Vision Res. Vol. 31, No. 6, pp. 983-998, 1991 Printed in Great Britain

0042-6989/91 $3.00+0.00 Pergamon Press ple

A T R A N S D U C E R MODEL FOR CONTRAST PERCEPTION MARKW. C A N N O N H. G. Armstrong Aerospace Medical Research Laboratory, A A M R L / H E F Wright-PattersonAir Force Base, O H 45433-6573,U.S.A. and STEVEN C. FULLENKAMP Logicon Technical ServicesInc.,Dayton, Ohio, U.S.A. (Received 6 June 1989; in revised form 16 August 1990)

Abstract--Multiplechannelmodelsof visualfunctionproposedto date havebeeneitherthreshold models or suprathresholdmodels.The transitionbetweendetectionand perceptionof contrast which involves the gradualdisappearanceof spatialresponsepoolinghas not beenaddressed.Weproposea modelthat allows us to simulatecontrastdetection,contrastperception and the gradualdisappearanceof spatial response pooling with contrast. The model successfullypredicts thresholdsand perceivedcontrast functionsfor several multiple componentstimuli. Contrast transducer

Contrastperception

Spatialfiltering Visualmodelling

their models are similar functions. While Legge (1981) had reservations about using the CTF to At the present time, the most successful models account for perceived contrast, Swanson et al. in spatial vision appear to be those with an (1984) used a version of the Wilson (1980) CTF initial processing stage of multiple spatial filter to account for suprathreshold contrast matches mechanisms, each of which is followed by a between stimuli of different spatial frequencies. non-linear transducer function. These models None of these models, however, can correctly have been used to quantitatively account for simulate the transition in the tbrm of spatial certain aspects of both threshold and supra- processing that occurs as stimulus contrast threshold vision. Legge and Foley (1980) used is increased from below threshold to high such a model to explain their contrast masking suprathreshold levels. Let us look at a specific example. It is well results. Wilson and his associates (Wilson & Bergen, 1979; Bergen, Wilson & Cowan, 1979; known that the threshold of a sine-wave grating Wilson, 1980; Swanson, Wilson & Giese, 1984) patch decreases as patch size is increased. This have used a multiple mechanism model to quan- has been attributed to spatial response pooling titatively simulate both contrast detection at among mechanisms that respond to the grating. threshold and apparent contrast at supra- It is also known that spatial response pooling threshold levels. Both of these models are one appears to be inactive for both contrast percepdimensional in the sense that the stimul~ and tion (Swanson et al., 1984; Cannon & Fullenfilters were only defined along one spatial axis. kamp, 1988) and contrast discrimination (Legge Two dimensional quantitative models have also & Foley, 1980) at high stimulus contrasts. The been published by Watson (1983) and by perceived contrast for a large grating patch is Cannon and Fullenkamp (1988). The Watson the same as that for a very small grating patch model is primarily a threshold model while of the same spatial frequency. What is not well the model of Cannon and Fullenkamp only known is that the disappearance of this spatial simulates perceived contrast for suprathreshold response pooling for apparent contrast is gradstimuli with contrasts above 0.1. Both the Legge ual. Spatial response pooling disappears at a and Foley model and the models generated by stimulus contrast of about 0.1 for 4c/deg Wilson and his associates use a contrast trans- grating patches (Cannon & Fullenkamp, 1988). ducer function (CTF) derived from contrast The original Cannon and FuUenkamp model discrimination data. Consequently, the CTFs in was set up to account for contrast perception at INTRODUCTION

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MARK W. CANNON and STEVENC. FULLENKAMP

contrasts above 0.1 and did not include spatial response pooling. The model was able to successfully simulate the relative perceived contrast of several multi-component stimuli with response pooling across mechanisms tuned to different spatial frequencies and orientations. The Swanson et al. model had CTFs that extended from below threshold to a contrast of 1.0, but the authors were forced to model threshold and suprathreshold data in two different simulations. Spatial response pooling was retained for threshold simulations and it was simply eliminated for suprathreshold simulations. Legge and Foley (1980) also chose to model their contrast masking data in two separate simulations; with a form of spatial response pooling for contrasts near threshold and without spatial response pooling at high contrasts. The transition between the contrast region where spatial response pooling is active and the region where it is inactive has not been addressed by either model. The Legge and Foley model was intended only to deal with contrast increment thresholds and will not be considered further since we have no data on the behavior of contrast discrimination for stimuli of different sizes through this transition region. The Wilson model, however, has been used to deal with apparent contrast at suprathreshold levels and thus merits further analysis. A possible reason for the difficulty in modelling the transition could be the way spatial response pooling is introduced into the model. In the Wilson model, threshold is mediated by the Quick (1974) summation formula applied to the outputs of all the active spatial filters after these outputs have been passed through the CTFs. In the original Wilson and Bergen (1979) model the result of this computation was a single number proportional to the threshold contrast sensitivity. Clearly, if the size of the stimulus is increased, more mechanisms will be excited and the Quick sum will also increase indicating a higher sensitivity and lower threshold. Unfortunately, even though the CTFs are non-linear, an increase in stimulus size also equates to an increase in sensitivity at high contrasts in the Swanson et al. model, unless some way can be found to reduce the effectiveness of the spatial response pooling portion of the computation as contrast increases. This is not as easy as it sounds since the Quick formula provides no allowance for this type of adjustment. Since our lab has been primarily involved in the study of suprathreshold vision we have a

strong interest in developing a model of contrast perception that accounts for the disappearance of spatial response pooling with contrast. We also wanted a model that was generalizable; that is, responses of individual filter mechanisms and their non-linear transducers must be calculable from any input pattern. After analyzing the Legge and Foley CTF it became obvious to us that there was a way to handle the disappearance of spatial response pooling within the framework of an existing CTF. The Legge and Foley function contains a parameter that can be manipulated to simulate the shape of perceived contrast functions obtained in magnitude estimation experiments (Cannon & Fullenkamp, 1988) for sine wave grating patches of different spatial extent. The Wilson CTF could almost certainly be adapted in the same way, but it contains many more terms and an interpretation of the meaning of those terms would not be as simple as in the Legge and Foley form of the CTF. Our formulation of the Legge and Foley function for contrast perception is given in equation (1) below: PC = F Ca+hi(Ca + Aa).

(1)

In our context we use the equation to describe a perceived contrast response, PC, computed from a non-linear function of contrast, C. If the function is plotted on double logarithmic coordinates, exponent b is the slope of the upper branch of a two branched response. Our suprathreshold data (Cannon & Fullenkamp, 1988; Cannon, 1985) from magnitude estimation experiments imply that this exponent should be 0.5. Exponent a represents the initial steep rise of the perceived contrast function and previous data indicated that it should be near 3.0. The term F is an amplitude adjustment factor. Note that these exponents are different from those derived by Legge and Foley for contrast discrimination. A typical function is shown as the solid curve in Fig. 1. An interesting feature of this function is that increasing A, while leaving F, a and b fixed, will change the position of the lower portion of the function (the dashed curves in Fig. 1), but will leave the upper portion of the function unchanged. A is related to the location of the knee of the function. When C is smaller than A the function rises steeply. As C becomes larger than A the function approaches a power function with exponent b and the term A has less and less effect on the shape of the function.

A transducer model for contrast perception

with other stimuli. In such a model we propose to use a modification of the Legge and Foley function to represent the contrast transducer function for each spatial filter mechanism. This modification is shown in equation (2),

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Fig. 1. Perceived contrast as simulated by the Legge and Foley transducer function for three values of parameter A. The horizontal axis represents the contrast of the stimulus. As parameter A is increased, the lower part of the curve shifts further to the right while the upper part remains relatively unchanged. This behavior is similar to that demonstrated by Cannon and Fullenkamp (1988) for the perceived contrast of sine wave grating patches as the spatial extent of the gratings was reduced. In such an interpretation, A would be inversely related to stimulus spatial extent.

The change in the shape of the Legge and Foley function produced by increasing A is very similar to the change in the shape of the perceived contrast function shown in Fig. 2 of Cannon and FuUenkamp (1988) as the size of the stimulus patch is reduced. If threshold is defined as the contrast at which our adaptation of the Legge and Foley function crosses some criterion level, increasing A would correspond to an increase in threshold. Thus, an increase in A corresponds to a decrease in spatial response pooling. By finding a functional relationship between A and the amount of spatial response pooling predicted by the Quick formula it would be possible to fit our previous perceived contrast data and in some way account for the behavior of the data. However, curve fitting for specific stimuli would not develop a model with the power to predict perceived contrast responses to other stimuli. We desired, instead, a model consisting of many spatial frequency and orientationally tuned mechanisms, the responses of which, when pooled in some way would produce an equivalent perceived contrast function. Once these filter mechanisms and their response pooling characteristics were adjusted to match our threshold and perceived contrast data for a specific set of sine wave grating patches, the predictive power of the model could be tested

(2)

In this interpretation, R is the output of the contrast transducer associated with a particular filter mechanism in response to some stimulus. F is again a scaling factor and CR is the response of a linear filter tuned to a particular spatial frequency and orientation. The amplitude of C R is proportional to the contrast of the stimulus being processed by the linear filter and depends on the shape of the spatial, spatial frequency and orientation weighting functions of the filter. Exponents a and b again describe the shape of the non-linear response, as in equation (1), and A is a term that specifies the threshold of the mechanism. Since A will determine the mechanism threshold, it must depend on local factors like the region of space and the spatial frequency to which the associated linear filter is tuned. According to our previous data (Cannon & Fullenkamp, 1988) A must also depend in some way on the global factor of response pooling among the filter outputs. Finally, we propose to use the same computation for model perceived contrast estimates that we used in our earlier study (Cannon & Fullenkamp, 1988), but now the non-linear filter mechanism responses are used as inputs to the computation instead of the original simple power functions. The specific model proposed will be discussed in detail below. METHODS

Three types of experimental paradigms were used to collect the psychophysical data required for setting up and testing the model. These were magnitude estimation and contrast matching for suprathreshold data and two interval forced choice for threshold measurements. Our magnitude estimation and threshold measurement techniques have been fully described in earlier publications (e.g. Cannon & Fullenkamp, 1988). The contrast matching experiment was used to verify our low contrast magnitude estimation results and will be described in detail. In each session of this experiment a single 4c/deg Gabor sine patch with a spatial halfwidth a, of 8.0, 2.0 or 0.5 was matched in apparent contrast to a fixed contrast full screen

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MARKW. CANNONand STEVENC. FULLENKAMP

sinewave grating pattern of the same spatial frequency. The contrast of the full screen comparison grating was set at one of six values that ranged between 0.006 and 0.095 inclusive in 0.2 log unit steps. The full screen pattern was always horizontal and the Gabor patches were vertical to minimize adaptation effects. A Gabor sine function is simply a sine wave grating multiplied by a two dimensional Gaussian envelope. The spatial halfwidth of the envelope at amplitude 1/e is given by sigma specified in terms of the number of cycles of the underlying sine wave. A sigma of 8.0 means the envelope reaches the 1/e point 8 cycles away from the origin. A modified method of constant stimuli was used to determine the contrast at which a G a b o r patch had the same apparent contrast as the full screen grating set at one of the 6 contrasts listed above. The Gabor patch test stimulus was presented at each of 8 contrast levels, with a spacing of 0.05 log units between levels, bracketing the contrast where the match was expected. (Preliminary runs were made for each subject to determine about where the matches would occur.) The test and comparison stimuli were presented for durations of 500 msec within two consecutive 1 sec intervals. The beginning of each interval was marked with an auditory tone. The contrast level of the test and the order of presentation were randomized. Subjects reported, by means of a two position switch, the interval which contained the stimulus of higher contrast. The next test-comparison pair were presented 1 sec after the subject indicated a choice from the previous pair. A complete set of data for each session consisted of responses to 30 trials at each of the 8 test contrast levels. Psychometric functions estimating the probability that the test had greater apparent contrast than the comparison were constructed from these data. The contrast at which this function crossed the 0.5 probability level was the contrast of perceptual equality (contrast match) between one of the Gabor patch sizes and the full screen grating at one of its 6 contrast levels. These contrast matching data were averaged across the four subjects to produce a mean matching contrast for each of the three sizes of Gabor patches at each of the 6 fixed grating contrast levels. It is this mean data that will be shown later in this paper. There are two ways to specify the contrast of the type of stimuli we used in this study. All of the stimuli are sine waves or combinations of sine waves multiplied by a Gaussian envelope.

One way is to specify the contrast o f the original grating pattern and let the actual contrast of the pattern change with the width of the Gaussian envelope (Watson, 1982). The other way, and the one we have chosen, is to specify the contrast from the maximum peak to trough differences in the patterns. Specificially, pattern contrast was defined as (Lmax- Lmin)/ (Lmax+ Lmin), where Lmax is the highest luminance and Lmin is the lowest luminance in the pattern. All stimuli were computer generated and transferred to a Grinnell video frame store device. Contrast of the video images was controlled by an analog multiplier built in-house. Images were displayed on a Conrac 2600 video monitor with a P4 white phosphor. The screen was masked to a size of 6.5 deg square by a large surround illuminated with the same average luminance as the screen, 80cd/m z. All the data shown in this paper were collected at this luminance level. DETERMINATION OF THE MODEL CONTRAST TRANSDUCER FUNCTION

It is evident from the form of the function given in equation (2) that thresholds increase as the magnitude of the term A in the denominator of the function increases. This term will be referred to as T H R E S H for the rest of the discussion, due to its obvious relationship to the stimulus threshold. As stated earlier one component of T H R E S H for a given mechanism must change as a function of the spatial frequency to which the unit is tuned. As size increases, threshold decreases, so T H R E S H must contain a component inversely related to stimulus size. In order to keep the model as simple as possible, we chose to use the Quick formulation of response pooling across mechanisms to mediate the dependence on stimulus size. Our initial modelling attempts made the response pooling component of T H R E S H only sensitive to response pooling across space. Further experiments and simulations indicated that T H R E S H should contain a term incorporating response pooling across space, spatial frequency and orientation. The final form for T H R E S H is given in the equation below: T H R E S H = K WT(f, O) (CRMAx/QPooL), (3) Parameter K and the function WT(f, O) are determined by matching model responses to experimental data. K is a scaling constant and is

A transducer model for contrast perception

SPATIAL FREQUENCY THRESHOLD wr~IGHTS

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Fig. 2. A two dimensional model for contrast perception. Spatial filtering and final processing are identical to similar stages in the suprathreshold model proposed by Cannon and Fullenkamp (1988). The contrast transducer function R at the bottom of the figure is new. This transducer is "a combination of global processing, CR~Ax/QPooL,and local processing, WT(f), that allows the model to simulate contrast perception from detection to high contrast levels. See text for details.

the same for all mechanisms. WT(f, O) is a value in the sum is proportional to contrast. In weighting function which depends on the spatial this initial study, we wanted a THRESH term frequency and orientation to which the mechan- that was independent of contrast in order to ism is tuned. Higher values of WT(f, O) imply preserve the form of the Legge and Foley funchigher thresholds. For example, mechanisms tion as stimulus contrast increased. This would tuned to 4 c/deg will have a smaller value of require a normalized QmoL. After trying several WT(f, O) than filters of higher or lower spatial normalization methods, we settled on using frequency. In most of the simulations to follow, the maximum filter output, CRMAxas our nordifferences in threshold due to orientation are malization term. Thus, the third term in ignored, so the weighting functions WT(f, O) equation 3, CRMAx/QPooL, is a ratio of two will be referred to as WF(f) for those simu- terms that are proportional to contrast and lations where orientation differences do not play therefore is independent of contrast. The magnia part. The term QmoL is computed by taking tude of the term is determined solely by the the Quick summation, with exponent 3.0, distribution of stimulus energy among the filter over all the linear filter outputs. For those not mechanisms. As stimulus size increases while familiar with the Quick summation formula for contrast is held constant, the magnitude of response pooling, it is given in equation (4), the ratio CRMAx/QPooL decreases reducing where CR is the output of a linear filter centered THRESH as desired. at location X, Y and tuned to spatial frequency f and orientation 0 INCORPORATION OF THE CTF INTO THE 2D MODEL

Note that QPooL increases in direct proportion to the contrast of the stimulus since every CR

The complete model is illustrated in Fig. 2. The input image is a 256 x 256 pixel array representing a 4 x 4 deg area. The filters are

988

MARKW. CANNONand S~VENC. FULLENKAMP

Gabor sine functions tuned to spatial frequen- puts from each spatial location are analyzed cies from 1 to 16 c/deg in half octave steps and separately. The outputs are pooled across to orientations from 0 to 180 deg in 15 deg steps. spatial frequency and orientation, at each Filter halfwidths at amplitude 1/e in the space spatial location, again using the Quick sumdomain are 0.707 in agreement with our earlier mation formula. In agreement with our earlier suprathreshold model (Cannon & Fullenkamp, model the exponent for this summation is 2.5. 1988). This gives a filter bandwidth of about 1 The decision algorithm assesses the Quick octave that is constant across all spatial frequen- summed outputs at all spatial positions and cies. All filters are set up so that they have the chooses the largest of these summed responses same mean square integral over space. They are to mediate the perception of contrast. The equal energy filters. The absolute values of the model contrast response, when discussed in filter outputs, CR, shown in the figure represent subsequent sections of this paper, will be refermodel responses at one spatial position x,y and ring to this maximum response. Finally, since at one orientation, 0i. A complete set of model the perceived contrast of stimuli below responses for an input stimulus would, of threshold is zero, we subtract a small constant course, include all spatial positions and orien- from our model contrast response function. tations. No attempt has been made to account This contrast represents the mean value of a for a possible difference in spatial sampling as a noisy threshold level in the contrast response function of the center frequency of the filter as computation and causes the perceived contrast proposed by Watson (1982). This model as- functions to become nearly vertical on a plot of sumes that all filters are equally dense in the log contrast response vs log contrast as perspace domain. Nothing is lost by this assump- ceived contrast approaches this level. This contion except the possibility of a more efficient stant was set at a value of 0.1 at the beginning computation. We have also not accounted for of our simulations and remained fixed through the increase in threshold with eccentricity in this all simulations described below. Random fluctuversion of the model. We feel that the effect of ations in this value would produce similar flucthis omission is minimal for three reasons. tuations in the position of the threshold along First, most of the stimuli are spatially localized the contrast axis. at fixation. Second, the perceived contrast computation depends on the response of those MODEL CALIBRATION mechanisms that respond maximally and for All simulations described in this paper were our stimuli those mechanisms are located at fixation. Third, since our stimuli generally carried out using a Data Translation D T 7020 decrease in contrast from fixation toward the array processor board mounted in a Zenith 248 periphery, the dominant contribution to the ( a PC-AT clone) microcomputer. There are Quick summation formula for response pooling three free parameters in the model that must be at threshold comes from the region near adjusted before the model can be tested. Two o f these were already mentioned in the discussion fixation. The response of each filter is passed through of equation (2); K, and WT(f). The third its own contrast transducer function (CTF). The parameter is the exponent a of the CTF. Expoform of each transducer function is determined nent b is fixed at 0.5 by the upper portions of our by a two step process. The ratio CRMAx/QPoo L perceived contrast data derived from magnitude is computed by the Global Processor and is fed estimation experiments. The first task is setting to all transducer functions where it multiplies K and the exponent a. In our first iterations WT(f) was set at a constant value for all the spatial frequency weighting function WF(f) associated with each transducer. This multipli- spatial frequencies, and values of K and a were cation gives the CTF a lower branch response found that gave a good fit to the magnitude that depends on both the spatial frequency of estimation data for a 4 c/deg Gabor patch with = 8.0. This is the solid curve in Fig. 3. The the input filter and the response pooling computation on the stimulus. The transducer function data points along the curve are means of magnioutputs are then used to determine perceived tude estimation data from 5 subjects for the contrast. The decision rules for contrast percep- same stimulus. Mean thresholds are the vertical tion are the same as those proposed and verified arrows at the bottom of the graph. The actual in our first suprathreshold model (Cannon & values of model parameters are given in the Fullenkamp, 1988). The non-linear CTF out- Appendix and need not concern us here.

A transducer model for contrast perception o ~ 4c/de9 ~ = 8 & - - 4c/deg ~ = 2 o-.- 4c/de9 ~=0.5

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Fig. 3. Perceived contrast data and model responses for three Gabor sine patches of different sizes. Measured thresholds are the vertical arrows along the contrast axis. Model parameters were adjusted to fit the e = 8.0 curve to the data. The (r = 2.0 and 0.5 curves were generated with no further parameter adjustment and are in fairly close agreement with the data. a-0.5

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With no further adjustment of K and a, model responses were determined for a a = 2.0 G a b o r and a a = 0.5 Gabor. The results of these tests are shown as the dashed and dot-dashed curves in Fig. 3. Model predictions of both perceived contrasts and thresholds are in good agreement with the data. The determination of WF(f) was done next by testing the model using Gabors with a = 0.5 at 2.0, 4.0, 8.0 and 16.0 c/deg and adjusting WT(f) until model thresholds matched experimental data. The perceived contrast functions produced by the model for a = 0.5 stimuli at the 4 spatial frequencies mentioned above are shown in the upper panel of Fig. 4. WT(f) was adjusted until each of the four perceived contrast functions crossed the 0.01 contrast response level adjacent to the vertical arrow representing its threshold contrast. The WT(f) function required is shown in the lower panel of Fig. 4. Once the values of WT(f) were set, the system was tested again with 4 c/deg, a = 8.0, 2.0 and 0.5 Gabors. The responses showed only insignificant deviations from the curves displayed in Fig. 3, so no further parameter adjustments were necessary. With two successful tests already accomplished (a = 2 . 0 and 0.5 at 4 c/deg), the model was subject to further testing with stimuli that were not used in the calibration process. TESTS OF MODEL PERFORMANCE

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frequency (c/deg)

Fig. 4. Parameter adjustment for thresholds at four spatial frequencies. Vertical arrows are the measured thresholds for small Gabor patches at the spatial frequencies. The curves in the upper panel show model responses to the same stimuli after the WT(f)s were adjusted to values defined by the function shown in the lower panel. See text for details.

The first of the two simulations described in this section of the paper were performed with no further change in model parameters since these tests were expected to account for experimental data collected from the same group of five subjects who produced the calibration data shown in Figs 3 and 4. The first test involved model predictions for changing size stimuli at a different spatial frequency. Magnitude estimation and threshold data were obtained in the same experimental sessions for a 4 c/deg a = 8.0 G a b o r and for 16c/deg Gabors with as of 8.0, 2.0 and 0.5. The mean perceived contrast estimates and model responses for these stimuli are illustrated in Fig. 5. Model predictions again provide a fairly good description of the data. The magnitude estimation data for the 4.0 c/deg, a = 8 stimulus are a bit more noisy than the similar data shown in Fig. 3, but the model response, with no parameter changes, is still in adequate agreement with the new data. An interesting feature of both data and model

990

MARK W. CANNON and Sa'EVENC. FULLENKAMP *-4c/deg D - - 16 c/deg =--- 16c/deg o . . . 16 c/deg

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Fig. 5. Perceived contrasts and mode] responses to three patch sizes at 16c/deg and one patch size at 4c/deg. Perceived contrast data for all stimuli were determined in the same experimental session. Model responses to the stimuli were computed without any parameter changes. Both model and data show the 16c/deg responses to be slightly smaller than the 4 c/deg responses. Agreement with thresholds (vertical arrows) is good.

predictions is that the 16 c/deg perceived contrast estimates are below the 4 c/deg perceived contrast estimate at high contrast levels. We first thought that this decrease, in the model, was due to the fact that there are no filters with center frequencies above 16c/deg and the 16 c/deg Gabor functions have considerable spectral energy at higher spatial frequencies. Consequently, we set up a model containing two more mechanisms with filters tuned to 22.6 and 32 c/deg and extrapolated our WT(f) function to these higher spatial frequencies. In this simulation we set up the model so the spatial sampling in pixels per cycle at 32c/deg was the same as that for 16c/deg in previous simulations. The results were essentially the same even though model responses did show a very small increase. While these higher frequency spectral components are present, their WT(f) values increase drastically with spatial frequency so their contributions to the suprathreshold response pooling equation are small relative to the response at 16c/deg. This reduced amplitude effect is further enhanced by the fact that the response pooling equation suppresses small mechanism responses relative to the peak response from the 16c/deg mechanism. Model behavior in Fig. 5 appears to be due primarily to the higher WT(f) values in the vicinity of 16 c/deg. Apparently, contrast constancy (Georgeson & Sullivan, 1975) for Gabor stimuli begins to

degrade near 16c/deg for both model and data. The model was derived primarily to account for differences in perceived contrast due to increases in stimulus size. However, a successful model should be able to account also for contrast perception of stimuli composed of several spatial frequency components at different orientations. Consequently, magnitude estimation and threshold data were obtained from a set of 4 new subjects for stimuli consisting of the sum of two orthogonal sinusoidal components at a spatial frequency of 4 c/deg. These stimuli had Gaussian envelopes with as of 8.0, 2.0 and 0.5. The stimuli are illustrated in Fig. 6. The magnitude estimation sessions consisted of presentations of the three stimuli mentioned above along with the tr = 8.0, 4 c/deg Gabor used in the initial calibration. This stimulus was used in case any re-calibration of the model was required due to different response characteristics of the new set of subjects. Indeed, the threshold for the tr = 8.0, 4 c/deg Gabor for this group of subjects was slightly higher than for the previous group. In order to match the model response for the a = 8.0, 4 c/deg stimulus to the data only a small increase in parameter K was required. This represents an overall decrease in mechanism sensitivity at low contrast, but does not alter the high contrast response. The adjustment produced the leftmost solid curve in Fig. 7. The other curves in Fig. 7 represent model responses to the orthogonal gratings with the new value for K, but with all the other parameters set at their previously fixed values. The model makes fairly accurate predictions of both threshold and perceived contrasts for all stimuli. VERIFICATION OF MODEL RESPONSES FROM MATCHING EXPERIMENTS

All of the suprathreshold data used in the previous examples was collected using the method of magnitude estimation. Since the method of magnitude estimation may not have completely unqualified support from all researchers, we decided to check on the accuracy of our model predictions by using contrast matching techniques described in the methods section of this paper. In these experiments we determined the contrasts required for 4 c/deg Gabor sine patches with as equal to 8.0, 2.0, and 0.5 to have the same apparent contrast as a full screen 4 c/deg sine wave grating, as this full screen grating took on various contrast

Orthogonat grating patches Sigma

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A transducer model for contrast perception • -u---a--o --

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Fig. 7. Model responses to both orthogonal sine stimuli and a single component sine patch. Again, all magnitude estimation data represent results from sessions where all four stimuli were presented. Different subjects were used in these experiments, resulting in a different threshold for the single component 4 c/deg patch. Parameter K was increased so the model response could match the single component threshold. The other model responses were determined with the new value of K, but all other parameters were held constant.

993

the solid curve (~ = 8.0) for perceived contrast in Fig. 3 to represent the perceived contrast of the full screen grating we can predict, from the other curves, what contrasts the smaller patches should have to produce the same perceived contrast as the solid curve. This is accomplished by finding the perceived contrast amplitude for the tr = 8.0 curve at a contrast of, say, 0.006 and extending a horizontal line at that amplitude to intersect the other two curves. The contrasts at which the intersections occur are the matching contrasts. The model predicted matching contrasts are the solid curves in Fig. 8. The mean contrast matches are the individual data points. Note that the vertical spacing between the a = 0.5 and a = 2.0 data is greater than the spacing between the a = 2.0 and a = 8.0 data, in agreement with model predictions and magnitude estimates. The ability of the model to predict contrast matching data provides further validation of model performance. DISCUSSION

Comparison with other models levels between 0.006 and 0.095. Our matching data showed that there was no signfiicant difference between the perceived contrast of the = 8.0 patch and the full screen grating over the contrast range we studied. Thus, if we take - ModeL predictions • • • Matching data

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Fig. 8. Comparison of contrast matching data with model predictions of the same data. Contrast matching experiments were conducted to verify the predictions of a model derived from the magnitude estimation data shown in Fig. 3. All three sizes of Gabor patch were matched in apparent contrast to a full screen sinewave grating which was set at seven different contrast levels. The matching data are the individual data points. The solid lines are model predictions derived from the model response curves in Fig. 3. Model predictions lie very close to the matching data.

It might be appropriate at this time to highlight the differences between the model presented here and the usual approach taken in multiple mechanism models. Most multiple mechanism models filter the input with linear filters, operate on the filter outputs with fixed non-linear CTFs and then perform response pooling on the outputs of the C T F stage. The linear filter gains in these models usually vary across spatial frequency to reflect the shape of the contrast sensitivity function. This type of model has been used successfully to predict thresholds, but does not transition correctly into the suprathreshold domain, where spatial response pooling disappears. These models must also vary the shape and amplitude of the C T F as a function of spatial frequency to account for contrast constancy (Georgeson & Sullivan, 1975; Swanson, Georgeson & Wilson, 1988) at high contrast. Our model filters have constant gain across spatial frequency. Response pooling of the filter outputs across spatial frequency, orientation and space, adjusts the form of the non-linear CTFs. This stimulus dependent form adjustment allows us to use the same function for the C T F at all spatial frequencies and orientations. The outputs of the C T F s are then response pooled across spatial frequency and orientation at each spatial position and the largest pooled

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MARKW. CANNONand STEVENC. FULLENKAMP

response is chosen to mediate perceived contrast. When this response exceeds some threshold level the stimulus is "seen" with a quantitatively definable perceived contrast. Thus, the model allows a smooth transition between the subthreshold and suprathreshold regimes. The magnitude of the sub-threshold response pooling computation delays or accelerates the growth of each mechanism's contrast response depending on the spatial extent of and the number of spatial frequency components contained in the stimulus. Large stimuli, for example, force the mechanism contrast responses to grow initially faster and, thus, exceed threshold at a lower contrast than smaller stimuli. Both model and data indicate that, for the type of stimuli used in these experiments, perceived contrast at high contrast levels is largely independent of the size or spatial frequency component content of the stimulus. It depends only on stimulus contrast which we have defined by using the luminance of the brightest peak and darkest trough (see Methods section). However, analysis of model responses with orthogonal gratings makes it clear that this is accomplished only by response pooling across spatial frequencies and orientations. For the case of the sum of two 4 c/deg orthogonal components (Fig. 6), each component has half the total photometrically measured contrast. Consider filters tur,ed to the spatial frequency of the grating components and oriented along the two stimulus axes. These are the filters that will produce the maximum responses to the stimulus. Each filter produces an output amplitude equivalent to half the response that one of these filters would produce if the input were a single component grating oriented along one of these axes, with the same spatial frequency and with a contrast equal to the contrast of the summed orthogonal pair. Thus, if we chose to avoid spatial response pooling by equating perceived contrast to the single maximum mechanism response, our computed perceived contrast for the orthogonal components grating would be half the experimental value. Response pooled across mechanisms tuned to different orientations and spatial frequencies, on the other hand, produces model responses in agreement with experimental data (Fig. 6). A stronger argument for response pooling across spatial frequencies as well as orientations can be supported by other data, not shown, where we determined perceived contrasts for a small patch containing orthogonal 2

and 4 c/deg components and a small 4 c/deg patch of the same peak to trough contrast. The perceived contrasts were equal at contrasts above 0.1, although model mechanisms tuned to the 2 and 4 c/deg components each showed half the response produced by the single component grating. Model responses, pooled across spatial frequencies and orientations agreed with experimental data. The similarity of the high contrast portion of perceived contrast functions for most of the stimuli illustrated above makes it clear that the main differences in perceived contrast for different stimuli occur for contrasts from threshold to about a log unit above threshold. Perceived contrast behavior at low contrasts in our model is determined almost entirely by the term labelled THRESH in equation (3). A major portion of THRESH is the ratio CRMAx/QPoo L. Under normal stimulus conditions, once the CRMAx/QPoo L computation is determined, both the contrast at which the model perceived contrast response intersects the threshold criterion level and the shape of the lower part of the response function is fixed. Thus, any stimuli having the same CRMAx/QPoo L should produce identical perceived contrast functions. Model responses and adaptation data

Georgeson (1985) studied the effect of adaptation on the apparent contrast of a test grating, where both test and adapting gratings had the same spatial frequency. He found that, in general, the apparent contrast of a test grating was significantly reduced for test contrasts below the adapting contrast, but was not affected for test contrasts above the level of the adapting contrast. This implies that perceived contrast functions generated by adapted test gratings may behave in a manner similar to the perceived contrast functions in Fig. 3 except that the threshold shift is due to adaptation instead of a change in stimulus size. Higher adaptation levels produce higher thresholds and the perceived contrast functions rise somewhat more steeply from the higher thresholds to join the unadapted perceived contrast function at a contrast near the adaptation contrast. Georgeson proposed to account for his data with a subtractive rule for adaptation. His attempt to predict the effect of a subtractive adaptation on contrast matching results (Fig. 2b, Georgeson, 1985) was suggestive, but did not accurately reflect the true form of curves drawn through his data.

A transducer model for contrast perception

995

While a mechanism to quantitatively simulate data. We used an adaptation stimulus with the adaptation has not yet been added to our model, same spatial properties as Georgeson's, but some insight into the reasons for the form of fixed the contrast at 1.0. We then adjusted Georgeson's adaptation data can be gained by ADAPT to shift the thresholds of the test a simple approximation. In the context of our stimuli to values similar to those obtained by model it seemed obvious that the effect of Georgeson for his four different adaptation adaptation should be incorporated into the conditions. Finally, we determined model perTHRESH term of the CTF. Any attempt to ceived contrast functions when ADAPT was reduce the gains of the linear filters would just fixed at each of those four levels. shift the entire perceived contrast function Georgeson used two different stimuli, but in toward higher contrasts, on a plot of log both cases he adapted and tested with the same perceived contrast vs log contrast, without stimulus. Both stimuli were 3 c/deg hard edged changing its shape. Such a shift would prevent vertical gratings. In one condition the grating adapted responses from reaching the same was 3.7 cycles wide by 4.5 cycles high. In the amplitude, at high contrasts, as the unadapted other condition the grating size was doubled responses. This is essentially the same con- to 7.4 x 9.0 cycles by moving the subject clusion reached by Georgeson (1985). Since we closer while keeping spatial frequency fixed. are not changing linear filter properties, CRMAX Georgeson performed the post adaptation and QI'OOL, which are computed on the linear matching experiment with subjects fixating on a filter outputs, remain the same. small dark point placed midway between the This leaves the W T ( f , O)terms. We have adapted and comparison gratings, so subjects reintroduced the orientation dependence of viewed both gratings simultaneously in the these terms, since adaptation produces an periphery. A vertical strip, 0.5 deg wide in one increase in mechanism thresholds that is depen, experiment and 1 deg wide in the other experdent on both the spatial frequency and orien- iment separated the two gratings. Both gratings tation of the adapting stimulus. We have not in the matching experiment had a zero of the collected data to determine exactly how the W T sine wave at the edge of the strip containing the terms are affected by various levels of adapting fixation point. The other end of the grating contrast, but we propose a simple linear model terminated in a partial cycle. Since we have not that can be tested. The proposed modification of included eccentricity effects in the W T weighting the W T terms to account for adaptation effects functions, the model cannot simulate this experis shown in equation (5) where the modified iment in complete detail. Thresholds for differweight is WTAD. ent spatial frequencies increase at different rates with eccentricity when eccentricity is expressed W T A D ( f , O) = W T ( f , O) in degrees (Robson & Graham, 1981). However, (1.0 + ADAPT C A ( f , 0)). (5) the main response activity is determined by the 3 c/deg mechanisms and the increase of The term C A ( f , O) is directly proportional to threshold with eccentricity for 3 c/deg is small at the activity induced in a linear filter tuned to 1 or 2 deg. Moreover, analysis by the model spatial frequency f and orientation 0 by the showed that the maximum of the contrast presence of the adapting stimulus. ADAPT is a response computations for the larger grating scaling factor that depends in some as yet occurred half a cycle from the edge of the unspecified way on the total activity induced in pattern that terminated at zero amplitude, both the filters by the adaptation stimulus. What this before and after adaptation. Consequently, in adaptation correction does in the model is to Georgeson's experimental configuration, both increase the THRESH term for each mechanism detection and contrast perception would be by an amount directly proportional to the mediated by mechanisms near 0.66 deg eccenproduct ADAPT x CA (f, 0). If no adapting tricity and peripheral threshold effects should be stimulus is present or if the adapting stimulus minor. lies outside the spatial, spatial frequency or Georgeson did not give the threshold values orientation passband of a mechanism, the W T for all the conditions that he tested, but he value for that mechanism is not changed. The proposed a function to describe both matching functional form of ADAPT is unknown, but and threshold data. This equation is s = t - ka. meaningful values of the product ADAPT x The term a is the adapting contrast and t is the C A ( f , O ) can be determined from available contrast required for the adapted test stimulus

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MARKW. CANNONand STEVENC. FULLENKAMP

to match an unadapted standard of contrast s. For threshold data, t is the threshold under the adapted conditions and s is the normal unadapted threshold. Georgeson proposed a fit to both matching data and threshold data with a k of 0.33. His threshold data fall somewhat below this curve (Georgeson, 1985, Fig. 3) so we fitted the threshold data with the same function, but with a k of 0.175. This provided a better fit to the threshold points at the upper part of the curve where most of our simulations were to be done. Thresholds were computed from this function for adaptation contrasts of 0.04, 0.08, 0.16 and 0.48, based on an unadapted threshold of 0.0033. The adapted thresholds were 0.01, 0.017, 0.031 and 0.09. The parameter K in our model was adjusted to give model response curve for the unadapted stimulus a threshold of 0.0033 while all other model parameters remained fixed. The A D A P T values required for the model responses to match the adapted thresholds were 0.038, 0.08, 0.185 and 0.7. Model perceived contrast curves for the 7.4 x 9.0 cycle patch are shown in Fig. 9A and are labeled with the contrast of the adapting stimulus associated with each. These curves now start from approximately the same thresholds as those produced by Georgeson's adaptation experiment. Let us see how well the shapes of these curves can account for Georgeson's adaptation data. The seven dashed horizontal lines intersect the unadapted curve at contrasts of 0.05, 0.1, 0.2, 0.4, 0.8, 0.16 and 0.32. The intersection of the lowest horizontal line with each of the adapted curves specifies the contrast required under each of these adapted conditions to achieve a perceived contrast equal to that produced by an unadapted stimulus at a contrast o f 0.05. F r o m the entire group of intersections specified in Fig. 9A we can determine the contrasts required for all the adapted stimuli to match the unadapted grating for seven different contrasts of the unadapted grating. This is the same type of information provided by Georgeson's matching data. The curves describing our matching stimulation are shown in Fig. 9B. Points determined from Fig. 9A, representing the contrasts required for the adapted stimuli to match the unadapted stimulus under four different adaptation conditions, were plotted in Fig. 9B and then connected with the four smooth curves. The contrast of the unadapted grating is on the vertical axis. The contrast of the adapted grating is on the horizontal axis.

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Fig. 9. (A) Model estimates of perceived contrast for grating stimuli of the type Georgeson (1985) used in his adaptation experiments. The leftmost curve is the model perceived contrast response with K adjusted to give a threshold of 0.333 and represents the unadapted condition. Other curves represent model perceivedcontrast responses with the ADAPT parameter adjusted so model thresholds equal Georgeson's adapted thresholds. Horizontal dashed lines represent perceived contrast levels produced by an unadapted stimulus presented at contrasts of 0.005, 0.01, 0.02, 0.04, 0.08 and 0.16. See text for details. (B) A simulation of Georgeson's (1985) matching experiment for determining the apparent contrast of suprathreshold stimuli after adaptation. Each of the smooth curves is derived from the intersections of the perceived contrast curves and the horizontal dashed lines in (A) and may be interpreted as follows. The horizontal coordinate of each curve represents the contrast required for an adapted test stimulus to match an unadapted comparison stimulus in apparent contrast. The contrast of the unadapted stimulus is specified by the vertical coordinate. The solid line at 45 deg represents the match of the unadapted grating to itself. The horizontal dashed line is the unadapted threshold. See text for details. Data points are Georgeson's results. The model responses account fairly well for Georgeson's matching data.

A transducer model for contrast perception The solid line at 45 deg represents matches of the unadapted grating with itself. The horizontal dashed line is the unadapted threshold. Georgeson's matching data for the identical conditions are plotted in the same figure. Model predictions capture the main features of the data and the fit to the data points is much better than that in Georgeson's Fig. 2b (Georgeson, 1985). Simulation curves show the approach to equality in apparent contrast for adapted and unadapted stimuli at contrasts near the adapting contrast as observed by Georgeson. The simulation also accounts for at least part of the change in slope near threshold noted by Georgeson. This change in slope occurs because all perceived contrast curves (Fig. 9A) become nearly parallel as they approach threshold. We performed further tests of the model using smaller 3.7 cycle wide stimulus and found that the perceived contrast curves generated were in agreement with Georgeson's Fig. 1. Thus, with a simple assumption about the effects of adaptation on the W T terms in the transducer function, the model provides a fairly accurate simulation of the main features of Georgeson's adaptation data. Two different contrast mechanisms?

The simulations presented above have shown that a model using the modified Legge and Foley CTF function can account for a wide variety of both threshold and suprathreshold data. However, there are other possible explanations for the gradual disappearance of spatial response pooling as contrast increases. One of these involves the possibility that there are two classes of mechanisms, one of which is sensitive to spatial response pooling and saturates at some intermediate contrast, the other of which has no spatial response pooling component and continues to increase with contrast. If perceived contrast was mediated by a summation of the outputs of these two mechanisms, response pooling would fade away gradually much as in the present model. Our previous publication (Cannon and Fullenkamp, 1988) showed an apparent notch in the tr = 8.0 perceived contrast function at a contrast of about 0.06, where the perceived contrast curves for a = 8.0, 2.0 and 0.5 come together. Inspection of all of the new data for 4c/deg tr = 8 stimuli show similar notches, but they are not in the same place on the contrast axis. We are reluctant to propose this as evidence for two mechanisms because of the inconsistency in position and because we V R 31/6--F

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cannot find a consistent notch in individual perceived contrast functions. We feel it may be due to the normalization technique used to bring perceived contrast curves generated with different numerical scales to a common mean (Cannon, 1984). Perceived contrast and contrast discrimination

Unfortunately, the model does not bridge the gap noted by Legge (1981) between CTFs for contrast perception and CTFs for contrast discrimination. Legge and Foley's (1980) derivation of a CTF was based on the hypothesis that the dip observed in contrast masking functions at low pedestal contrasts was caused by an accelerating contrast transducer function. The CTF used for contrast perception, even without subtracting the constant criterion threshold level discussed earlier, is significantly steeper at low contrasts than the CTF described by Legge and Foley (1980) for contrast discrimination. The dip near detection threshold in the dipper shaped contrast discrimination function would thus be deeper than most of those observed experimentally (Legge & Foley, 1980; Legge & Kersten, 1987; Bradley & Ohzawa, 1986) if contrast discrimination functions were computed from the slope of the perceived contrast CTF. Recent results from Georgeson and Georgeson (1987), however, indicate that the dipper shaped part of the contrast discrimination function may not be attributable to a fixed contrast transducer function. In light of their findings our inability to find a correspondence between the perceived contrast CTF and the hypothetical CTF derived from masking experiments may not be surprising. The model is attractive from a computational, image processing, point of view in that it uses self similar filters and the same CTF across the entire spatial frequency range studied. The self similarity of the filters may have to be modified at low spatial frequencies, during future simulations, since Wilson, McFarlane and Phillips (1983) claim that filter bandwidths below 2 c/deg are significantly wider than those above 2 c/deg. Future studies must also introduce the dependence of W T terms on eccentricity and determine the relationship between the adapting contrast and the ADAPT term. However, over the range of stimulus values tested, the model proved to be generalizable in that it could account for perceived contrasts and thresholds of several stimuli not in the original calibration set.

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REFERENCES Bergen, J. R., Wilson, H. R. & Cowan, J. D. (1979). Further evidence for four mechanisms mediating vision at threshold: Sensitivities to complex gratings and aperiodic stimuli. Journal of the Optical Society of America, 69, 1580-1587. Bradley, A. & Ohzawa, I. (1986). A comparison of contrast detection and discrimination. Vision Research, 26, 991-997. Cannon, M. W. (1984). A study of range effects in free modulus magnitude estimation. Vision Research, 24, 1049-1055. Cannon, M. W. (1985). Perceived contrast in the fovea and periphery. Journal of the Optical Society of America, A 2, 1760-1768. Cannon, M. W. & Fullenkamp, S. C. (1988). Perceived contrast and stimulus size: experiment and simulation. Vision Research, 28, 695-709. Georgeson, M. A. (1985). The effect of spatial adaptation on perceived contrast. Spatial Vision, l, 103-112. Georgeson, M. A. & Georgeson, J. M. (1987). Facilitation and masking of briefly presented gratings: Time-course and contrast dependence. Vision Research, 27, 369-379. Georgeson, M. A. & Sullivan, G. D. (1975). Contrast constancy: Deblurring in human vision by spatial frequency channels. Journal of Physiology, London, 252, 627-656. Legge, G. E. (1981). A power law for contrast discrimination. Vision Research, 21, 457-467. Legge, G. E. & Foley, J. M. (1980). Contrast masking in human vision. Journal of the Optical Society of America, 70, 1458-1471. Legge, G. E. & Kersten, D. (1987). Contrast discrimination in peripheral vision. Journal of the Optical Society of America, A 4, 1594-1598. Quick, R. F. (1974). A vector magnitude model of contrast detection. Kybernetik, 16, 65-67. Robson, J. G. & Graham, N. (1981). Probability summation and regional variation in contrast sensitivity across the visual field. Vision Research, 21, 409-418. Swanson, W. H., Georgeson, M. A. & Wilson, H. R. (1988). Comparison of spatial contrast responses across spatial mechanisms. Vision Research, 28, 457-459. Swanson, W. H., Wilson, H. R. & Giese, S. C. (1984). Contrast matching data predicted from contrast increment thresholds. Vision Research, 24, 63-75. Watson, A. B. (1982). Summation of grating patches indicates many types of detector at one retinal location. Vision Research, 22, 17-26. Watson, A. B. (1983). Detection and recognition of simple spatial forms. In Braddick, O. J. & Sleigh, A. C. (Eds.), Physical and biological processing of images (pp. 100- I 14). Berlin: Springer.

Wilson, H. R. & Bergen, J. R. (1979). A four mechanism model for threshold spatial vision, Vision Research, 19, 19-32. Wilson, H. R. (1980). A transducer function for threshold and suprathreshold human vision. Biological Cybernetics, 38, 171-178. Wilson, H, R., McFarlane, D. K. & Phillips, G. C. (1983). Spatial frequency tuning of orientation selective units estimated by oblique masking. Vision Research, 23, 873-882.

APPENDIX

Model Parameters Model responses from each spatial filter are computed from the equation

R = S CRa+b/(CR a + THRESHa). In turn, THRESH is given by the equation: THRESH = K W T ( f )/( CRMAx/QPool).

CRMAx is the maximum filter response and Qmor is the Quick sum with exponent 3.0 across all filter outputs. As defined in the paper, Qw~oL would be zero when there is no input. Under the no input condition, CR~Ax would also be zero, so the ratio CRMAx/Qr~ooL would be undefined. This can be easily fixed by assigning a small residual value to both QmoL and CRMAx which may be equivalent to some internal noise in the circuits which compute them. Parameters are K = 7.5, a = 3.0, b = 0.5, S = 1.6. Methods for assigning values to K, a, and b are described in the text. S is a scaling factor, common to all filters, that was set rather arbitrarily to scale the outputs to the values shown in the figures. The function W T ( f ) is defined only at the center frequencies of the spatial filters. These center frequencies and the function values are given below:

f 1.000 1.414 2.000 2.828 4.000

WT

f

WT

3.0 2.2 1.6 1.25 1.2

5.656 8.000 11.312 16.000

1.2 1.45 2.0 2.9

CRMAx and QpooL are determined from filter responses, so all inputs to the model are processed first by the array of linear filters. As mentioned in the test, filters are Gabor sine functions with a spatial halfwidth at amplitude I/e of 0.707 cycles. This gives the filters a spatial frequency bandwidth of about 1 octave.