- Email: [email protected]
Spatial pooling properties deduced from the detectability of FM and quasi-AM gratings: A reanalysis
Pbion Rzs. Vol. 16. pp. IO?1 to 1026. Pergamon Press 1976. Pnnted in Great Bntain.
LETTER SPATIAL POOLING DETECTABILITY
TO THE EDITORS
PROPERTIES DEDUCED OF FM AND QUASI-AM A REANALYSIS
FRO>1 THE GRATINGS:
(Rrceiced 28 October 1975; in recked form 16 Febrtrury 1976)
In a thought-provoking paper. Stromeyer and Klein (1975) study several models of spatial pattern detection. The class of models they consider (Campbell and Robson, 1968; Thomas, 1970; Kulikowski and KingSmith, 1973) postulates the existence, at some stage in the visual system, of many different sizes of receptive field centered on every retinal point. Within this “multiple channels” context, the questions they address directly are: (1) What is the bandwidth, or range of spatial frequencies. to which these receptive fields respond? and (2) What kind of pooling, if any, exists among receptive fields located at different spatial positions across the visual field? As Granger (1973) pointed out in a somewhat different context, bandwidth estimates in many situations depend on the spatial pooling mechanism assumed. As a tool to disentangle these two factors, Stromeyer and Klein propose the use of three different types Gf grating: sinusoidal, frequency-modulated, and quasLamplitude-modulated. This interdependence of bandwidth and spatial pooling estimates is evident in the interpretation of the sine-plus-sine experiments (Sachs, Nachmias and Robson, 1971; Lange, Sigel and Stecher, 1973; Quick, 1973). In these experiments, the detectability of a grating composed of two sinusoids (called a “quasi-amplitude-modulated grating” by Stromeyer and Klein) is compared to the detectability of each sinusoid by itself. Stromeyer and Klein suggest that channel bandwidth may well have been underestimated in the original interpretation of these experiments. The small amount of measured facilitation between sinusoids of nearly equal frequency, they argue, may not necessarily imply narrow bandwidth channels. For, if there is spatial pooling, the contrast attenuation inherent in the “beating” of such sinusoids with each other may diminish the facilitation effect, giving the illusion of narrowly-tuned channels. Using frequency-modulated gratings, which are by definition of constant contrast across the visual field, Stromeyer and Klein attempt to provide evidence against the existence of these narrow-band channels in favor of medium-band channels. They compare the psychophysical detectability of sinusoidal, frequency-modulated (FM), and quasi-amplitude-modulated (quasi-AM) gratings. From this comparison they draw several conclusions concerning bandwidth and spatial pooling. There are several points, both about the theory and the data, which Stromeyer and Klein have over‘The contrasts in the two component sinusoids were not exactly equal. The contrasts were adjusted to make equal the amount by which each component’s contrast exceeded its own threshold. Since the thresholds for frequencies near 4 cjdeg were all nearly equal, the contrasts used were nearly equal.
looked. When these points are taken into account, the conclusions that can be drawn from their study are substantially changed, especially those about spatial pooling. Because both the questions asked by this study and the method used are interesting. it seemed worthwhile to consider these points in detail. DAT.4
Stromeyer and Klein’s theoretical arguments rest on the assertion that three particular gratings are all equally detectable: a simple sinusoidal grating, a quasi-AM grating. and an FM grating. The simple sinusoidal grating, the reference grating, was of a frequency near 4.0 c deg and a contrast of 10 units. (The particular units are not important for this argument.) The quasi-AM grating consisted of two sinusoidal components, one component of a frequency near 4 c/deg and the other of a frequency 4/j of that. Each component had a contrast of 8 units. The FM grating consisted of three major frequency components: a center frequency (carrier) near 4 c/deg and two sideband frequencies spaced about 1.5 c/deg above and below the center frequency. The overall contrast was 10 units; the contrast of each of the three components was 5.5 units. Data for rhe FM
grating
The main empirical contribution of Stromeyer and Klein’s study is the comparison of the detectabilities of the simple sinusoidal and FM gratings described above. Over all their experiments (see thiir Table 1) the average ratio of d’ values for the two patterns was 0.96. where a value less than 1 indicated that the FM grating was less detectable than the simple sinusoid. They conclude that the two gratings are ap proximately equally detectable. The variability of the data, however, should not be forgotten. The standard error associated with the detectability ratio above is 0.08 (see their Table 1). The usual confidence range (52 S.E.) is thus 0.96 + 0.16, or 0.80-1.12. Data for the quasi-AM
grating
Evidence for the assertion that the quasi-AM and simple sinusoidal gratings described above are equally detectable was not collected by Stromeyer and Klein but is loosely attributed by them to Sachs er al. (1971) and Quick (1973). Sachs et al. (1971) did not study the range of frequencies used by Stromeyer and Klein. An examination of Quick’s data (which is also presented in Quick and Reichert. 1975) does not support the contention of equal detectability. Quick and Reichert used quasi-AM gratings containing two sinusoidal components of approximately equal contrast.’ They present their data in terms of
Letter ICIthe Editors
1032
a retarire sensiticiry measure. This measure can be
defined in terms of contrast. When the sinusoidal and the quasi-AM gratings have been adjusted to be equally detectable, the relative sensitivity equals the contrast of the sinusoidal grating divided by the contrast in one component of the quaCALM grating. For a quasi-AM grating like that discussed by Stromeyer and KIein, Quick and Reichert found relative sensitivity to be approx 1.6, 1.5 and 1.35 for their three subjects EQ, FQ and JH (see their Figs. 3 and 4). In other words. for these three subjects, a sinusoidal grating having a contrast of 10 units was just as detectable as a quasi-AM grating having components of 6.25, 6.6 or 7.3 units of contrast. For these subjects, therefore, the quasi-AM grating discussed by Stromeyer and Klein (having 8 units of contrast in each component) would be more detectable than the sinusoid. It will be convenient later to present Quick and Reichert’s data in the following form: their three subjects were 1.X (which is 8 divided by 6.25), 1.20 and 1.08 times more sensitive to the quasi-AM grating discussed by Stromeyer and Klein than to the sinusoid. The three different numbers given for the three different subjects provide some measure of the variability in Quick and Reichert’s study, both intra- and inter-subject variability. There is another source of variability, however, that is more difficult to deal with. That if the possible difference between the subjects used in Quick and Reichert’s experiments and the subjects used in Stromeyer and Klein’s. Ideally. the same subjects would have viewed all three grating types. THEORY
Stromeyer and Klein assume that neural units having similar best frequencies also have similar bandwidths. Although some investigators have proposed models in which there are multiple bandwidths at any given frequency (Kulikowski and King-Smith, 1973). Stromeyer and Klein’s assumption of single bandwidth per frequency is not unusual. Since they are working within a small range of frequencies. their further assumption that all neural units have the same bandxvidth expressed in log frequency is not critical. .S_rmmerrJproperries. Stromeyer and Klein explicitly assume that the neural units have receptive fields of many different symmetries. That is, one receptive field might have inhibitory sections on two sides of an excitatory section. Another might have only one inhibitor]; section next to one excitatory section. Stromepr and Klein assume that there is a very large number of ditrerent symmetries (indexed by a parameter 1) for every possible best frequency and spatial location. Other investigators (e.g. Sachs er ul., 1971; Graham. 1976) have assumed a11receptive fields have the same symmetry. Srimularion maps. The sensitivity of a neural unit to a pattern is a function of its receptive field’s best frequency cf,), location (.u,) and symmetry (z). This sensitivity is denoted S&,r,x,) by Stromeyer and Klein. but this quantity rarely appears in their paper. According to their Appendix A, it is not necessary to keep track of all the different symmetries of receptive field. For each combination of best frequency and location. one can simply work with the one receptive fiefd that has the optimal symmetry for the pattern under consideration, i.e. the receptive field that is most sensitive to that pattern. The sensitivity of this optimal unit is
This section will attempt to clarify the terms and concepts used by Stromeyer and Klein. Properties of neural units’ receptice fields
Like many other investigators, Stromeyer and Klein are interested in a class of models which postulates the existence of neural units having receptive fields of many different sizes. They index size by the spatial frequency to which the receptive field responds best, called 1,. Each of the neural units is presumed to behave linearly over the small range of near-threshold intensities tested. Since the stimuli in this study are ail stationary, achromatic, vertical patterns, the receptive fields’ temporal, wavelength-selective and orientation-selective characteristics can be ignored. They further assume that each size of receptive field exists in many locations uniformly spread across the retina, and they index location by the position of the central point of the receptive field, x,. This disregard for retinal inhomogeneity could be justified on the grounds that all the patterns they consider are distributed quite widely across the retina. ’ The words “channel” and “mechanism” have been avoided in this discussion and will continue to be. There is some inconsistency among authors and some difficulty in Stromeyer and Klein’s paper itself. It appears. for example. that they sometimes use the word mechanism to mean any single receptive field and sometimes to mean the receptive field of symmetry that produces the maximal response.
Working with these quantities S&f&, .u,) produces the same predictions as working with the sensitivities of all the neural units. It is these quantities that are plotted in the stimulation mnps shown in Stromeyer and Klein’s Figs. 5-9. For each of the three gratings {the sinusoidai, FM, and quasi-AM gratings described earlier) two stimulation maps are presented, one for each of two possible bandwidths of neural unit. One bandwidth is very narrow (approx : octave at half amplitude) and the other medium (approx 1 octave at half amplitude).” Theorerical
detection rules and spatial pooling
To complete the specification of a particular version of the multiple-channels model, a detection rule is necessary to specify how the responses of the neural units lead to the detection of a pattern. Here the question of spatial pooling is important. Prirr peak detection. Under a very simple detection rule called peak detection by Stromeyer and Klein, the sensitivity of the observer to the pattern is simply equal to the sensitivity of the most sensitive of all the neural units (the “peak”). This sensitivity can easily be found by picking the biggest number out of the stimulation maps. According to this rule, there is no pooling of information from different neural units. In particular. there is none from units at different spatial positions.
Letter to the Editors Probabiliry summation. Suppose. however. that the variability in the observer’s responses is due to variability in the neural units’ responses. Suppose also that the observer detects a pattern whenever the response of one or more neural units is larger than some criterion. As has been explained more fully elsewhere (Graham, 1976), probability summation will then occur across different neural units. The sensitivity of the observer will depend not just on the sensitivity of the one most sensitive unit, but on the sensitivity of many units. Whether this pooling will occur over all neural units or not depends on the assumption one makes about the correlation of the variability in different neural units. If the variability in all neural units is assumed to be independent, then the probability summation pooling will occur over all the neural units. This case will be called pure proba~ilir~ su~arion. If, however, the variability in all units of the same best frequency is perfectly correlated while the variability in units of different best frequency is independent, then pooling will not occur over all units. Instead, within each group of neural units having the same best frequency there will be peak detection, and the peak sensitivities from different best frequencies will form a pool over which probability summation occurs. Or, to say it in other words, there will be probability ~rn~t~on across difirent frequencies but not across different spatial locations or symmetries. (Sachs et (I/., 1971, worked with a model of this kind.) Other possibilities exist that do not involve probability summation across frequencies: probability summation across space but not across different best frequencies or symmetries, for example. However, since Sachs et al. (1971) provided good evidence that there is probabiiity summation across frequencies, these other possibilities seem less interesting. Linear pooling. Another kind of detection rule that Stromeyer and Klein consider is linear pooling. Here the sensitivities of neural units in the pool simply add. Only one neural unit for each spatial position is in the pool. the unit of whichever best frequency responds maximally (Klein, personal communication). The pooling formula. As Quick (1974) has pointed out, there is an easy method for calculating the results of probability summation, based on a particularly convenient analytic form for psychometric functions. Accordingly, the sensitivity of an observer when there is probability summation equals a nonlinear summation of the individual sensitivities in the probability summation pool. Let these individual sensitivities be called s(i), where i indexes individual members of the pool. Then the sensitivity of the observer will equal
li
)
where k. the exponent, is in the range from 4 to 8, and the summation is done over all members in the pool. The exact value of the exponent depends on the steepness of the psychometric function. Although we have introduced formula (1) in the context of probability summation, it can be viewed as a general pooling formula. The peak detection case can be shown to be equivai~t to this formula with an exponent equal to infinity.‘ An exponent in the range from 4 to 8 leads to probability summation
1023
predictions. An exponent of :! produces power summation. An exponent of 1 produces linear pooling. Snomerer and Klein’s pooling method. Stromeyer and Klein’s discussion (their Appendix 3) of the pooling formula is somewhat misleading. They pool over the stimulus envelope rather than over the sensitivities of neural units. (Compare the last formula of their Appendix B with formula (1) above. Alternately, notice that in their formula for Quick’s ~ychome~c function there is no sensitivity factor; this would be the correct formula if all neural units detected all patterns of one unit contrast with a probability of onehalf.) They may have done this because they were using the pooiing formula only for the broadest bandwidth neural units they considered (the medium bandwidth). These units are, to a first approximation, uniformly sensitive to all the Frequencies present in the patterns used. If the units actually were uniformly sensitive, Stromeyer and Klein’s pooling formula would reduce to the correct one. But, since the medium bandwidth neural units are not all quite uniformly sensitive, the ratios of “effective contrast” given in the table in Appendix B, are not quite correct even for the medium bandwidth case. For narrower bandwidths, the ratios analogous to those in the table certainlv must be computed from formula (1) and are quite different from those in the table. RESULTS
Stromeyer and Klein’s conclusions
Stromeyer and Klein arrive at several conclusions. (1) They reject pure peak detection (an exponent of infinity in the pooling formula) because @. 906) “the peak detection model predicts that medium band mechanisms respond too strongly to the quasi-AM grating and. . .narrowband mechanisms respond too weakly to the FM grating”. (2) They reject probability summation pooling because “probability summation does not sufficiently decrease the detectability of quasi-AM gratings, assuming medium-band mechanisms are used” and ‘-does not sufficiently increase the detectability of FM gratings using narrowband mechanisms for detection” (p. 907). (3) Assuming medium-band channels, “linear pooling appears to account for the facts in detectability of both FM and quasi-AM gratings, and narrowband visual mechanisms may be unnecessary” (p. 908). Taking into account the empirical and theoretical points described above. we arrive at rather different conclusions, even if we restrict ourselves to the same model versions discussed by Stromeyer and Klein. Theoretical predictions
Figure 1, which is a complicated figure that will be described in parts, shows the predictions of various versions of the model for the trio of patterns discussed by Stromeyer and Klein (the particular sine, quasi-AM and FM patterns described earlier). The predictions are plotted as sensitivity relative to the sine-i.e. as sensitivity of an observer to one pattern (either the quasi-AM or the FM) divided by the sensitivity of the observer to the sine. Since Stromeyer and Klein suppose the three chosen patterns are equally detectable to a human observer, they ask of a successful model that it predict relative sensitivities
I I)31
Letter to the Editors
equal to 1.0 for both the quasi-AM and the FM gratmg. As discussed earlier however, the data suggest that these three chosen patterns are probably not equally detectable. The relative sensitivity of the chosen quasi-AM grating was 1.25. 1.20 and 1.05 for Quick’s three observers. The variability in Stromeyer and Klein’s own data leads to a possible range of 0.8-1.11 for the relative sensitivity of the FM grating.3 These data are plotted on the far right-hand side of the ngure. Relative sensitivity- for the quasi-AM grating rs indicated by filled circles and relative sensitivity for the FM grating by open circles. The other four panels of the figure contain the predictions calculated from four different detection rules described earlier: pure peak detection (far left): probability summation across frequencies with peak detection across spatial Locations and symmetries (second to left); pure probability summation across al1 neural units (middle): and linear pooling (second to right). Using the pooling formula (I), the predictions in Fig. 1 were calculated” from the stimulation maps of Stromeyer and Klein. For each kind of detection rule, the predictions are shown for several bandwidths of channel (horizontal axis). The widths labelled N (n~row) and %I (medium) are those for which the stimulation maps were given in Stromeyer and Klein (1975). The intermediate bandwidth (labelled I) is slightly greater than the narrow one.’ The stimulation maps for this case
were also calculated by Stromeyer and Klein (personal communication). In the panels for pure probability summation and linear pooling there is one extra point for quasi-AM gratings which is plotted just to the right of the medium bandwidth points. These extra points are the predictions computed by Stromeyer and Klein (Appendix B) in the manner discussed in the earlier section “Stromeyer and Klein’s pooling method”. The points are plotted to the right because they are, in effect, predictions for a bandwidth even wider than the medium one. In both of the paneis involving probability summation, there is one extra point for FM gratings which is plotted to the left of the narrow bandwidth points. These extra points are predictions assuming that the bandwidth is so narrow that there is no neural unit which responds more to the FM grating than to one of its components alone. These extra points and the points for the peak detection case are the only predicted values expiicitly discussed by Stromeyer and Klein. Considering the full range of predictions and data leads to conclusions different from those reached by Stromeyer and Klein.
’ To obtain this range of relative sensitivities, it was necessarv to chanoe the ratios of cf’values into ratios of semitici&. (Sen&ity is the reciprocal of the contrast
!c~= infinity and L, = 5. For the pure probability summation case. f;, = i;: = 5. For linear pooling, the observer’s sensitivity equals Ix k
necessary to produce a criterion response.) To make this change requires knowing the transducer function. a function relating stimulus contrast to d’. Stromeyer and Klein 11974) and Nachmias and Sansbury (1974) have suggested that R’ is some low power {second or third) of contrast. If so. the ratios of sensitivities would be closer to one than the ratios of ri’ values. With a linear transducer function, on the other hand, the ratios of sensitivities would be exactly equal to the ratios of d’ values. Since the question of a transducer function is far from settled, it seems reasonable to use the larger confidence range based on the linear transducer function. If, however, a smaller confidence range were used. the regions of acceptable predictions would be diminished. If the transducer function were nonlinear enough, there might be no range of acceptable predictions from the first detection rule (pure peak detection). A Formally, the calculations were done as follows. The observer’s sensitivity to a particular pattern assuming a particular bandwidth was calcufated from the appropriatz
srrmulation map for that pattern and bandwidth. For the first three detection rules. the observer’s sensitivity equals
where
The exponent k, determines the pooling over spatial position and symmetry: the exponent k2 determines the pooling over frequency. For the pure peak detection case. li, = k, = infinity. For the probability summation across frequency with peak detection across space and symmetry.
Compnring predictions
to dutu
The shaded areas in the figure contain regions of acceptable predictions, the regions where both the predictions for the quasi-AM grating and the predictions for the FM grating are consistent with the data (far right panel).
.+ Lf
P
where i( equals infinity. The su~at~on was done over ail j, in the stimulation maps. There were !1 different& spaced uniformly on a linear frequency axis. The second summation was done over 12 x, spaced uniformly across one period of the reference sinusoid’s frequency. (The published stimulation maps for the quasi-AM and FM gratings show these I:! different positions. The published map for the sinusoid does not show all 12 because for a given & the response at all positions is constant.) Using only the sensitivities of units with optimal symmetry is presumably equivalent to assuming that the distribution of values of the symmetry parameter is uniform over its range of 0.0-1.0. Ideally, one should do the calculations weighting in the differential sensitivities at different frequencies, since the contrast sensitivity function is not perfectly flat (although it was assumed to be so in making up the stimulation maps). Also one should consider frequencies higher than those given in the stimulation maps. We could not do this, unfortunately. since the necessary information was not available. One might also want to consider other models in which, for example. the summation is done over frequencies spaced uniformly on a logarithmic frequency axis rather than a linear frequency axis. It is not clear how sensitive the predictions are to these details. There may be round-off errors in our calculations since the numbers given in the published stimulation maps are rounded off to the nearest integer. 5The parameter sigma in formula (6) of Stromeyer and Klein (1975) equalled. 0.20 for the intermediate case and 0.17 for the narrow case.
Letter to the Editors [Peak
drtsctlon
I
Prob.summotion (fr.qu*ncy only)
PUr* pIpI_ summation
_inaar pooling
-
N
I
M
N I
M Bandwidth
N I
M
N
I
M
FM OuosiAM Pattern
Fig. 1. The predictions of four spatial pooling models (the four left panels) for sensitivity to FM (open circles) and quasi-AM (filled circles) gratings relative to the sensitivity to the reference sinusoidal grating. These predictions are shown for several bandwidths (N = narrow. I = intermediate. M = medium). The measured psychophysical relative sensitivity is shown in the right panel. The data for FM gratings are from Stromeyer and Klein (1975); the data for quasi-AM gratings are from Quick and Reichert (1975). The shaded areas indicate the bandwidths for which both the predicted quasi-AM and the predicted FM sensitivities are consistent with the data.
Consider the peak detection case first (far left). Although for both the narrow and medium bandwidths, the predictions are inconsistent with the data, the full graph suggests that some bandwidth between these extremes might produce predictions consistent with the data. The acceptable range of bandwidths is very small, however, and leads to predictions in the extreme ends of the data ranges (high for quasi-AM, low for FM). Thus one might accept, with reservations, Stromeyer and Klein’s first conclusion. Once probability summation is introduced, either just across frequencies (second to left) or over all neural units (middle), somewhat larger ranges of bandwidth produce predictions consistent with the data. With probability summation across frequencies only, a range of narrow bandwidths produces acceptable predictions. Here again, the predictions are in the extremes of the data ranges. With probability summation across all units, however, a large range of medium bandwidths produces predictions consistent with the data (and not just with the extremes of the data ranges). It seems reasonable to conclude, therefore, in direct contradiction to Stromeyer and Klein’s second conclusion, that a model postulating probability summation, particularly probability summation across all of a collection of medium bandwidth neural units, is consistent with the data. Finally, in contradiction to Stromeyer and Klein’s third conclusion, there is no range of bandwidths for the linear pooling case which leads to acceptable predictions for both the quasi-AM and FM gratings (second to right). Too little sensitivity to the FM grating is always predicted. All the above conclusions, unfortunately, rest on the assumption that Quick’s subjects and Stromeyer and Klein’s subjects do not differ in some systematic way. Other models exist, of course, that Stromeyer and Klein did not consider. In particular, there are models which postulate only one symmetry of receptive field, CONCLUSlONS
Although the results of Stromeyer and Klein’s study are somewhat disappointing in that they do not
allow very strong conclusions, they are suggestive. As is shown in the predictions plotted in Fig. 1, which were based on the stimulation maps calculated by Stromeyer and Klein, the comparison of data obtained with FM gratings, quasi-AM gratings and sine-wave gratings does allow some discrimmation among models having different bandwidth and spatial pooling properties. More extensive data might be able to make better use of this discrimination. Not all possible models, or even all possible versions of models having multiple kinds of receptive fields, have been considered here. But in a theoretical framework with many unknowns, one cannot hope to decide all questions at once. Of the models Stromeyer and Klein consider, a model in which there are medium bandwidths and probability summation across all neural units seems to provide the best fit to the data. Certainly the study cannot be taken, as they did, as evidence for a linear pooling model and against a probability summation model. Two other recent studies also tend to support a model in which there are medium bandwidths and a spatial pooling mechanism like probability summation across space (King-Smith and Kulikowski, 1975; Mostafavi and Sakrison, 1975). It does not yet seem possible, however, to conclusively rule out other models, including ones with narrower bandwidths and no spatial pooling. Dept. of Psychology, Columbia Uniwrsiry, New York, NY 10027, U.S.A.
NORXL~GFUHLCI BERNICEE. ROCXWITZ
REFERENCES Campbell F. W. and Robson J. G. (1968) Application of Fourier analysis to the visibility of gratings. J. Physiol., Lond. 197, 551466. Graham N. (1976) Spatial frequency channels in human vision: detecting edges without edge detectors. In Visual Coding and Adaptability (Edited by Harris C.). Erlbaum. &anger E. (1973) An alternative model for grating detection. Paper presented at the meetings of the Association for Research in Vision and Ophthalmology. Sarasota, Fla., May, 1973.
IO26
Letter
to the Editors
King-Smith P. E. and Kulikowski J. J. (1974) The detection of gratings by independent activation of line detectors. J. Phpiol., Lond. 247, 237-271. Kulikowski J. I. and King-Smith P. E. (1973) Spatial arrangement of line, edge, and grating detectors revealed by subthreshold summation. Vision Rex 13. 1455-1478. Lange R., Sigel C. and Stecher S. (1973) Adapted and unadapted spatial-frequency channels in human vision. Vision Res. 13, 2139-2143. ‘Mostafavi H. and Sakrison D. (1975) On the behavior and modeling of a single channel in the visual system. Paper presented at the meetings of the Association for Research in Vision and Ophthalmology. Sarasota, Ha., May. 1975. Nachmias J. and Sansbury R. B. (1974) Grating contrast: discrimination may be better than detection. Vision Res. 14, 1039-1042. Quick R. F. (1973) Spatial-frequency selectivity in human vision: the frequency response of contrast detection
channels. Ph.D. thesis. Carnegie-Mellon University. Pittsburgh Pa. Quick R. F. (197-t) A vector-magnitude model of contrast detection. Kybrmetik 16. 64-67. Quick R. F. and Reichert T. A. (1974) Spatial-frequency selectivity in contrast detection. Vision Rrs. 15, 637-6-13. Sachs M. B., Nachmias J. and Robson J. G. (1971) Spatial frequency channels in human vision. J. opt. Sot. Am. 61, 1176-l 186. Stromeyer C. F. and Klein S. (1974) Spatial frequency channels in human vision as asymmetric (edge] mechanisms. Vision Rcs. 14. 1409-1420. Stromeyer C. E. 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. Thomas J. P. (1970) Model of the function of receptive fields in human vision. Psphol. Rec. 77. 121-134.
LETTER SPATIAL POOLING DETECTABILITY
TO THE EDITORS
PROPERTIES DEDUCED OF FM AND QUASI-AM A REANALYSIS
FRO>1 THE GRATINGS:
(Rrceiced 28 October 1975; in recked form 16 Febrtrury 1976)
In a thought-provoking paper. Stromeyer and Klein (1975) study several models of spatial pattern detection. The class of models they consider (Campbell and Robson, 1968; Thomas, 1970; Kulikowski and KingSmith, 1973) postulates the existence, at some stage in the visual system, of many different sizes of receptive field centered on every retinal point. Within this “multiple channels” context, the questions they address directly are: (1) What is the bandwidth, or range of spatial frequencies. to which these receptive fields respond? and (2) What kind of pooling, if any, exists among receptive fields located at different spatial positions across the visual field? As Granger (1973) pointed out in a somewhat different context, bandwidth estimates in many situations depend on the spatial pooling mechanism assumed. As a tool to disentangle these two factors, Stromeyer and Klein propose the use of three different types Gf grating: sinusoidal, frequency-modulated, and quasLamplitude-modulated. This interdependence of bandwidth and spatial pooling estimates is evident in the interpretation of the sine-plus-sine experiments (Sachs, Nachmias and Robson, 1971; Lange, Sigel and Stecher, 1973; Quick, 1973). In these experiments, the detectability of a grating composed of two sinusoids (called a “quasi-amplitude-modulated grating” by Stromeyer and Klein) is compared to the detectability of each sinusoid by itself. Stromeyer and Klein suggest that channel bandwidth may well have been underestimated in the original interpretation of these experiments. The small amount of measured facilitation between sinusoids of nearly equal frequency, they argue, may not necessarily imply narrow bandwidth channels. For, if there is spatial pooling, the contrast attenuation inherent in the “beating” of such sinusoids with each other may diminish the facilitation effect, giving the illusion of narrowly-tuned channels. Using frequency-modulated gratings, which are by definition of constant contrast across the visual field, Stromeyer and Klein attempt to provide evidence against the existence of these narrow-band channels in favor of medium-band channels. They compare the psychophysical detectability of sinusoidal, frequency-modulated (FM), and quasi-amplitude-modulated (quasi-AM) gratings. From this comparison they draw several conclusions concerning bandwidth and spatial pooling. There are several points, both about the theory and the data, which Stromeyer and Klein have over‘The contrasts in the two component sinusoids were not exactly equal. The contrasts were adjusted to make equal the amount by which each component’s contrast exceeded its own threshold. Since the thresholds for frequencies near 4 cjdeg were all nearly equal, the contrasts used were nearly equal.
looked. When these points are taken into account, the conclusions that can be drawn from their study are substantially changed, especially those about spatial pooling. Because both the questions asked by this study and the method used are interesting. it seemed worthwhile to consider these points in detail. DAT.4
Stromeyer and Klein’s theoretical arguments rest on the assertion that three particular gratings are all equally detectable: a simple sinusoidal grating, a quasi-AM grating. and an FM grating. The simple sinusoidal grating, the reference grating, was of a frequency near 4.0 c deg and a contrast of 10 units. (The particular units are not important for this argument.) The quasi-AM grating consisted of two sinusoidal components, one component of a frequency near 4 c/deg and the other of a frequency 4/j of that. Each component had a contrast of 8 units. The FM grating consisted of three major frequency components: a center frequency (carrier) near 4 c/deg and two sideband frequencies spaced about 1.5 c/deg above and below the center frequency. The overall contrast was 10 units; the contrast of each of the three components was 5.5 units. Data for rhe FM
grating
The main empirical contribution of Stromeyer and Klein’s study is the comparison of the detectabilities of the simple sinusoidal and FM gratings described above. Over all their experiments (see thiir Table 1) the average ratio of d’ values for the two patterns was 0.96. where a value less than 1 indicated that the FM grating was less detectable than the simple sinusoid. They conclude that the two gratings are ap proximately equally detectable. The variability of the data, however, should not be forgotten. The standard error associated with the detectability ratio above is 0.08 (see their Table 1). The usual confidence range (52 S.E.) is thus 0.96 + 0.16, or 0.80-1.12. Data for the quasi-AM
grating
Evidence for the assertion that the quasi-AM and simple sinusoidal gratings described above are equally detectable was not collected by Stromeyer and Klein but is loosely attributed by them to Sachs er al. (1971) and Quick (1973). Sachs et al. (1971) did not study the range of frequencies used by Stromeyer and Klein. An examination of Quick’s data (which is also presented in Quick and Reichert. 1975) does not support the contention of equal detectability. Quick and Reichert used quasi-AM gratings containing two sinusoidal components of approximately equal contrast.’ They present their data in terms of
Letter ICIthe Editors
1032
a retarire sensiticiry measure. This measure can be
defined in terms of contrast. When the sinusoidal and the quasi-AM gratings have been adjusted to be equally detectable, the relative sensitivity equals the contrast of the sinusoidal grating divided by the contrast in one component of the quaCALM grating. For a quasi-AM grating like that discussed by Stromeyer and KIein, Quick and Reichert found relative sensitivity to be approx 1.6, 1.5 and 1.35 for their three subjects EQ, FQ and JH (see their Figs. 3 and 4). In other words. for these three subjects, a sinusoidal grating having a contrast of 10 units was just as detectable as a quasi-AM grating having components of 6.25, 6.6 or 7.3 units of contrast. For these subjects, therefore, the quasi-AM grating discussed by Stromeyer and Klein (having 8 units of contrast in each component) would be more detectable than the sinusoid. It will be convenient later to present Quick and Reichert’s data in the following form: their three subjects were 1.X (which is 8 divided by 6.25), 1.20 and 1.08 times more sensitive to the quasi-AM grating discussed by Stromeyer and Klein than to the sinusoid. The three different numbers given for the three different subjects provide some measure of the variability in Quick and Reichert’s study, both intra- and inter-subject variability. There is another source of variability, however, that is more difficult to deal with. That if the possible difference between the subjects used in Quick and Reichert’s experiments and the subjects used in Stromeyer and Klein’s. Ideally. the same subjects would have viewed all three grating types. THEORY
Stromeyer and Klein assume that neural units having similar best frequencies also have similar bandwidths. Although some investigators have proposed models in which there are multiple bandwidths at any given frequency (Kulikowski and King-Smith, 1973). Stromeyer and Klein’s assumption of single bandwidth per frequency is not unusual. Since they are working within a small range of frequencies. their further assumption that all neural units have the same bandxvidth expressed in log frequency is not critical. .S_rmmerrJproperries. Stromeyer and Klein explicitly assume that the neural units have receptive fields of many different symmetries. That is, one receptive field might have inhibitory sections on two sides of an excitatory section. Another might have only one inhibitor]; section next to one excitatory section. Stromepr and Klein assume that there is a very large number of ditrerent symmetries (indexed by a parameter 1) for every possible best frequency and spatial location. Other investigators (e.g. Sachs er ul., 1971; Graham. 1976) have assumed a11receptive fields have the same symmetry. Srimularion maps. The sensitivity of a neural unit to a pattern is a function of its receptive field’s best frequency cf,), location (.u,) and symmetry (z). This sensitivity is denoted S&,r,x,) by Stromeyer and Klein. but this quantity rarely appears in their paper. According to their Appendix A, it is not necessary to keep track of all the different symmetries of receptive field. For each combination of best frequency and location. one can simply work with the one receptive fiefd that has the optimal symmetry for the pattern under consideration, i.e. the receptive field that is most sensitive to that pattern. The sensitivity of this optimal unit is
This section will attempt to clarify the terms and concepts used by Stromeyer and Klein. Properties of neural units’ receptice fields
Like many other investigators, Stromeyer and Klein are interested in a class of models which postulates the existence of neural units having receptive fields of many different sizes. They index size by the spatial frequency to which the receptive field responds best, called 1,. Each of the neural units is presumed to behave linearly over the small range of near-threshold intensities tested. Since the stimuli in this study are ail stationary, achromatic, vertical patterns, the receptive fields’ temporal, wavelength-selective and orientation-selective characteristics can be ignored. They further assume that each size of receptive field exists in many locations uniformly spread across the retina, and they index location by the position of the central point of the receptive field, x,. This disregard for retinal inhomogeneity could be justified on the grounds that all the patterns they consider are distributed quite widely across the retina. ’ The words “channel” and “mechanism” have been avoided in this discussion and will continue to be. There is some inconsistency among authors and some difficulty in Stromeyer and Klein’s paper itself. It appears. for example. that they sometimes use the word mechanism to mean any single receptive field and sometimes to mean the receptive field of symmetry that produces the maximal response.
Working with these quantities S&f&, .u,) produces the same predictions as working with the sensitivities of all the neural units. It is these quantities that are plotted in the stimulation mnps shown in Stromeyer and Klein’s Figs. 5-9. For each of the three gratings {the sinusoidai, FM, and quasi-AM gratings described earlier) two stimulation maps are presented, one for each of two possible bandwidths of neural unit. One bandwidth is very narrow (approx : octave at half amplitude) and the other medium (approx 1 octave at half amplitude).” Theorerical
detection rules and spatial pooling
To complete the specification of a particular version of the multiple-channels model, a detection rule is necessary to specify how the responses of the neural units lead to the detection of a pattern. Here the question of spatial pooling is important. Prirr peak detection. Under a very simple detection rule called peak detection by Stromeyer and Klein, the sensitivity of the observer to the pattern is simply equal to the sensitivity of the most sensitive of all the neural units (the “peak”). This sensitivity can easily be found by picking the biggest number out of the stimulation maps. According to this rule, there is no pooling of information from different neural units. In particular. there is none from units at different spatial positions.
Letter to the Editors Probabiliry summation. Suppose. however. that the variability in the observer’s responses is due to variability in the neural units’ responses. Suppose also that the observer detects a pattern whenever the response of one or more neural units is larger than some criterion. As has been explained more fully elsewhere (Graham, 1976), probability summation will then occur across different neural units. The sensitivity of the observer will depend not just on the sensitivity of the one most sensitive unit, but on the sensitivity of many units. Whether this pooling will occur over all neural units or not depends on the assumption one makes about the correlation of the variability in different neural units. If the variability in all neural units is assumed to be independent, then the probability summation pooling will occur over all the neural units. This case will be called pure proba~ilir~ su~arion. If, however, the variability in all units of the same best frequency is perfectly correlated while the variability in units of different best frequency is independent, then pooling will not occur over all units. Instead, within each group of neural units having the same best frequency there will be peak detection, and the peak sensitivities from different best frequencies will form a pool over which probability summation occurs. Or, to say it in other words, there will be probability ~rn~t~on across difirent frequencies but not across different spatial locations or symmetries. (Sachs et (I/., 1971, worked with a model of this kind.) Other possibilities exist that do not involve probability summation across frequencies: probability summation across space but not across different best frequencies or symmetries, for example. However, since Sachs et al. (1971) provided good evidence that there is probabiiity summation across frequencies, these other possibilities seem less interesting. Linear pooling. Another kind of detection rule that Stromeyer and Klein consider is linear pooling. Here the sensitivities of neural units in the pool simply add. Only one neural unit for each spatial position is in the pool. the unit of whichever best frequency responds maximally (Klein, personal communication). The pooling formula. As Quick (1974) has pointed out, there is an easy method for calculating the results of probability summation, based on a particularly convenient analytic form for psychometric functions. Accordingly, the sensitivity of an observer when there is probability summation equals a nonlinear summation of the individual sensitivities in the probability summation pool. Let these individual sensitivities be called s(i), where i indexes individual members of the pool. Then the sensitivity of the observer will equal
li
)
where k. the exponent, is in the range from 4 to 8, and the summation is done over all members in the pool. The exact value of the exponent depends on the steepness of the psychometric function. Although we have introduced formula (1) in the context of probability summation, it can be viewed as a general pooling formula. The peak detection case can be shown to be equivai~t to this formula with an exponent equal to infinity.‘ An exponent in the range from 4 to 8 leads to probability summation
1023
predictions. An exponent of :! produces power summation. An exponent of 1 produces linear pooling. Snomerer and Klein’s pooling method. Stromeyer and Klein’s discussion (their Appendix 3) of the pooling formula is somewhat misleading. They pool over the stimulus envelope rather than over the sensitivities of neural units. (Compare the last formula of their Appendix B with formula (1) above. Alternately, notice that in their formula for Quick’s ~ychome~c function there is no sensitivity factor; this would be the correct formula if all neural units detected all patterns of one unit contrast with a probability of onehalf.) They may have done this because they were using the pooiing formula only for the broadest bandwidth neural units they considered (the medium bandwidth). These units are, to a first approximation, uniformly sensitive to all the Frequencies present in the patterns used. If the units actually were uniformly sensitive, Stromeyer and Klein’s pooling formula would reduce to the correct one. But, since the medium bandwidth neural units are not all quite uniformly sensitive, the ratios of “effective contrast” given in the table in Appendix B, are not quite correct even for the medium bandwidth case. For narrower bandwidths, the ratios analogous to those in the table certainlv must be computed from formula (1) and are quite different from those in the table. RESULTS
Stromeyer and Klein’s conclusions
Stromeyer and Klein arrive at several conclusions. (1) They reject pure peak detection (an exponent of infinity in the pooling formula) because @. 906) “the peak detection model predicts that medium band mechanisms respond too strongly to the quasi-AM grating and. . .narrowband mechanisms respond too weakly to the FM grating”. (2) They reject probability summation pooling because “probability summation does not sufficiently decrease the detectability of quasi-AM gratings, assuming medium-band mechanisms are used” and ‘-does not sufficiently increase the detectability of FM gratings using narrowband mechanisms for detection” (p. 907). (3) Assuming medium-band channels, “linear pooling appears to account for the facts in detectability of both FM and quasi-AM gratings, and narrowband visual mechanisms may be unnecessary” (p. 908). Taking into account the empirical and theoretical points described above. we arrive at rather different conclusions, even if we restrict ourselves to the same model versions discussed by Stromeyer and Klein. Theoretical predictions
Figure 1, which is a complicated figure that will be described in parts, shows the predictions of various versions of the model for the trio of patterns discussed by Stromeyer and Klein (the particular sine, quasi-AM and FM patterns described earlier). The predictions are plotted as sensitivity relative to the sine-i.e. as sensitivity of an observer to one pattern (either the quasi-AM or the FM) divided by the sensitivity of the observer to the sine. Since Stromeyer and Klein suppose the three chosen patterns are equally detectable to a human observer, they ask of a successful model that it predict relative sensitivities
I I)31
Letter to the Editors
equal to 1.0 for both the quasi-AM and the FM gratmg. As discussed earlier however, the data suggest that these three chosen patterns are probably not equally detectable. The relative sensitivity of the chosen quasi-AM grating was 1.25. 1.20 and 1.05 for Quick’s three observers. The variability in Stromeyer and Klein’s own data leads to a possible range of 0.8-1.11 for the relative sensitivity of the FM grating.3 These data are plotted on the far right-hand side of the ngure. Relative sensitivity- for the quasi-AM grating rs indicated by filled circles and relative sensitivity for the FM grating by open circles. The other four panels of the figure contain the predictions calculated from four different detection rules described earlier: pure peak detection (far left): probability summation across frequencies with peak detection across spatial Locations and symmetries (second to left); pure probability summation across al1 neural units (middle): and linear pooling (second to right). Using the pooling formula (I), the predictions in Fig. 1 were calculated” from the stimulation maps of Stromeyer and Klein. For each kind of detection rule, the predictions are shown for several bandwidths of channel (horizontal axis). The widths labelled N (n~row) and %I (medium) are those for which the stimulation maps were given in Stromeyer and Klein (1975). The intermediate bandwidth (labelled I) is slightly greater than the narrow one.’ The stimulation maps for this case
were also calculated by Stromeyer and Klein (personal communication). In the panels for pure probability summation and linear pooling there is one extra point for quasi-AM gratings which is plotted just to the right of the medium bandwidth points. These extra points are the predictions computed by Stromeyer and Klein (Appendix B) in the manner discussed in the earlier section “Stromeyer and Klein’s pooling method”. The points are plotted to the right because they are, in effect, predictions for a bandwidth even wider than the medium one. In both of the paneis involving probability summation, there is one extra point for FM gratings which is plotted to the left of the narrow bandwidth points. These extra points are predictions assuming that the bandwidth is so narrow that there is no neural unit which responds more to the FM grating than to one of its components alone. These extra points and the points for the peak detection case are the only predicted values expiicitly discussed by Stromeyer and Klein. Considering the full range of predictions and data leads to conclusions different from those reached by Stromeyer and Klein.
’ To obtain this range of relative sensitivities, it was necessarv to chanoe the ratios of cf’values into ratios of semitici&. (Sen&ity is the reciprocal of the contrast
!c~= infinity and L, = 5. For the pure probability summation case. f;, = i;: = 5. For linear pooling, the observer’s sensitivity equals Ix k
necessary to produce a criterion response.) To make this change requires knowing the transducer function. a function relating stimulus contrast to d’. Stromeyer and Klein 11974) and Nachmias and Sansbury (1974) have suggested that R’ is some low power {second or third) of contrast. If so. the ratios of sensitivities would be closer to one than the ratios of ri’ values. With a linear transducer function, on the other hand, the ratios of sensitivities would be exactly equal to the ratios of d’ values. Since the question of a transducer function is far from settled, it seems reasonable to use the larger confidence range based on the linear transducer function. If, however, a smaller confidence range were used. the regions of acceptable predictions would be diminished. If the transducer function were nonlinear enough, there might be no range of acceptable predictions from the first detection rule (pure peak detection). A Formally, the calculations were done as follows. The observer’s sensitivity to a particular pattern assuming a particular bandwidth was calcufated from the appropriatz
srrmulation map for that pattern and bandwidth. For the first three detection rules. the observer’s sensitivity equals
where
The exponent k, determines the pooling over spatial position and symmetry: the exponent k2 determines the pooling over frequency. For the pure peak detection case. li, = k, = infinity. For the probability summation across frequency with peak detection across space and symmetry.
Compnring predictions
to dutu
The shaded areas in the figure contain regions of acceptable predictions, the regions where both the predictions for the quasi-AM grating and the predictions for the FM grating are consistent with the data (far right panel).
.+ Lf
P
where i( equals infinity. The su~at~on was done over ail j, in the stimulation maps. There were !1 different& spaced uniformly on a linear frequency axis. The second summation was done over 12 x, spaced uniformly across one period of the reference sinusoid’s frequency. (The published stimulation maps for the quasi-AM and FM gratings show these I:! different positions. The published map for the sinusoid does not show all 12 because for a given & the response at all positions is constant.) Using only the sensitivities of units with optimal symmetry is presumably equivalent to assuming that the distribution of values of the symmetry parameter is uniform over its range of 0.0-1.0. Ideally, one should do the calculations weighting in the differential sensitivities at different frequencies, since the contrast sensitivity function is not perfectly flat (although it was assumed to be so in making up the stimulation maps). Also one should consider frequencies higher than those given in the stimulation maps. We could not do this, unfortunately. since the necessary information was not available. One might also want to consider other models in which, for example. the summation is done over frequencies spaced uniformly on a logarithmic frequency axis rather than a linear frequency axis. It is not clear how sensitive the predictions are to these details. There may be round-off errors in our calculations since the numbers given in the published stimulation maps are rounded off to the nearest integer. 5The parameter sigma in formula (6) of Stromeyer and Klein (1975) equalled. 0.20 for the intermediate case and 0.17 for the narrow case.
Letter to the Editors [Peak
drtsctlon
I
Prob.summotion (fr.qu*ncy only)
PUr* pIpI_ summation
_inaar pooling
-
N
I
M
N I
M Bandwidth
N I
M
N
I
M
FM OuosiAM Pattern
Fig. 1. The predictions of four spatial pooling models (the four left panels) for sensitivity to FM (open circles) and quasi-AM (filled circles) gratings relative to the sensitivity to the reference sinusoidal grating. These predictions are shown for several bandwidths (N = narrow. I = intermediate. M = medium). The measured psychophysical relative sensitivity is shown in the right panel. The data for FM gratings are from Stromeyer and Klein (1975); the data for quasi-AM gratings are from Quick and Reichert (1975). The shaded areas indicate the bandwidths for which both the predicted quasi-AM and the predicted FM sensitivities are consistent with the data.
Consider the peak detection case first (far left). Although for both the narrow and medium bandwidths, the predictions are inconsistent with the data, the full graph suggests that some bandwidth between these extremes might produce predictions consistent with the data. The acceptable range of bandwidths is very small, however, and leads to predictions in the extreme ends of the data ranges (high for quasi-AM, low for FM). Thus one might accept, with reservations, Stromeyer and Klein’s first conclusion. Once probability summation is introduced, either just across frequencies (second to left) or over all neural units (middle), somewhat larger ranges of bandwidth produce predictions consistent with the data. With probability summation across frequencies only, a range of narrow bandwidths produces acceptable predictions. Here again, the predictions are in the extremes of the data ranges. With probability summation across all units, however, a large range of medium bandwidths produces predictions consistent with the data (and not just with the extremes of the data ranges). It seems reasonable to conclude, therefore, in direct contradiction to Stromeyer and Klein’s second conclusion, that a model postulating probability summation, particularly probability summation across all of a collection of medium bandwidth neural units, is consistent with the data. Finally, in contradiction to Stromeyer and Klein’s third conclusion, there is no range of bandwidths for the linear pooling case which leads to acceptable predictions for both the quasi-AM and FM gratings (second to right). Too little sensitivity to the FM grating is always predicted. All the above conclusions, unfortunately, rest on the assumption that Quick’s subjects and Stromeyer and Klein’s subjects do not differ in some systematic way. Other models exist, of course, that Stromeyer and Klein did not consider. In particular, there are models which postulate only one symmetry of receptive field, CONCLUSlONS
Although the results of Stromeyer and Klein’s study are somewhat disappointing in that they do not
allow very strong conclusions, they are suggestive. As is shown in the predictions plotted in Fig. 1, which were based on the stimulation maps calculated by Stromeyer and Klein, the comparison of data obtained with FM gratings, quasi-AM gratings and sine-wave gratings does allow some discrimmation among models having different bandwidth and spatial pooling properties. More extensive data might be able to make better use of this discrimination. Not all possible models, or even all possible versions of models having multiple kinds of receptive fields, have been considered here. But in a theoretical framework with many unknowns, one cannot hope to decide all questions at once. Of the models Stromeyer and Klein consider, a model in which there are medium bandwidths and probability summation across all neural units seems to provide the best fit to the data. Certainly the study cannot be taken, as they did, as evidence for a linear pooling model and against a probability summation model. Two other recent studies also tend to support a model in which there are medium bandwidths and a spatial pooling mechanism like probability summation across space (King-Smith and Kulikowski, 1975; Mostafavi and Sakrison, 1975). It does not yet seem possible, however, to conclusively rule out other models, including ones with narrower bandwidths and no spatial pooling. Dept. of Psychology, Columbia Uniwrsiry, New York, NY 10027, U.S.A.
NORXL~GFUHLCI BERNICEE. ROCXWITZ
REFERENCES Campbell F. W. and Robson J. G. (1968) Application of Fourier analysis to the visibility of gratings. J. Physiol., Lond. 197, 551466. Graham N. (1976) Spatial frequency channels in human vision: detecting edges without edge detectors. In Visual Coding and Adaptability (Edited by Harris C.). Erlbaum. &anger E. (1973) An alternative model for grating detection. Paper presented at the meetings of the Association for Research in Vision and Ophthalmology. Sarasota, Fla., May, 1973.
IO26
Letter
to the Editors
King-Smith P. E. and Kulikowski J. J. (1974) The detection of gratings by independent activation of line detectors. J. Phpiol., Lond. 247, 237-271. Kulikowski J. I. and King-Smith P. E. (1973) Spatial arrangement of line, edge, and grating detectors revealed by subthreshold summation. Vision Rex 13. 1455-1478. Lange R., Sigel C. and Stecher S. (1973) Adapted and unadapted spatial-frequency channels in human vision. Vision Res. 13, 2139-2143. ‘Mostafavi H. and Sakrison D. (1975) On the behavior and modeling of a single channel in the visual system. Paper presented at the meetings of the Association for Research in Vision and Ophthalmology. Sarasota, Ha., May. 1975. Nachmias J. and Sansbury R. B. (1974) Grating contrast: discrimination may be better than detection. Vision Res. 14, 1039-1042. Quick R. F. (1973) Spatial-frequency selectivity in human vision: the frequency response of contrast detection
channels. Ph.D. thesis. Carnegie-Mellon University. Pittsburgh Pa. Quick R. F. (197-t) A vector-magnitude model of contrast detection. Kybrmetik 16. 64-67. Quick R. F. and Reichert T. A. (1974) Spatial-frequency selectivity in contrast detection. Vision Rrs. 15, 637-6-13. Sachs M. B., Nachmias J. and Robson J. G. (1971) Spatial frequency channels in human vision. J. opt. Sot. Am. 61, 1176-l 186. Stromeyer C. F. and Klein S. (1974) Spatial frequency channels in human vision as asymmetric (edge] mechanisms. Vision Rcs. 14. 1409-1420. Stromeyer C. E. 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. Thomas J. P. (1970) Model of the function of receptive fields in human vision. Psphol. Rec. 77. 121-134.