsm’ : and the natural pupil was used. Both a simple grating and a modulated grating were used in each run. and the contrast of both gratings were identical. Thus the data points for each run are plotted as pairs. with open and closed circles separated horizontally by a tiny amount for clarity. Figure 4 shows similar data, where the gratings were centered at 45 cideg; the mean luminance was raised to 75 cd/m’; and the artificial pupil was used. Table 1 shows the average ratio of detectability d’ of modulated versus simple gratings and + 1.0 S.E. of these averages. A value greater than one indicates that the modulated grating was more detectable than the simple grating. The average ratio is given for each subject separately and for each of the two luminance levels. The overall average is close to one, 0.96 -t_0.08. The FM gratings were chosen so that the central frequency component would be near the peak of the subjects’ contrast sensitivity functions and thus more visible than the sideband components: Thus the FM gratings should not be more detectable than the simple gratings merely because one of the sideband components stimulates a more sensiriue channel than does

An ROC curve was fitted to the confidence ratings obtained in each run. The curve was fitted by a maximum likelihood technique similar to that of Ogiliie and Creelman (1968). but where the signal and noise distributions were assumed to be Gaussian (Dorfman and AIf. 1969). The technique finds the values of parameters which were most likely to have generated the data. The parameters were the criteriorl let:& (between the rating categories). the d’ value of the test pattern. and the slope of the ROC curve. The d value is the measure of detectabilitv used for all results. It is explicitly defined to be the hor&ontal intercept of the ROC curve plotted on double-axis -_score (standard score) paper. To make this clear, imagine that the subject has a 50 per cent chance of correctly identifying a signal (hit rate), then the I score of the false alarm rate is -d’. or equivalently, the -_score for the correct rejection of blanks is d’. For example. a 50 per cent hit rate and 84 per cent correct rejection of blanks, gives d = 1. The slope parameter, Z, is related to the widths of the signal and noise distributions by aJa. = 1 + ad’. The slope parameter, unlike the other parameters, should be fairly constant from run to run (Nachmias and Kocher, 1970). The first half of the data was analyzed with a free to vary and then all data were analyzed with z fixed. The average computed value was x = 0.27, which is in agreement 2.1 -

RESULTS

2.0 -

TEST

CONTRAST

I%)

Fig. 3. Detectability d’ of simple sinusoidal gratings (open circles) versus modulated gratings (closed circles). Both gratings were centered at 4.0 c/deg, mean luminance was 5.0 cd/ m’, and the natural pupil was used. A simple and modulated grating of identical contrast were used in a single run. Thus the data are plotted as pairs, with open and closed circles separated by a small horizontal distance for clarity of presentation. The bars show k 1.0 SE.

00

c

’

Fig. 4. Same as Fig. 3, but here the gratings are centered at 4.5 c/deg and mean luminance is 75 cd/m2. Subjects CFS and SK used artificial pupils of 2.3 and 3.0 mm respectively. Again the open circles represent simple gratings and closed circles reDreSent modulated gratings.

The detectability of Table 1. The average ratio

versus sinusoidal grat-

these averages. Values

Subject

Ham

luminance 5.0 edfm2

Hem

luminance 75 cd/m2

C.F.S.

1.27 + 0.14

0.75 + 0.13

S.K.

1.11 f: 0.23

0.83 + 0.07

Overall Ratio 0.96

+

0.08

the simple gratings. Tables 2 and 3 show in f&t that the sideband frequencies are not more detectable than the center frequency. For the lower luminance level (Table 2), the central component at 4-Oc/deg was compared within a single run, to a grating of 2.7 c/deg of identical contrast. (The main side-band components are at 2.7 and 5.3 c/deg-see Fig. 2, top.) A 4.0 c/deg was also compared to a 5.0 c/deg grating, where each grating, in this case, was used in separate runs close together in time. The average difference in d for these pairs of gratings and f 1.0 S.E. of the average are shown in the right-most column of Table 2. A positive value indicates that the 4-Oc/deg grating was the more detectable member of the pair. Table 3 shows similar data for the higher luminance level, where the central component at 4.5 c/deg is compared to a grating of 3-Oor 6-Oc/deg within single runs--see the spectrum in Fig. 2, bottom. The contrast sensitivity at the side-band frequencies is not greater than the sensitivity at the central frequency. Thus, the similar detectability of simple and frequency modulated gratings cannot be accounted for by a more sensitive channel peaked at one of the sideband frequencies.

DISCI-SSION

Our basic finding is that an irregular frequency modulated grating and a regular sinusoidal grating of identical contrast are about equally detectable. The sinusoidal grating however was somewhat more detectable than the FM grating at the higher luminance level (Table 1). This may well be due to the fact that at the higher luminance level. the subjects’ contrast sensitivity was less at both sideband frequencies (Table 3) than at the center frequency. The present results imply that the bandwidth of detection mechanisms near 3-6 c/deg may span one octave, whereas other studies on grating detection. discussed earlier, imply that the bandwidth may be considerably narrower. This apparent conflict will be considered in several steps: (1) First, calculations are presented to show the degree to which different mechanisms are stimulated by FM and quasi AM gratings. And the implications of a peak detection model are considered. (2) The role of probability summation is then examined, and it is concluded that this factor is too small to resolve the conflict over bandwidth.

Table 2. Detectability d and + I.0 SE. for gratings of 4.0 c/deg vs 2.7 and 5.0 c/deg. The average differences in d’ for pairs are given in the right column, where positive values indicate that the 4.0 c/deg grating is more detectable. The contrast of all gratings was 0.363%; mean luminance 5.0 cd/m2

Subject C.P.S. 2.1 cldcg

4.0 cldag

0.76 + 0.33

0.70 + 0.34

0.39 + 0.31

0.60 + 0.32

5.0 c/da&

Average difference

0.08 2 0.32

1.26 + 0.41

-0.08 + 0.28

1.62 + 0.48

0.28 + 0.31

1.34 + 0.37

Subject

S.K.

0.49 + 0.37

0.62 + 0.39

0.912

0.39

0.17 + 0.29

0.97 + 0.42

1.00 + 0.42

0.76 2 0.38

0.87 + 0.40

1

I

-0.16 + 0.26

1

1.67 + 0.48

0.81 + 0.48

2.05 + 0.55

0.312

1.28

V.R.,5-x/9--8

1.28

.+

0.44

C'.F. STROMEVER IIIand S.

904

KLEIS

Table 3. Detectability d’ and + I.0 SE. for gratings of 4.5 c,‘deg vs 3.0 and 60 cldeg. The average differences in & for pairs are given m the right column. where positive values indicate that the 4.5 c/deg grating is more detectable. Mean luminance 75 cd:m’ Subject C.F.S. 3.0

4.5 cldeg

0.222

1.29 L 0.41

1.49 + 0.54

0.333

0.37 + 0.37 2.58 + 0.65 0.82 0.38

1.12 _ + 0.37

6.0 c/de&

Avere~e difference

contrast (Z)

0.366 0.200

a.44 + 0.31

2.615 1.23 + 0.66 0.41

i

0.200

1.14 + 0.40

0.66 + 0.35

0.244

0.50 + 0.36

0.45 2 0.36

0.366 0.200

3.70 + 0.41 0.86 1.20

2.20 0.56 + 0.60 0.32

0.47 + 0.29 1

Subject S.K.

0.200

0.95 + 0.45

0.002

0.222

1.21 2 0.38

0.92 + 0.36

0.222

0.79 + 0.36

0.36 + 0.34

0.244

1.04 + 0.37

0.58 + 0.35

0.244

2.92 + 0.84

1.09 2 0.42

(3) Finally, possibilities are considered of how the responses of medium-band mechanisms can be linearly pooled to account for the detectability of both FM and quasi AM gratings.

Quantitative estimates of how various gratings stimulate mechanisms of different bandwidth will now be considered. Two bandwidths are examined. (1) A narrow-band mechanism, N, given by @NffP/_O= exp’, - C($&) - f12/a’: (6) where f, indicates peak frequency and 0 = 0.17. The bandwidth is chosen to agree with the data of Sachs et al. (1971) near 14 c/deg and the data of Quick (1973) near 5 c/deg-a full bandwidth of about one quarter octave at half amplitude. Kulikowski and King-Smith (1973) found a narrower channel with u = 0.11 near 5 #deg. However, the value Q = 017 will be used here. (2) A medium-band mechanism, M, given by ,-UfJ/Y

%f(fpplf)=

[

e-k

_ e-w.r”v)2 _

e-4A

1

0.27

0.53 + 0.21

octave bandwidth observed in adaptation studies is apparently not due to physiological non-linearities caused by high-contrast patterns, for the same bandwidth is observed with adapting patterns that are nearthreshold (Stecher, Sigel and Lange, 1973). Adaptation studies also show that the channels may be somewhat narrower at higher spatial frequencies than those used in the present study (Blakemore and Campbell. 1969: Stecher et al., 1973). For example, Klein, Stromeyer and Dawson (in preparation) showed, with a signal detection paradigm, that a grating of 64 cjdeg produced no threshofd change at 10.1 c/deg (3/4 octave above the adapting frequency). and thus channels near 10 c/deg are not affected by a 6 cideg grating. (Detection studies with quasi AM grating also suggest that spatial frequency bandwidths are narrower at higher frequencies--see Introduction.) The shape of the receptive field (line spread function) of a mechanism is given by the Fourier transform of the spatial frequency function @&!fl of the mechanism

2

(7)

where k = 0462 is chosen so that the function peaks at f = f,. Blakemore and Campbell (1969) used this expression, which has a one octave full bandwidth at half amplitude, to fit their adaptation data. Further evidence for medium-band channels is provided by studies on apparent contrast following adaptation (Blakemore. Muncey and Ridley, 1973), simultaneous noise masking (Stromeyer and Julesa, 1972), the McColIough effect (Stromeyer, Lange and Ganz., 1973; Stromeyer, Lange and Dawson, in preparation) and facilitation (Stromeyer and Klein, 1974). The one

RF,,,, (s

-

s,) =

Tdf@(lp/l’)

i0

cos k[f(s

- s,) - 2-j.

Each mechanism can be characterized by its ccltter position x,, peak spatial frequency &. and phase J. For an on-center receptive field, ‘Y= 0: for an antisymmetric mechanism, u = + l/45 and for an offcenter mechanism x = l/2. The receptive field of the medium-band mechanisms resembles, for example, the symmetric bar-shaped or antisymmetric edge-shaped receptive fields of simple cells (Stromeyer and Klein, 1974). The receptive field of the narrow-band

The detectability of frequency modulated gratings

Spatial frequency

905

Spatial position

i .A

-L

XC

0.30 040 0.50 060 0.70 0.80 090 1GO 1.10 1.20 1.30 140 1.50 160

0 0 0 0 0 3 7 10 7 3 0 0 0 0

0 0 0 0 0 3 7 10 7 3 0 0 0 0

0 0” 0 0 3 7 10 7 3 0 0 0 0

0 0 0 0 0 3 7 10 7 3 0 0 0 0

0 0 0 0 0 3 7 10 7 3 0 0 0 0

0 0 0 0 0 3 7 10 7 3 0 0 0 0

1

0 0 0 0 0 3 7 10 7 3 0 0 0 0

2 3 5 7 9 IO 10 10 9 8 6 5 4

1 2 3 5 7 9 10 10 10 9 8 6 5 4

1 2 3 5 7 9 10 10 10 9 8 6 5 4

1

I

2 3 5 7 9 10 10 IO 9 8 6 5 4

2 3 : 9 10 10 10 9 8 6 5 4

1 2 3 5 7 9 10 10 10 9 8 6 5 4

1 2 3 5 7 9 10 10 10 -9 8 6 5 4

Fig. 5. A stimulation map which shows the amount of stimulation of mechanisms whose peak spatial frequencies are indicated on the ordinate and whose spatial positions are indicated on the abscissa. The stimulus here is a sinusoidal grating of frequency 100 and contrast 10. On the left side of the figure the mechanisms are assumed to have the narrow bandwidth found by Sachs et al. (1971) and by Quick (1973) as given in equation (6) of our discussion; on the right side, they are assumed to have the medium bandwidth found by Blakemore and Campbell (1969), as given in equation (7). The mechanisms are assumed to be equally sensitive at all spatial positions, and thus stimulation is constant across the grating. (See Discussion.)

mechanism resembles a grating of regularly spaced, alternating excitatory and inhibitory bands (see Introduction). The stimulation of a mechanism by a pattern P(x) is determined by the overlap of the receptive field and the pattern (Appendix A). Only mechanisms with the same orientation as the grating will be considered here. Figures 5 to 9 show stimulation of both the narrowband and medium-band mechanisms by particular sinusoidal, FM, and quasi AM gratings. Each entry in these stimulation maps indicates the amount of stimulation of a mechanism whose peak spatial frequency, fY is specified on the ordinate and whose center position at a particular place on the grating, x, is specified Spatial frequency r, 0.30 040 0.50 060 @70 0.80 0.90 1Gil 1.10 I.20 1.30 140 1.50 160

Spatial position XC

on the abscissa. The entry pertains only to the mechanism with the optima1 phase, a (where x can range from 0 to 1). For example, the optimal mechanism centered on a bright bar is an on-center cell (a = 0) that fires to a bright bar, and the optimal mechanism centered on a dark bar is an off-center cell (a = l/2) that fires to a dark bar. The optima1 mechanism centered at a zero crossing has an antisymmetric edge-shaped receptive field (a = f l/4). (Details of the calculations are given in Appendix A.) The left and right halves of Fig. 5 show, respectively. stimulation of the narrow-band and medium-band mechanisms by a sinusoidal grating at frequency I.0 and contrast 10 (units arbitrary). Figures 6 and 7 show stimulation of the narrowband and medium-band mechanisms, respectively, by the FM grating used in the present experiment. The grating has contrast 10, as does the sinusoidal grating (Fig. 5), and the three central spectra1 components fall

2222222222222 1111111111111 1111111111111 4444444444444 5555555555555 0112233322110 4444444444444 6666555556666 6654322234566 5554332334555 6665555555666 6665544455666 5 5 4 4 3 22234455 4332211122334

Fig. 6. Stimulation map for the narrow-band mechanisms. The stimulus is the sinusoidally frequency modulated sinusoidal grating used in the present experiments. The grating is centered at frequency lG0 and the contrast is 10. The modulation index of the grating m is 1.434 and the ratio of the central frequency to the modulating frequency f/f_ is 3. The abscissa in all stimulation maps show spatial positions that cover one repetition period of the grating.

Spatial frequency f. 030 040 0.50 060 0.70 0.80 0.90 IQ0 1.10 I.20 1.30 1.40 1.50 160

Spatial position XC 11223343322 0 02-3 4 5 5 5 4 3 2 1235667665321 1357777777531 1478888888741 35898878898 67898767898 8998765678998 10 10 1110 11 10 11 10 10 10 10 9

9 8 6 4 9 7 5 3 9 6 4 2 8 6 3 2 8 _5 3 1 7 5 3 1

3 4 6 2 3 5 1 2 4 0 2 3 1 1 3 1 1 3

11 0 0

53 76

8 9 10 10 7 9 IO II 6 9 10 II 6 8 10 II 5 8 10 10 5 7 9 10

Fig. 7. Stimulation map for the medium-band mechanisms. The stimulus is the same frequency modulated pattern that was used in computing the map in Fig. 6.

C. F. STROME~TER III and S. KLEW

906

Spatial frequent!

Spatial

f .P @30 0.40 0 50 0.60 @70 0.80 @90 i%l 2.10

I.20 1.30 1.40 1.50 160

ill11

position s,

If

1111

4433221223343 7165432345671 10 IO 9 7 5 13 I? 11 9 7 15 14 13 II 7 15 IS 13 11 8 15 14 13 10 7

3 4 4 4 4

13 13 11 9 7 4 11 11 10 8 6 4 998754345 7766433346677 555443334455s 444332223

2 3 2 4 1 4 0 4 I 4

5 7 7 9 7 11 8 ii 7 10

2 4 7 3 4 6

11

9 10 10 II 13 13 13 14 15 13 15 15 13 14 1.5

9 11 13 13 8 10 11 11 7899

3444

Fig. 8. Stimulation map for the medium-band mechanisms. The stimulus is a superposition of two gratings of frequencies I.00 and 0.80 both of contrast 8 (quasi amplitude modulated pattern). This stimulus is similar to the pattern used by Sachs et at. (1971)and Quick (1973). at frequencies 0.66. l*OO,and 1,33--all with 55 units of contrast. The weaker components of the grating as shown in Fig. 2, were also used in calculating the stimulation maps. The narrow-band mechanisms are stimulated somewhat weakly at all positions across the grating, whereas the medium-band mechanisms are stimulated moderately well at most positions. The modulated grating has bars that vary in width, and thus for each different position, a mechanism of different peak frequency will be maximally stimulated. Figures 8 and 9 show stimulation of the mediumband and narrow-band mechanisms, respectively, by a quasi AM grating consisting of the superposition of two sinusoidal gratings with frequencies O+?and 1.0, each of 8 units of contrast. This pattern is similar to the patterns used by Sachs et al. (1971) and by Quick (1973). The contrast values for all the stimulation maps were chosen to represent stimuli that have been shown to be about equally detectable. To predict the visibility of these patterns, it is necessary to know how the outputs of the mechanisms com-

Spatial frequency f, @30 0.40 0.50 0.60 @70 0.80 0.90 IGO 1.10 1.20 1.30

1.40 1.50

1.60

Spatial position s, 00000000000 00000000000 00000000000 11111111111 55554444455 10 10 9 8 7 6 10 IO 9 7 5 3 99888777888 66646666666 2 2222222222 00000000000 00000000000 00000000000 00000000000

00 00 00

11 5s 6 6 7 8 9 10 10 1 3 5 7 9 10 10 99 66 22 00 00 00 00

Fig. 9. Stimu~tion map for the narrow-band mechanisms. The stimulus is the same quasi amplitude modulated pattern that was used in computing the map in Fig. 8.

bine. The peak detection model assumes the pattern is detected by the mechanism that responds maximally. Implications of this model can be seen with Figs. S-9. Assume that a pattern is detected when any mechanism is stimulated to a level of 10. Figure 6 shows that the FM grating stimulates the narrow-band mechanisms only to level 6 (that is, only 60 per cent of the amount required to make the FM grating as detectable as the sinusoidal grating-Fig. 5), whereas the medium-band mechanisms are stimulated to level 11 (Fig. 7)_near the threshold level of 10. Figure 8 shows that the quasi AM grating stimulates the medium-band mechanisms to level 15 (actually 15.3-or fuIly 50 per cent above the amount required to detect the sinusoidal grating). However, the narrow-band mechanisms are stimulated to level 10 (Fig. 9), (Indeed, the ban&i&h of the narrow-band mechanisms was ~ulcu~ured by others so that the peak stimulation level would be at threshold.) Thus the peak detection model predicts that the mediumband mechanisms respond too strongly to the quasi AM grating and that narrow-band mechanisms respond too weakly to the FM grating. To resolve this conflict, a different detection model is needed. Next we consider whether probability suction can sufbciently increase the response of narrow-band mechanisms to the FM grating or sufficiently decrease the response of medium-band mechanisms to the quasi AM grating so that a single bandwidth mechanism accounts for all results.

Detection may be based on the average of the response of many mechanisms, rather than on simply the peak response. Probability summation provides one possibility for combining the responses in a weighted average. For an ideal observer, acting according to signal detection theory, probability suction is calculated by the square root of the sum of squares of the detectabilities of independent channels (Green and Swets, 1966). Thus, if two patterns are each detected by different independent channels, then the detectability of the combined pattern is d’ = (d’,’ f d2*)*. To predict detectability, we must know the function that relates dete~tab~ity d’ to the stimuIation magnitude (which itself, is linearly related to contrast, c). This function, called the transducer function. has the form d’ = kc’ for grating detection. Stromeyer and Klein (1974) found a transducer exponent t of about 3 for a 9 c/deg grating and Nachmias and Sansbury (1974) obtained an exponent between 2 and 3. In extensive m~surements (Klein et al., in preparation) a single test pattern with three possible contrast values was randomly presented with blank trials during each session. One of the parameters of the minimum chisquare analyses of the signal detection data was the exponent r, where a value t = 2.5 gave the best fit. Thus. the response to the superposition of several gratings whose spatial frequencies are sufficiently separated to stimulate different independent channels, is given by d’ = (Td;‘)in

= k(;cji)“’

(8)

where ci is the contrast of the ith component. This formula for combining contrasts does not depend on signal detection theory-the formula is derived in Appen-

The detectability of frequency modulated gratings

dix B from probability summation. as applied, to the psychometric function (obtained with method of constant stimuli). To predict the detectability of the FM grating by narrow-band mechanisms, using probability summation, we assume for simplicity a bandwidth that is narrow enough so that each of the three central components of the grating stimulate different independent mechanisms. (The narrow-band mechanisms which have a 50 per cent fall off 20 per cent away from the peak are too narrow to produce significant summation between the central components that are separated 33 per cent in spatial frequency.) The three central components have contrast O-55 times the contrast of the entire grating. (The other components contribute little to detection because of the large exponent f of the transducer function.) Thus d mod =

k[k;Q”2.

where c,~ refers to the amplitude of the spectral components of the modulated grating. The ratio of the predicted detectability of the modulated versus unmodulated sinusoidal grating can be calculated:

cod

-

4in

k[3cL]“2 =

kc,,

r,n

=

31’2(cmod/C~inY

=

J3(Oqf

0.4 * 0.1

907

The amplitude of the envelope of this quasi AM pattern, expressed in the arbitrary contrast units. is thus 16lcosOl 27~x1. A model based on medium-band mechanisms. with probability summation. must consider not just the peak contrast of 16. but the contrast across the entire grating. The table in Appendix B shows that probability summation yields an effective contrast of 0.81 of the peak contrast of 16 units, or 13.0 eficriue contrast units. This reduction in effective contrast is not a sufficient reduction to account for the empirical observation that the quasi AM grating is only as detectable as a sinusoidal grating of 10 contrast units-the quasi AM grating is thus too detectable according to probability summation. Probability summation also can not account for the observation (Hoekstra et al., 1974) that the contrast threshold for sinusoidal gratings of 1 to 7 c/deg (L, above 25 cd/m’) is about halved when the number of cycles is increased from two to six. Probability summation predicts (Appendix B) that tripling the number of cycles reduces the threshold by only 3”” = 3’;’ = 1.3. A two fold reduction would require a 32-fold increase in the number of equally activated, independent channels. King-Smith and Kulikowski (1974) claim that the sensitivity of the grating detector is due largely but not entirely (King-Smith and Kulikowski, 1975) to probability summation among several line detectors. However, the sharp, individual lines that they use as stimuli have low frequency components (which are not present in gratings) that may aid detectability of the lines, and thus it is questionable whether the detectability of gratings can be predicted from the detectability of individual lines. To conclude, probability summation does not sufficiently increase the detectability of FM gratings, using narrow-band mechanisms for detection. Also, probability summation does not sufficiently decrease the detectability of quasi AM gratings, assuming mediumband mechanisms are used.

(9) where the exponent t of the transducer function is assumed to be 25 + O-5.The observed overall ratio in the present experiment is O-96 f 008 (Table 1). Probability summation is thus not sufficient to make the FM grating as detectable as the sinusoidal grating, assuming that the three grating components stimulate different channels. This argument can be made still stronger, for the factor J3 in equation (9) may be too Linear pooling large, since it describes the ideal observer with uncorreProbability summation predicts that the mediumlated noise between independent detection channels. A band mechanisms will yield too high a detectability for real observer may not be able to attend equally to all quasi AM ratings, because the large exponent p used channels, and thus the noise would be correlated. in probability summation (Appendix B) weights too Studies on the detection of two superposed sinusoidal heavily the response of mechanisms at the optimal gratings well separated in spatial frequency (Graham position. And thus the rather weak responses of and Nachmias, 1971; Pantle, 1973) suggest that probmechanisms at non-optimal positions (at regions of the ability summation is not fully effective. This may be due to signal uncertainty, which would also reduce grating where contrast is lowest) do not significantly the detectability of the FM gratings below the level decrease the response of the total pool of mechanisms. A pooling mechanism will now be considered which predicted from the present calculations of probability linearly summates the response of spatially contiguous summation. Can probability summation, however, predict the medium-band mechanisms. The effective contrast of the detectability of the quasi AM grating if medium-band quasi AM grating [represented by equation (lo)], for this pooling mechanism, is 64 per cent of the peak conmechanisms are used for detection? This is now consitrast, as may be seen in the Table of Appendix B-for dered. The implicit argument for narrow-band mechanisms is that medium-band mechanisms re- the exponent p = 1. This pooling reduces the effective spond too strongly to quasi AM gratings according to contrast to IO.2 contrast units (64 per cent of the 16 the peak detection model. The quasi AM grating with peak units), which implies that the quasi AM grating should be about as detectable as a sinusoidal grating frequency components 0.8 and 1.0, each with contrast of locontrast units-the two’gratings are in fact about 8. (used to construct Figs. 8 and 9) can be represented equally detectable. In addition, by limiting the spatial by a trigonometric identity. extent of the pooling to a fixed number of cycles (depending on luminance), the data of Hoekstra et al. P(X) = 8 (cos 0.8 2n.u + cos 1.0 2nx) (1974) on grating detection as a function of number of = I6 cos 0.127~ cos 09 27~. (IO) cycles can readily be accounted for. =

908

C. F. STROMEYER II1 and S. KLEIN

The linear pooling may be carried out by a mechanism which has inputs from several mediumband mechanisms of similar peak spatial frequencies and different contiguous center positions. The medium-band mechanisms are assumed to respond approximately linearly as a function of contrast. at least within a small range above threshold, and the higher pooling mechanism may linearly summate these responses. (The non-lineartty represented by the transducer function. that relates detectability d’ to contrast. is thus assumed to occur after the pooling mechanism.) The pooling mechanism has no unique receptive field, for it responds about equally to a bar placed at any position within its large receptive field, and thus a regular grating and a somewhat irregular (FM) grating will produce similar responses. The mechanism does not have a narrow-band frequency response, but rather a response similar to the first-stage medium-band mechanisms. The receptive field of the pooling mechanism is not the Fourier transform of the frequency response since the first-stage mechanisms introduce a nonlinearity by acting as rectifiers (due to their low spontaneous firing rate). Other schemes for linear pooling can readily be imagined-besides the present physiological model. Linear pooling appears to account for the facts on the detectability of both FM and quasi AM gratings, and narrow-band visual mechanisms may be unnecessary. The present study provides initial evidence that narrow-band visual mechanisms do not exist. Narrowband mechanisms might serve little purpose since they would poorly encode phase, or position, information, whereas medium-band edge mechanisms could ideally perform this function. AckIlowledgernents-We thank Professors Karl Pribram and Leo Ganz for their generous support and Cecilia Bahlman and Jean Norris for their help with the manuscript. REFERENCES Abadi R. V. and Kulikowski J. J. (1973) Linear summation of spatial harmonics in human vision. Vision Res. 13, 1625-1628. Black H. S. (1953) Modulation Theory. Van Nostrand New York. Blakemore C. and Campbell F. W. (1969) On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images. J. Physiol., Land. 203,237-260. Blakemore C.. Muncey J. P. J. and Ridley R. M. (1973) Stimulus specificity in the human visual system. Vision Res. 13, 1915-1931. Campbell F. W. and Green D. G. (1965) Optical and retinal factors affecting visual resolution. J. Physiol., Lond. 181, 576-593. Campbell F. W. and Robson J. G. (1968) Application of Fourier analysis to the visibility of gratings. J. Physiol.. Land. 197, 551-566. Dorfman D. A. and Alf A. E. Jr. (1969) Maximum-likelihood estimation of parameters of signal-detection theory and determination of confidence intervals--rating-method data. J. math. Psychol. 6, 487-496. Egan J. P.. Schulman A. 1. and Greenberg G. Z. (1959) Operating characteristics determined by binary decisions and by ratings. J. acoust. Sot. Am. 31, 768-773. Flamant l? (1948) Variation du diametre de la pupille de l’oeil en fonction de la brillance. Rev. onr. 27. 751. Graham N. and Nachmias J. (1971) Detection of grating patterns containing two spatial frequencies: a comparison

of single-channel and multiple-channels models. )/I.SIO~I Rcs. 11, 251-259. Green D. M. and Swets J. A. (1966) Signal Dercctlorr Tkeor!, and Ps_vchophysics. Wiley, New York. Hoekstra J.. van der Goot D. P. J.. van den Brink G. and Bilsen F. A. (1974) The influence of the number of cycles upon the visual contrast threshold for spatial sine wave patterns. l’ision Rcs. 14, 365-368. King-Smith P. E. and Kulikowski J. J. (1974) The detection of gratings by independent activation of line detectors. Paper presented to Meeting of Association for Research in Vision and Ophthalmology. King-Smith P. E. and Kulikowski J. J. (1975) The detection of gratings by independent activation of line detectors. J. PhJsiol.. Land. (in press). Kulikowski J. J. and King-Smith P. E. (1973) Spatial arrangement ofline, edge and grating detectors revealed by subthreshold summation. Vision Res. 13, 1455-1478. Lange R. V.. Sigel C. and Stecher S. (1973) Adapted and unadapted spatial frequency channels in human vision. b’ision Rcs. 13. 2139-2144. Nachmias J. and Kocher E. C. (1970) Visual detection and discrimination of luminance increments. J. opt. Sot. Am. 60,382-389.

Nachmias J. and Sansbury R. V.-(1974) Grating contrast: discrimination may be better than detection. Vision Res. 14, 1039-1042. Nes F. L. van and Bouman M. A. (1967) Spatial modulation transfer in the human eye. J. opt. Sot. Am. 57, 401-406. Ogilvie J. C. and Creelman C. D. (1968) Maximum-likelihood estimation ofreceiver operating characteristic curve parameters. J. mar/t. Psychol. 5. 377-391. Pantle A. (1973) Visual effects of sinusoidal components of complex gratings: independent or additive’! Visiotl Res. 13, i195-2204. . Ouick R. F. Jr. (1973),.Soatial-freauencv selectivity in human . vision: the frequency response 01 contrasi detection channels. Doctoral dissertation, Carnegie-Mellon University, Pittsburg, Pennsylvania. Quick R. F. Jr. (1974) A vector-magnitude model of contrast detection. Kybernetic 16,65-67. Sachs M. B., Nachmias J. and Robson J. G. (1971) Spatialfrequency channels in human vision. J. opt. Sot. Am. 61, 1176-1186. Stecher S., Sigel C. and Lange R. V. (1973) Spatial frequency channels in human vision and the threshold for adaptation. Vision Res. 13, 1691-1700. Stromeyer C. F. III and Klein S. (1974) Spatial frequency channels in human vision as asymmetric (edge) mechanisms. Vision Res. 14, 1409-1420. Stromeyer C. F. III and Julesz B. (1972) Spatial-frequency masking in vision: critical bands and spread of masking. J. opt. $0~. Am. 62.1221-1232. _ Stromeyer C. F. III, Lange A. F. and Ganz L. (1973) Spatial frequency phase effects in human vision. Vision Res. 13, 2345-2360.

APPENDIX A The stimulation map is calculated from the overlap of the stimulus pattern P(x) and the receptive tield RF)p,.(~ - x,). The receptive field (line spread function) can be written in terms of the mechanism sensitivity curve @(&/j)

RFl&

- xc) =

= df @U-p/f)COSZx[/(x - x,) - %] i0

where x, is the central position of the mechanism and n is the phase of the mechanism. For a bright line mechanism x = 0; for an antisymmetric edge mechanism I = +_l/4: and for a dark line mechanism 3~= l/2. Since all stimulus patterns used in the present experiment are symmetric periodic functions. they can be expanded in

909

The detectability of frequency modulated gratings the series

To evaluate the product. we will use a parameterization of the psychometric function proposed by Quick (1974) p(c) = 2-c*

P(X) = x a, cos Znrt/‘+ “=I3 where II is an integer andfo is the repetition frequency of the grating. For simplicity, a flat contrast sensitivity function is assumed. The stimulation of a mechanism is given by the overlap between the receptive field of the mechanism and the pattern. The stimulation of the mechanism characterized by position X, peak frequencyf,, and phase I is given by L dl RFJp,& - s,) P(s) S(Jp. 1. .u,) = I -1

where p = 5 is gotten by directly fitting the psychometric functions found by Sachs er al. (1971). nhe value p = 7.8 which Quick uses does not fit the data of Sachs et al. (1971) unless an additional correction for guessmg is assumed.] The contrast c is in units of threshold contrast, since P( 1) = 0.5--the threshold contrast is defined as 50 per cent probability of seeing on the psychometric function. Using this parameterization. the probability that the pattern is not detected by any of several independent channels is

The term c,rl is the effective contrast of a complex pattern and is

= f. a. @(fpbtfo)cos2Ncf0.~ + d.

Ccff =

S(_/j&x,) = cos 2tr(B - +34&J,). The stimulation magnitude SM&x,) SMcf,x,) = C(~U&))’ +

1 1.P

Algebraic manipulations enable the dependence upon the mechanism phase to be separated out as follows

1 [.

cf

The signal detection theory procedure for combining the output of independent channels gives the same pooling formula since

is

w&~cN211’2

with

AUp,xc) = i

a,

@(fp/flfo)cos2n(nfo x,)

Relating d; to ci using the transducer function gives

“=O

c1

112

B(f,, x,) = f a, @(fp/nfolsin WI/, x,1.

c:rr =

“=O

L

The angle B is given by

B(&x,) = a@U,/f) sin 211(&x,). Thus

SM.&J,) = M_fdA B = f0.L For a simple grating the stimulation magnitude has become independent of the position x,. For general patterns, .W(&K,) is a slowly varying function of I,. The function SM(j&,) provides information on how strongly different patterns stimulate the different mechanisms. and this information is represented in the stimulation maps. B

Probability summation can be applied to independent detection channels. Let P&J be the probability that channel i detects a pattern of contrast ci, then the probability that channel i does not detect the pattern is P,(q) = 1 - P&J. The function P(c) is the psychometric function and is obtained with the method of constant stimuli, in which the observer maintains a constant, low false alarm rate. Probability summation states that the probability that one or more channels detects a pattern is

-=,-I

ceff(sin)

c,,,(quasi AM)/c,,(sine):

I

IJ 1j-.Wxj

with the integral extending over the region of probability summation. For the quasi AM pattern given by equation (IO) of the Discussion A(x) = lcos 01 2~x1. The effective contrast of this grating relative to an unmodulated sinusoidal grating of the same peak amplitude and spatial frequency is calculated and shown in the table for different values of the pooling exponent p.

P(c) = I -J-pi(Ci,. Pooling exponent p Model

.

J

Thus the pooling exponent is related to the transducer exponent by p = 2t. Consider, for example, a pattern of two superposed sinusoidal gratings that are well separated in spatial frequency in order to stimulate different, independent detection channels. If each component is at its own detection threshold, then the effective contrast of the compound grating is ccn = (1 + 1)“p = 2°‘20 = 1.15. Thus by probability summation, the effect of adding the second grating to the first grating, increases detectability by the same amount as would occur if the second grating were not added, but rather the contrast of the first grating was increased by 15 per cent. As another example, assume that independent detection channels at different spatial positions are stimulated with a grating A(x) cos 2x$x, whose amplitude A(x) is slowly varying across the grating. The ratio of the effective contrast of this modulated grating to the effective contrast of an unmodulated sinusoidal grating of amplitude A0 is l-r 1 I/l, 4(x)p ds c&nod)

tan B = W~~J/d_&J The roles of SM(&x,) and fl can be clarified by examining the response to a simple grating of frequency f and amplitude a : AC&x,) = aN&/f) cos Zn(_&)

APPENDIX

cc:

z

Linear summation

Energy (power) summation

064

0.7 I

5 Probability summation

8 Probability summation (Quick)

P:k detection

0.81

092

I.0

C. F. STROMEYER III and S. KLEW

910

Both the linear summation and the energy summation models are compatible with the premise that quasi AM gratings are detected by medium-band mechanisms (see DiscussionI.

APPENDIX

e-!.i : d)

(c s -m

(C-4)

C

Hoekstra er al. (1974) measured the contrast threshold of sinusoidal gratings as a function of number of bars and luminance. The threshold decreased as the number of cycles was increased up to a saturation point, which depended on luminance but which was independent of spatial frequency within the range tested-l-7 c/deg. The bandwidth of the detection mechanism can be estimated from the data, assuming a peak detection model in which the threshold is determined by the mechanism that responds maximally. Let it be assumed that the sensitivity S of the mechanism to a pattern P(x) is proportional to the overlap of the pattern and the receptive field RF of the detection mechanism. S&.Xx

to a very large grating can be expressed

PF,,,(x)P(N

d.x.

(C-l)

If the frequency selectivity of the mechanism is

(pub/Y7 = expi - L&l - Il’/~‘l

G-2)

where& is the peak frequency of the mechanism [same as equation (6) of Discussion], then the receptive field [the Fourier transform of equation (C-2)] is RF&(x) = ~0s 2x(&x i- a)E(x)

(C-3)

where the envelope is E(x) = exp - (x&u)~. The sensitivity of the mechanism to a sinusoidal grating of n cycles relative

(C-3

where

and Thus 22

d = c = 0.45 r/n. Ii,‘2 x

(C-7)

For z = 1.1 which from equation (C-5) gives SJS = 0.7. we get 20 = l.O/ll@,. (C-8) Thus the full bandwidth. 2~. is the reciprocal of the number of cycles (~a,,) needed to obtain 70 per cent of the maximal possible sensitivity value. Inspection of Hoekstra et at.‘s (1974) data shows that ~7 = 2.3. 3.6, 50, 5-5 at mean luminance 2, 25, 165 and 600 cd/m2 respectively. Equation (C-8) gives the respective values u = 0.22, 0.14, 0.10, @09. The bandwidths at the three higher luminance levels are compatible with that found by Kulikowski and King-Smith (1973). Hoekstra et al. (1974) calculated bandwidths which were narrower assuming that the detection mechanism responds to a narrow Jrequency range of the power spectrum of the stimulus. Experiments with FM gratings throw considerable doubt upon these methods of estimating the frequency selectivity of visual mechanisms.