VisionRes.Vol. 30, No. 3, pp. 449-461. 1990 Printed in Great Britain. All rightsresewed

0042~6989/9053.00 + 0.00 Copyright 0 1990 PergamonPressplc

CONTRAST DISCRIMINATION CANNOT EXPLAIN SPATIAL FREQUENCY, ORIENTATION OR TEMPORAL FREQUENCY DISCRIMINATION F. Bowux

SAMUEL

Smith-Kettlewell

Eye Research Institute, 2232 Webster Street, San Francisco, CA 94115, U.S.A.

(Received 17 April 1989; revised 13 July 1989) --Current models of spatial frequency (SF) and orientation discrimination arc based on contrast discrimination data. In these “error propagation” models, the precision of all discrimination tasks is limited by “peripheral” noise in contrast-sensitive channels. Therefore, all discrimination thresholds should be proportional to the contrast Weber fraction AC/C.To test this prediction, increment thresholds were measured for contrast, SF, orientation and temporal frequency (TF) for contrasts ranging from 2 to 50%. All measurements used the same stimuli, procedures and observers. For contrasts of 2% and higher, the contrast discrimination threshold AC risea with approximately the 0.6 power of contrast, while SF and TF discrimination arc independent of contrast. Furthermore, orientation discrimination is nearly independent of contrast at a SF of 4 cpd. No error-propagation model can explain these results. Therefore, SF and TF discrimination, and orientation discrimination at 4cpd arc limited by contrast-independent central noise. Discrimination

Contrast

Transducer

Masking

INTRODUCI’ION

Consider a psychophysical contrast discrimination task. The observer is shown two patterns with different contrast levels, and must choose that pattern with higher contrast. Observers are not perfect at this task, but make. errors, especially if the contrast difference is small. The observer’s performance is usually analyzed using signal detection theory (Green & Swets, 1966), assuming that the observer’s responses are based on a “decision variable”, which is a monotonically increasing function of contrast. The decision variable is subject to additive noise, which causes the observer to make errors. Contrast discrimination is then interpreted as a measure of the signal-to-noise ratio of the decision variable. What relationship might be expected between contrast discrimination and other discrimination tasks, such as spatial frquency (SF) discrimination? In the simplest model, all discrimination tasks are based on the same decision variable. In this model, a single measurement such as contrast discrimination would be sufficient to determine the decision variable’s signal-to-noise ratio, and thus predict performance at other discrimination tasks. Modem models of discrimination (Wilson, 1986; Watt & Morgan, 1984; Klein & Levi, 1985; Regan & Beverley, 1985) include a num449

ber of mechanisms, usually tuned to a particular region of location, SF and orientation. Each of these mechanisms are subject to “peripheral” noise, and the decision variable for a discrimination task is constructed from one or more of their outputs. The process of combining mechanism outputs to create the decision variable is assumed to be noise-free, as well as all later processes. The actual situation in human vision is probably more like that shown in Fig. 1. The stimulus is subjected to two sequential processes, each of which adds noise. The first process, “filtering”, is shared by all discrimination tasks. Many filters respond in parallel to the stimulus, and the filter outputs are selective for spatial position, SF, orientation, temporal frequency (TF), and other stimulus parameters. Each filter’s mean output is a monotonically increasing function of contrast, and the variance of the output is caused by additive peripheral noise. The second process, combination of channels, uses the outputs of one or more filters to calculate an estimate of a stimulus parameter such as contrast, SF, orientation, or TF. Separate ‘central noise” sources perturb the output of each combination process. Note that the terms “central noise” and “peripheral noise” refer to sources of random variability,

SAMLEL F. BOWNE

450

not other types of inefficiency such as nonoptimal combination rules. There is evidence to support the inclusion of central noise sources in psychophysical models. Barlow (1977) has proposed that detection of dim flashes and dot density discrimination are limited by central noise, and Burgess and Barlow (1983) have shown that the effect of central noise on dot density discrimination can be explained using a model with additive Gaussian noise. The observer’s decision variable for contrast discrimination is the output of the second stage, and performance is limited by two sources of noise: peripheral noise added at the filtering stage, and central noise added after the combination process. Other discrimination tasks, such as SF discrimination, share the same filters and thus the same peripheral noise but may have a different amount of central noise. If the central noise limits performance, contrast discrimination measurements might have no relevance to otb.er discrimination tasks. Many models of discrimination (Wilson, 1986; Klein & Levi, 1985; Regan & Beverley, 1985) have assumed that all suprathreshold discrimination tasks share a common noise source, neglecting the central noise shown in Fig. 1. In these models, the observer’s behavior is limited by the noise added at the filtering stage, propagated through the decision rule. I will call this class of models “error propagation models”. Such models are attractive because one type of discrimination (such as contrast discrimination) may be used to predict another (such as SF discrimination). In fact, all suprathreshold discriminations should have the same contrast dependence. The purpose of this study was to test that prediction by measuring the contrast dependence of several discrimination tasks.

METHODS

Stimuli were generated on a Tektronix 608 CRT with P31 phosphor using an In&free Picasso pattern generator. The refresh rate was 100 Hz and the mean luminance of the screen was 18 cd/m’. Luminances and contrasts were calibrated using a Pritchard photometer. The Picasso permitted effectively continuous variation of contrast, SF and TF, and the orientation could be rotated through integral multipks of 0.35”. All stimuli were presented for Momsec with abrupt onset and offset, in a circular region 4.5” in dia with a dark surround. Some 4cpd

The Central Noise Model

Fig. 1. A realistic model of discrimination, including both peripheral and central noise sources. The stimulus is first passed through a number of filters with outputs which depend on the SF, TF, orientation and contrast of the stimulus. One filter’s center-surround spatial sensitivity profile is shown. “Peripheral” noise is added to each filter’s output. The outputs of these filters enter combination proceslce, which extract estimates of stimulus parameters such as SF, TF. orientation and contrast. Central noise is added to each of these estimates to form the decision variables for discrimination tasks. Because the central noise may be different for each task, measurement of contrast dia&nination does not provide enough information to predict other discrimination thresholds. The box at the bottom labelled “homunculus” represents all higher mental process which use the decision variable to select a response. Any noise added by the homunculus is assumed to be negligible compared to the explicit noise sources shown.

stimuli were presented in a rectangular region 6 x S”, ad since control experiments showed no difference between the rectangular and circular field results, the thresholds were averaged together. All reference patterns had sine-wave luminance profiks with a SF of 1 or 4 cpd and vertical orientation. SF and orientation discrimination tasks used static targets, while TF discrimination used a target drifting rightward at a TF of 5 Hz (speed = 5 deg/sec). A small dark fixation dot was continuously present at the center of the display. The two observers viewed the displays binocularly with natural pupils at a distance of 114 cm in a dimly lit room. Observer SFB is an astigmatic myope with spectacle correction, and observer SPM is an uncorrected hyperope with 20/20 acuity at the viewing distance used in this study. Detection and discrimination thresholds were determined using two temporal intervals and a Quest 2AFC staircase procedure (Watson & PeIIi, 1983). One randomly chosen temporal interval contained a reference sinewave, and the other interval contained a sinewave with a small

Contrast independent feature discrimination

increment in contrast, SF or TF, or a small clockwise rotation. A blank screen at the mean luminance was shown during the 500 msec interstimulus interval. After each pair of presentations, the observer pressed one of two buttons, attempting to select that interval containing the increment. The observer was given audible feedback after each incorrect response. The staircase estimated the 92% correct point on the psychometric function (d’ = 2.0). The 92% correct criterion was used because it is the most efficient estimator of thresholds, in the sense that the smallest number of trials are required to reach a given precision (Watson & Pelli, 1983). The spatial phase of the target was randomized between presentations, to prevent local cues at the edges of the screen, or near the fixation dot, from providing useful information about SF or orientation. Each staircase lasted for 36 trials, and was repeated at least 3 times, so the resulting mean discrimination thresholds were based on at least 108 trials. Contrast discrimination, SF discrimination, and the other tasks performed required the observer to concentrate on a particular aspect of the stimulus. Interleaving trials with different tasks within the same staircase would have been confusing, so instead each threshold was measured separately. For example, the observer might first measure a contrast increment threshold with a 36trial staircase, and then measure a SF increment threshold with a sep arate 36trial staircase. The observer always knew which type of increment to look for, and each threshold was measured at least three times in a random order. Note that previous measurements have frequently used a lower criterion for threshold than the d’ = 2 level used here, which must be taken into account when comparing these data with literature values. The procedures used to compare such data are explained in Green and Swets (1966), and are briefly summarized below. Assuming that detectability d’ is proportional to the contrast, SF, orientation or TF increment for the suprathreshold discrimination tasks reported here, thresholds obtained by other techniques must be multiplied by a constant factor to obtain comparable d’ = 2 estimates. For example, McKee, Silverman and Nakayama (1985) reported thresholds defined as the distance between 50 and 75% “yes” responses in a yes/no task, which corresponds to a d’ = 0.67, so their thresholds should be multiplied by

451

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Contmst (XI Fig. 2. Contrast detection and discrimination thresholds arc shown as a function of pedestal contrast, using 1 cpd gratings presented for SM mace. The dritIing grating moved rightward at a TF of 5 Hz (speed = 5 deg/scc). Roth curves have the chamctcrktic “dipper” shape mported by Lcggc and Foley (1980). Solid lines show kast-squares fits of the data in the region 2-W% contrast to a power law AC a cm. The value of m is shown, with the standard error in pamnthcses. Error bars repmsmt I SEM.

2/0.67 = 3 for comparison with thresholds reported in this paper. Legge (1981) used a 3down, l-up staircase with two intervals which converges to the 79% correct point on the psychometric function, giving a d’ of 1.14. Legge’s suprathreshold contrast increment thresholds should therefore be multiplied by 2/1.14 = 1.75 to obtain d’ = 2 estimates. As noted in the Results section, all the discrimination thresholds reported in this paper are in good agreement with comparable literature values. The assumption that d’ is linearly proportional to stimulus increment has been tested by Got&d-Smith and Thomas (1989), who found deviations too small to affect the results presented here significantly. Their results are discussed in greater detail below. RESULTS

Figure 2 shows the contrast increment thresholds for static and drifting 1 cpd gratings. The threshold contrast increment is shown as a function of the reference or ‘pedestal” contrast. These results agree qualitatively with previous measurements by Legge and Foley (1980),

452

SAMUEL

showing the characteristic “dipper” shape. The high-contrast data (2-50%) have been fitted by a power law, shown as a solid line. For a static grating, the slope of the fitted line is 0.52 for observer SFB and 0.64 for observer SPM, SO the mean slope across subjects is 0.58. Both the slope and the numerical values of the contrast increment thresholds agree with those reported by Legge (1981), who used a 200 msec presentation of a static 2 cpd grating and obtained a slope of 0.59. In making this comparison, the discrimination thresholds reported by Legge were multiplied by 1.75 to obtain d' = 2 thresholds, as discussed in the Methods section. This correction does not affect the slope, but only the absolute contrast increments. Comparable slopes from contrast masking experiments include 0.55 for static sixth-derivative of Gaussian (D6) patterns (Wilson, McFarlane & Phillips, 1983), 0.56 for counterphase flickering D6 patterns (Lehky, 1985), and 0.78 for 1 cpd 8 Hz drifting sinewave gratings (Anderson & Burr, 1989). Another way to plot contrast discrimination thresholds is as a contrast Weber fraction AC/C, as shown in Fig. 3 for a 4 cpd target. Because contrast increment thresholds grow more slowly than Weber’s law, i.e. with log-log slope < 1, the contrast Weber fraction decreases with contrast. Error propagation models predict that all discrimination tasks should improve in direct proportion to the contrast Weber fraction, as will be shown in the Discussion. The SF increment thresholds shown in Fig. 3 do not improve with contrast, but are nearly independent of contrast. The discrepancy between contrast discrimination and SF discrimination is large; raising the contrast from 4 to 50% decreases the contrast Weber fraction by a factor of 3 or more, but has almost no effect on SF discrimination. Similar findings have been reported by Regan, Bartol, Murray and Beverley (1982) and Thomas (1983), who found contrast-independent SF discrimination above 3-5% contrast using 4 cpd gratings, with ASF/SF near 0.08 (scaled to d' = 2). Similar SF discrimination thresholds for 4 cpd gratings have been reported by Hirsch and Hylton (1982), Bradley and Skottun (1984), Richter and Yager (1984) and Mayer and Kim (1986), and for 5 cpd gratings by Burbeck and Regan (1983). The orientation discrimination thresholds shown in Fig. 3 are nearly independent of contrast, in qualitative agreement with Regan and

F. BOWNE

Contrast dircrlmlnotionfor 4 1.00

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Fig. 3. Contrast dcpcndcnce of discrimination tWs ace shown for 3 tasks and 2 subjects, using a 4 cpd target. Mid lines show the best-fitting power laws, and the exponent m is indicated with its standard error in jWanthesas. C0nttest discrkdnation greatly improvesat hi* ContcWs (exponent mar -0.5). but SF and orientation discrknination do not (exponent near 0).

Beverley (1985) who found contrast-independent orientation discrimination thresholds using a 12 cpd grating, for contrasts from 20 to 60%. eat contrasts of 32 and 50%, the discrir&u&.m thresholds are near 0.9” for both observers. Comparable literature values are 0.7” for a spatial frequency of Scpd (Burbeck 8t Regan, 1983), and 1-1.4” for a spatial frequency of 4cpd (Bradley & Skottun, 1984), scaled to d’ = 2. Figure 4 shows contrast, SF and orientation discrimination thresholds for a spatial frequency of 1 cpd. The contrast discrimination data are. taken from the upper two grapIy in Fig. 2. The contrast discrimination thnshokis are similar to those found at 4 cpd, except for a minor slope change in subject SPM. Similar slope changes with SF have been o&served before by Lehky (1985) in one of his two subjects. The SF discrimination thre&oIds are similar to those found at 4cpd, but the orientation discrimination thresholds are more

Contrast independent feature discrimination

contrast-dependent. The mean ASF/SF was about 0.07 for both subjects, in good agreement with the previous measurements at 1 cpd of 0.10-o. 13 (Regan et al., 1982),0.08-O.11 (Hirsch & Hylton, 1982), and 0.07-O.10 (Richter & Yager, 1984), all scaled to d’ = 2. The orientation discrimination thresholds at 32 and 50% contrast are near 1” for both observers, in agreement with literature values using 1 cpd (Bradley & Skottun, 1984) and 2cpd (Burbeck & Regan, 1983) gratings, scaled to d’ = 2. Figure 5 shows contrast discrimination and TF discrimination thresholds for a 1 cpd grating drifting at 5 Hz. The results are virtually identical to those found for SF discrimination; contrast discrimination improves at higher contrasts, while TF discrimination does not. The contrast independence of speed discrimination has been reported previously by McKee et al. (1986). The mean ATF/TF discrimination thresholds in Fig. 4 are 0.07 for SFB and 0.10

Contraat

discrimination for 1 cpd static grating

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Contrast (lb) Fig. 5. Contrast discrimination and TF discrimination thresholds are shown as a function of pedestal contrast, using 1 cpd gratings drifting with a base TF of 5 Hz. Contrast discrimination improves at high contrasts, while TF discrimination daes not.

for SPM. Literature values include 0.15-0.20 for a 1 cpd grating drifting at 5 Hz (McKee et al., 1986) and 0.045-0.060 for a 0.6cpd grating drifting at 5 Hz (Pantk, 1978). These literature values have been multiplied by 3 to obtain d’ = 2 estimates from the reported d’ = 0.67 estimates (see Methods).

frrquoncy dircrimlnatlon DI!XUSSION

Orientation discrimination 10.0

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cantmrt(Xl Fig. 4. Contrast dependence of discrimination thresholds for a I cpd grating. The contrast discrimination data are replotted from Fig. 2. As in Fig. 3. contrast discrimination is much more contrastdependent than SF discrimination. However, a comparison of Figs 3 and 4 shows that orientation discrimination is more contrastdependent at the lower spatial frequency.

A schematic diagram of the error-propagation model of discrimination is shown in Fig. 6. The stimulus could be any pattern, but this discussion is restricted to sine wave patterns for clarity. The stimulus is filtered by a number of channels which give a response which depends on SF, TF, orientation and contrast. The response of a channel is usually bandpass or lowpass in SF, TF, and orientation. However, all the responses are monotonically increasing functions of contrast and approach a power law dependence at high contrasts (Legge & Foley, 1980; Legge, 1981). Each channel has a certain amount of added peripheral noise, which is assumed to be zero-mean unit-variance Gaussian noise. The outputs of one or more of these channels are combined in some noise-free manner to produce an estimate of a stimulus parameter such as contrast, SF, orientation of TF. The accuracy of discrimination is then determined by the propagation of noise from

SAMUEL F. RO~NE

454

the filters through the combination rule, and the same peripheral noise limits contrast discrimination and all other discrimination tasks. Because all discrimination tasks are limited by the same peripheral noise in the error propagation model, all tasks have similar contrast dependence. The amount of propagated error is particularly easy to calculate at high contrast, where contrast discrimination thresholds obey a power law ACCCC~ where the exponent

(1)

f is typically between 0.5

and 0.7. Equation (1) shows that the absolute contrast increment threshold rises at high contrast, but the increment threshold relative to the base contrast (the “contrast Weber fraction”) is a decreasing function of contrast: AC/Ccc c -(I -‘).

(2)

This contrast dependence can be explained by a compressive nonlinearity in the filter outputs, as shown by Legge and Foley (1981), rilX Cl-J

The Error-Propagation

Model

T FlIta,‘

SL

Fig. 6. A schematic diagram of the error-propagation class of models, which includes the models of WiIson (1986). Klein and Levi (1985) and Regan and Revedey (1985). A numbar of filters respond to the stimulus, of which one is shown. The tilters are bandpass or lowpass in SF, TF, and orientation, and their responses are monotonically increasing functions of contrast. Gaussian noise is added to tbe output of each filter. A noiseless combination process is used to compute estimates of SF, TF, orientation and contrast from the filter responses. Since all discrimination tasks are limited by the noise added to the filter outputs, they will all have the same contrast dependena, as shown in the text.

(3)

model uses a more complex transducer than that given in equation (3), but it is instructive to first where ri is the response of channel i. In order to calculate the dependence of consider the high-contrast limit in which all other discrimination tasks on contrast, we need mechaaisms obey equation (3). If both patterns to know the combination rule. The next four have the same contrast c, this relation may be sections present the calculation of discrimi- si@ified using equation (3), nation thresholds for the Quick (1974) rule, the 'IQ best single unit rule (Klein & Levi, 1985),a ratio c’-J (5) D = z[r,(p,, I)-ri(pz, l)lQ of responses rule (Regan & Beverley, 1985),and the multiple-channel model of Wilson (1986). where r,(p,, 1) is the response of channel i to pattern pI at unit contrast. Discrimination The Quick rule improves as contrast increases, because D is The Quick rule was originally proposed as a multiplied by the increasing function c’ -J. model for contrast detection by Quick (1974) Consider SF discrimination using the Quick and subsequently applied to SF discrimination rule. If the filters have a continuous response by Wilson and Gelb (1984) and orientation profile as a function of SF, such as those discrimination by Phillips and Wilson (1984). In proposed by Wilson and Gelb (19&4x each their formulation, the probability of correctly channel’s response depends linearly on SF for discriminating between two patterns (No. 1 small SF increments. and 2) is a monotonically incma&g function of r&l, 1) - r&z, 1) = AiASF a measure of the difference between the two (6) patterns, denoted by D. The larger D is the more where ASF = SF, - SF, and Ai is independent easily the two patterns are d&iminated. Two of contrast. The SF dif%rence threshold is that patterns are said to be separated by a threshold value of ASF which makes D = 1, and may be amount when D = 1. D is calculated as follows, determined by substituting equation (6) into equation (5),

(i

&jF

where r,(p, , cl) is the response of channel i to pattern p, at contrast cl, and t,(p2, q) is its response to pattern p2 at contrast c, . The Wilson

= &

-(I -J)

(7) -I/Q

Contrast independent feature discrimination

where B is independent of contrast. The spatial frequency discrimination threshold falls in the same manner as the contrast Weber fraction shown in equation (2). In particular, for f = 0.6, for both contrast discrimination and SF discrimination should fall with log-log slope -0.4. This argument also applies to other discrimination tasks such as orientation and TF discrimination. Klein -Levi model

Klein and Levi (1985) proposed a very simple combination rule; each discrimination task is performed using the output of the single most useful unit as the decision variable. For example, the most useful unit for contrast discrimination might be the unit most sensitive to the stimulus, i.e. the unit with peak SF and orientation equal to the SF and orientation of the stimulus. The most useful unit for SF discrimination would be a different unit, with peak sensitivity roughly 1 octave lower than the stimulus SF, because the output of that unit depends strongly on the stimulus SF. One virtue of this model is that performance is independent of the exact number of filters available, as long as these filters are numerous enough so that a near-optimal filter is available for each task. The Klein-Levi model can be regarded as a special case of the Quick rule combination, for large values of the exponent Q. When Q is large, the main contribution in equation (4) comes from the single unit most sensitive to the difference between patterns 1 and 2, and other contributions become negligible. Therefore, if all units are operating in the high-contrast region, and therefore obey equation (3), the Klein-Levi model has the same prediction as the general Quick rule discussed above. All discrimination tasks improve as contrast increases, with the same exponent. In their 1985 paper, Klein and Levi used a logarithmic transducer which does not obey equation (3), but leads to contrast-independent *The assumption of uncomlated errors is not naxssary; even if the two responses r, and r, are partially comlated. equation (I I) still holds (Meyer, 1975). However, a small-errors assumption is needed; equation (9) is exact only if the variations in r, are much smaller than the mean value r,. This condition was satisfied for all the discrimination thresholds shown in Figs 3-5. since the thresholds ate smaller than the widths of SF, TF or orientation channels as measured by masking or sub threshold summation. In any case, all ratio models must make small-error assumptions to avoid dividing by zero.

455

Weber fractions Ae/c at high contrast. Our contrast discrimination data shown in Figs 2-4, and similar studies in the literature (Legge and Foley, 1980; Legge, 1981; Anderson Bt Burr, 1989) obey equation (3) at high contrast, and cannot be explained by the logarithmic transducer of Klein and Levi. Ratio of responses

As Regan and Beverley (1985) have pointed out, a contrast independent estimate of a parameter such as orientation may be constructed by taking the ratio of the responses of two channels. A measure of the orientation 8 of a pattern is given by the ratio R of two orientation-selective responses rI and r,. R(B) = r,@, CM&

~1.

(8)

Substituting equation (3) into equation (8), we see that the mean value of R is indeed independent of contrast. This contrast independence makes R very appealing as a decision variable for discrimination. If discriminations were mediated by mechanisms which calculate such ratios, discriminations of orientation would not be perturbed by random variations in contrast. Previous studies have shown that contrast variations do not impair judgements of orientation (Regan & Beverley, 1985) or speed (McKee et al., 1986). In a similar manner, taking the ratio of two channels with different orientation tunings but the same SF tuning would render orientation judgements independent of random SF variations. Many studies have shown such independena of different stimulus parameters: TF discrimination is unaffected by variations of orientation (Welch, 1989), speed discrimination is unaffected by SF variations (McKee et al., 1986), orientation discrimination is unaffected by SF variations (Bradley & Skottun, 1984; Burbeck & Regan, 1983) and vice versa (Burbeck & Regan, 1983). However, discrimination thresholds are still dependent on contrast in the ratio model. The discussion on the previous paragraph considered only the mean value of R, but discrimination thresholds are limited by the variance of R. Although the mean value of R is contrastindependent, its variance is not, because the underlying channel responses rl and r2 have higher signal-to-noise ratios at higher contrast. How does the noise in rl and r2 affect R? If the two channels are statistically independent, the standard formula for propagation of errors in a ratio* is applicable (Meyer, 1975).

456

%MlJEL

AR/R = ((Ar,/r,)* + (Ar,/r,)*)“*.

(9)

Here AR is the standard deviation of R and Ari is the standard deviation of ri, which is the standard deviation of the added noise in channel i. Since each channel has independent noise with standard deviation 1, Ar, = Ar, = 1. AR/R = [(l/r, )* + ( 1/r2)2]“2.

(10)

The contrast dependence of this quantity may be found by substituting from equation (3). AR = EC -(I -J)

(11)

E is a contrast-independent constant. The orientation discrimination threshold is proportional to AR, so the final contrast dependence of discrimination in the ratio model is the same as that for the Quick model; orientation discrimination should improve with increasing contrast. Again, this argument is general and applies to SF and TF discrimination as well. The Wilson model Wilson and his collaborators have presented an error-propagation model which has been applied to SF discrimination (Wilson Bi Gelb, 1984), orientation discrimination (Phillips & Wilson, 1984) and hyperacuity (WiIson, l!M). This model contains six spatial sensitivity profiles (SF channels), each of which is a di&rence of two or three Gaussians. The peak sensitivities of these filters range from 0.8 to 16cpd. Seven filters with each spatial sensitivity profile are included in the model, differing from one another by either translation or rotation in the image plane. Therefore, the total number of mechanisms contributing to psychophysical performance is 6 SF channels x 7 mechanisms per channel = 42. Here is a list of the 7 mechanisms within each SF channel. First, a verticallyoriented mechanism is centered on the contrast peak of the stimulus cosine wave. The second and third mechanisms are similar to the first, but displaced slightly leftward and rightward. The fourth through seventh mechanisms are related to the first mechanism by a rotation about its center through f one angle step or f

*Wilson and Gelb (1984) and Wilson (1986) used an iterative procedure to find the stimuIur inuement required to make D = I in equation (5) (they USCthe symbol AF instead of D). which correqxxuis to the 75% correct point on the psychometric function (d’ = 0.95). To obtain 92% correct (S-2) eatimatea, the same iterative procedure was used to converge on the value D = 2.10.

F.

BOWNE

two angle steps. The angle steps range from 30‘ for the lowest two mechanisms to 15” for the two highest mechanisms. Each of these mechanisms consist of a linear filter followed by a nonlinear transducer to produce an output ri given by this equation, ri = [(Sic)’ + K,(S,C)~-~~]/[K, + (Sic)?]

(12)

where Si is the contrast sensitivity of channel i and Ki and ei are contrast-independent constants. At very high contrasts, the first terms in the numerator and denominator become negligible, and the response becomes a power law, obeying equation (3) with f = ei. However, the (SC)~ term in the numerator, which is not present in the transducer used by Legge and Foley (1980), causes the response to differ significantly from the power-law limit even for contrasts as high as 30 times detection threshold. The Wilson model is therefore more complicated than the simple Quick rule model discussed above, in which all filters obeyed equation (3). Discrimination thresholds in the Wilson model depend on the pooled responses of many filters, some which are operating near their detection thresholds, and others which are far above threshold. In addition, the exponent ei is different for different SF channels. However, the final result is qualitatively unchanged by these complications, as shown in Fig. 7. The predictions of the Wilson model with pooling exponent Q = 2 are shown as solid lines, along the corresponding data from Figs 3 and 4. The contrast discrimination prediction is close to the observed thresholds, which is reasonable considering that the parameters of the Wilson model were primarily based on masked contrast detection data (Wilson et al., 1983). However, the orientation and SF discrimination predictions fall far from the data, and are too contrast-dependent. Nearly all the model’s parameters were taken from Wilson and Gelb (1984) and Wilson (1986), and the 92% correct point on the psychometric function was calculated as described in these papers .* The only parameters which differed from the previously published values were the peak sensitivities of the lowest two mechanisms. The peak contrast sensitivities of the mechanisms peaking at 0.8 and 1.7 cpd were raised from 30 and 70 (Wilson, 1986) to 110 and 120. The increased sensitivity to lower SFs is required to model the contrast sensitivity function for abrupt-onset 500 msec stimuli

Contrast independent feature discrimination

(Wilson, personal communication). Calculations with pooling exponents Q of 4 and 6 resulted in higher thresholds for all tasks, with similar contrast dependencies. It should be mentioned that a different spatial task, vernier offset detection, has contrastdependent thresholds more closely in accord with the Wilson model (Wilson, 1986;Bradley & Freeman, 1985; Krauskopf & Farell, 1990).

457 Wilson modal prdietiona lcpd Contmrt

Saturating transducers

4cpd discrimination

Spatial troquoney dhcrimimtlon

relationship between contrast discrimination and other discrimination tasks has been analyzed previously by Thomas (1983) for SF discrimination, Nakayama and Silverman (1985) for direction-of-motion discrimination, and Go&d-Smith and Thomas (1989) for SF and orientation discriminations. These authors explain the contrast independence of discrimination by using a saturating transducers* of this form The

r = c”/(F + c”)

0.01

I

1111111111 Oriantotion discrimination

(13)

instead of equation (3), where n is between 2 and 2.9, and F is a contrast-independent constant. While this function has been used to describe the firing rates of cortical cells in physiological measurements (Albrecht & Hamilton, 1982), it does not agree with the power-law dependence of contrast discrimination in psychophysical experiments (Legge & Foley, 1980, Legge, 1981; Anderson & Burr, 1989). At high contrast, the response predicted by equation (13) is independent of contrast, leading to a sharp rise in contrast discrimination thresholds. The psychophysical transducer may differ from the physiological transducer because of combination of information from many neurons, or because of gain control mechanisms (Bonds, 1988). Whatever their neural origin, our contrast discrimination measurements and those of Legge and Foley (1980) and Legge (1981) are explained by the transducer in equation (3), and cannot be explained by the transducer in equation (13). *Thomas (1983) and Goukd-Smith and Thomas (1989) considered two models: a constant-noise model in which the noise was contrast-independent and an incrasingnoise model. The principal difference between the two models is in the shape of the psychometric function. Throughout this paper, the curvature of d’ is neglected, so that the signal-to-noise ratio r determines performana, and the two models have the same predictions. This assumption is justified in the section entitled “the shape of the psychometric function”.

Contmd (XI

Fig. 7. Experimental discrimination thresholds from Figs 3 and 4 are compared with the predictions of the Wilson (1986) model. The contrast dkimination prediiions are best, but the predicted SF thresholds an too contrastdependent, and the prediction orientation thresholds arc too high.

The shape of the psychometric function The use of staircase procedures to determine increment thresholds involves an implicit assumption that the shape of the psychometric function is the same for all suprathreshold contrast levels and all tasks studied. A common assumption is that the signal-to-noise ratio of the decision variable (d’) is proportional to the increment being detected. With this linearity assumption, the psychometric function is a cumulative normal curve for yes/no methods, or a Weibull curve with slope parameter /I = 1.25 with 2AFC methods, as shown by Pelli (1987). However, neither models using a nonlinear contrast transducer and contrast-independent noise (Legge & Foley, 1980; Wilson, 1986) nor models using contrast-dependent noise (Thomas, 1983)predict perfect linearity between d’ and stimulus increment. In principle, if the shape of the psychometric function changed greatly with task or contrast, the contrast

SAMUELF.BOWNE

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dependence of different discrimination tasks might differ even in the absence of central noise. The shape of the psychometric function has been studied by Gouled-Smith and Thomas (1989), who compared contrast discrimination, SF discrimination and orientation discrimination as a function of contrast. They considered two models: the increasing-noise model and the constant-noise model. The primary difference between the models lies in the shape of the psychometric function, and amounts to a 25% difference in d’ ratio between easy and hard contrast discriminations (Gouled-Smith & Thomas, 1989, Fig. 7). They rejected the increasing-noise model, and found some support for the constant-noise model. In any case, none of the deviations they found from d’ linearity were large enough to explain the discrepancy between contrast discrimination and the other discrimination tasks shown in Figs 3-5 of this paper. Further evidence in support of the d’ linearity assumption comes from the agreement between the data in this paper and literature values measured using different procedures and other criterion d’ levels. In addition, the Wilson model includes the effects of nonlinear transducers exactly, and it still fails to explain the contrast independence of spatial tasks. In conclusion, shape variations in the psychometric function are not adequate to explain the contrast independence of the discrimination tasks shown in Figs 3-5; central noise is required. Mirage: a model in&ding

central noise

Watt and Morgan (1984, 1985) have proposed a multiple-channel model named Mirage which differs in important ways from the other models discussed above. The stimulus is passed through an array of filters which are derivatives of Gaussians, and chosen from 4 SF bands. Although the sampling in SF is coarse (as in the Wilson model), the spacing between filters positions is much finer, and treated as e&ctively continuous. The responses of all the filters with the same SF profile are plotted as a function of the spatial position of the filter’s peak response, resulting in four convolved image functions, one for each SF band. Peripheral noise is added to each function, and the result is then half-wave rectified and summed across SF bands, producing two rectified functions separrrtely representing the positive and negative excursions of image contrast.

All detection and discrimination tasks are then performed using localized zero-bounded regions of excitation, which may be larger or smaller than actual stimulus features at low contrast. However, for contrasts greater than about 3 times detection threshold, Mirage reaches its high-contrast limit and the zerobounded regions of excitation correspond to bright or dark bars in the image. The remainder of this discussion considers only the highcontrast limit of Mirage, which should be appropriate for all the data shown in Figs 3 and 4. In Mirage, contrast judgements are based on “mass”, the total amount of excitation in a zero-bounded response distribution, while edge position judgements are based on the centroid (center of mass) of the response distribution. Localization thresholds are limited by propagated noise in Mirage, and improve with increasing contrast (Watt & Morgan, 1984). Contrast discrimination thresholds calculated using propagated noise are independent of pedestal contrast at high contrast (Watt & Morgan, 1985, line E,,, in Figure 1I), in disagreement with the data of Legge (198 1). To resolve this discrepancy, Mirage includes a “Weber’s law for mass” term, in other words, contrastdependent central noise affecting only contrast discrimination. This represents a completely different approach from that used by Legge (1981), Wilson and Gelb (1984) and Klein and Levi (1985), who regard contrast discrimination as the measurement which most directly reflects primary filter properties. The transducer function of equation (3), which plays a central role in all other discrimination theories discussed, does not appear at all in Mirage. Indeed, Mirage is not an error-propagation model of the type shown in Fig. 6, but instead includes both peripheral and central noise sources as shown in Fig. 1. Mirage includes three types of noise: peripheral noise (“spatially uncorrelated white noise” in Watt & Morgan, 1984), central noise in contrast discrimination (“Weber’s law for mass” in Watt & Morgan, 1985) and central noise in localization judgements (“Weber’s law for distance” in Watt & Morgan, 1983). Consequently, contrast discrimination and spatial localization discrimination thresholds are not closely related in Mirage. Mirage is, therefore, in qualitative agreement with the main result of this paper, that contrast discrimination and SF discrimination (localization) are limited by different sources of noise.

Contrast independent feature discriminatidn

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However, Mirage fails in detail, because it predicts contrast-dependent localization discrimination thresholds (Watt & Morgan, 1984)which are inconsistent with the contrast-independent SF discrimination thresholds shown in Figs 3 and 4.

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SF discrimination, orientation discrimination (at 4 cpd) and TF discrimination are nearly independent of contrast for contrasts above 2 or 4%, while the contrast Weber fraction declines substantially over the same contrast region. Therefore, the error propagation model shown in Fig. 6 is wrong; SF, orientation and TF discriminations are not limited by the noise of the front-end filters. This conclusion holds for all types of error propagation models studied: Quick (1974) rule combination, the single best mechanism rule proposed by Klein and Levi (1985), the ratio rule proposed by Regan and Beverley (1985), and the Wilson (1986) multichannel model. All these models must be rejected. Instead, the model shown in Fig. 1 must be used, including both “peripheral noise” at the filtering stage and “central noise” at later stages of processing. Mirage, the multichannel model of Watt and Morgan (1984, 1985) is an example of a model including both peripheral and central noise, but it will also require modification to explain contrast-independent SF discrimination. Klein, Stromeyer and Ganz (1974) provided further evidence that SF judgements and contrast judgements involve different processes. Their experiments showed that a surrounding grating which has no effect on detection has the same effect on the perceived SF of a grating as adapting to a grating which raises the detection threshold by more than a factor of 2. Their theoretical analysis also showed that adaptation-induced shifts in perceived SF cannot be explained by simple pooling of the responses of SF-tuned channels. There is a simple way of explaining contrastindependent SF discrimination qualitatively with a central noise model. First, a contrastindependent estimate of SF is computed, then contrast-independent central noise is then added to that estimate. One possible contrastindependent estimate is the spatial separation between two features. The contrast-independent “central noise” might reflect an observer’s uncertainty about the absolute position of the

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Fig. 8. Width discrimination and cd~ blur discrimination from previously published data. The top graph shows fixtioMl width incrcmmt thtesholds for two tinea separated by a mean distance of lo’, as reported by Morgan and Rcgan (1987). With discrEnation is indcpcmknt of contrast, and may therefore be M on absolute position cues. The lower graph shows edge blur diination for Gaussian-blurred edges, with the edge position randomized to remove position CUCI,as reported by Watt and Morgan (1983). Edge Mur d&rimination improves at high contmsts, unlikewidth dmxin&ution, * and is prestmubiy baaed on the relative responses of SF-td tiler& The SF discrimination thrrrhoids in Figs 3 and 4 arc quite similar to the width disctimination data, which suggests that SF discrimination is based on ab8olutc position cues. The d&a points were taken from graphs in the indicated publications, transformed to W&r fractions. and doubled to obtain d’ = 2 estimates.

(Levi, Klein & Yap, 1987), which would affect spatial position judgements much more than contrast judgements. In this case, the term “central noise” might be misleading, since the error in absolute position is present at the earliest filter stage, but simply has a greater effect on localization judgements than on contrast judgements. The term “task-dependent noise” might be better. Hirsch and Hylton (1982) compared SF discrimination for sinewave targets with two-line separation discrimination (often called “width discrimination”) and found similar Weber fractions for both tasks. Furthermore, Burbeck (1987) showed that discrimination of the separation between two widely separated targets is the same whether the targets are rectangular bars or patches of high-frequency grating. She

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also showed that discrimination does not improve for contrasts in excess of 5 x detection threshold. These results suggest an interpretation of the contrast-independent SF discrimination as a separation discrimination. In this interpretation, the observer judges the location of two features in the grating, such as adjacent contrast peaks, and uses the separation of these two features as an estimate of SF. If the precision of localization is limited by errors in the position of local filters, or some other contrastindependent “central” noise source, SF discrimination will be independent of contrast. Indeed, as shown in the top graph of Fig. 8, Morgan and Regan (1987) found that separation discrimination judgements are independent of contrast for a mean separation of 10’. This separation roughly corresponds to the “absolute position cue” or “local sign” region of Klein and Levi (1985) and Yap, Levi and Klein (1988). What happens to SF discrimination when absolute position cues are removed? Watt and Morgan (1983) performed edge blur discrimination with randomly jittered edge positions, and found that blur discrimination falls as contrast is increased as shown in the lower graph of Fig. 8. Evidently, when position cues cannot be used, the observer is forced to use the relative responses of local filters with different SF tunings to estimate edge blur. The responses of the local filters are more accurate at high contrast, leading to more precise localization. In other words, SF judgements without local position cues are indeed limited by peripheral noise, and therefore might be well described by error propagation models. If propagated-noise models are abandoned, does that render psychophysical modelling a hopeless task? If the visual system is as complicated as shown in Fig. 1, with a different amount of central noise added to each task’s decision variable, there might be no relationship at all between different psychophysical tasks! However, an examination of a variety of spatial discrimination tasks reveals a promising regularity. Some tasks, such as SF discrimination and two-line separation discrimination, are independent of contrast, and may be limited by a single common source of central noise such as absolute position errors in the filters. Many other tasks fall with contrast with a slope near -0.5 on log-log coordinates, in&ding vernier offset detection (Krauskopf & Farell, 1990; Wilson, 1986; Bradley & Freeman, 1985), edge

F. E~CJWNE

blur discrimination (Watt & Morgan, 1983) and stereo disparity detection (Legge & Gu, 1989), and may be limited by the sort of peripheral noise sources included in propagated-noise models (Wilson, 1986; Klein & Levi, 1985; 1985). Although the Regan & Beverley, propagated-noise models cannot explain all discrimination tasks, it is possible to include both peripheral and central noise in a model of discrimination, following the example of Mirage (Watt & Morgan, 1985). The viewprint model of Klein and Levi (1985) also includes a spatial Gaussian blurring operation on the filter outputs, which has effects similar to including central noise in the positions of the filters. Wilson has recently described extensions of his model to include spatial position uncertainty and undersampling (Wilson, 1990), which might improve its performance. In conclusion, propagated-noise models are incomplete, but may serve as a useful first stage for a more complete model including both peripheral and central noise. Future research should reveal how many different sources of noise must be included for a satisfactory description of human discrimination. Acknowledgemnrs-The author is grateful to Suzanne McKee for helpful discussions, advice and serving as an observer, and to Ken Nakayama whose insights were important in developing these experiments. This paper was also greatly improved by helpful and insightful comments from Stan Klein, Gordon Lqge.. Michael Morgan, Jim Thomas, Roger Watt, Hugh Wilson and one anonymous reviewer. Hugh Wilson also provided substantial aid in calculating the predictions shown in Fig. 7, including parameter values, programs, and independent confirmation of the results. This research was supported by NIH grants F32 EYO6506, ROl EY06644 and 5 P-30-EY00186.

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