Vision Res. Vol. 31, No. 6, pp. 106%1072, 1991 Printed in Great Britain. All rights reserved
0042-6989/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press pie
A MODEL FOR PERCEIVED SPATIAL FREQUENCY SPATIAL FREQUENCY DISCRIMINATION
AND
DEAN YAGER Department of Vision Sciences, State University of New York, College of Optometry, 100 E. 24th Street, New York, NY 10010, U.S.A. PATRICIA KRAMER* Department of Psychology, College of the Holy Cross, Worcester, MA 01610, U.S.A. (Received 25 August 1985; in revised form 26 September 1989) Abstract--The responses of labelled spatial frequency channels are combined to generate an index of perceived spatial frequency. Spatial frequency discrimination thresholds are shown to be inversely related to the slope of the function of the index vs spatial frequency, when it is plotted on log-log co-ordinates. Spatial frequency
Theory
Channels
Discrimination thresholds
discrimination by the "line-element" model introduced by Helmholtz, and developed by Two visual stimuli must be different on at least Schr6dinger, Stiles, MacAdam, and others. one perceptual dimension if an observer can (Thorough reviews of this approach are given discriminate between them. The discrimination in Wyszecki & Stiles, 1967, and Bouman & task may have many different forms. A few of Walraven, 1972.) In general, these models the possibilities are these: a criterion d' with a postulate that the outputs of three types of "same/different" judgment; criterion proportion photorec~ptors are represented as points in correct responses in ordering the stimuli along three-dimensional space, and discrimination a physical dimension; and criterion proportion threshold is reached when the difference in correct responses in a two-alternative forced- receptor activities caused by two different color choice matching procedure where a third stimuli result in the two stimuli being separated stimulus is identical to one of the alternatives. by a criterion distance in the receptor-response One theoretical approach to discrimination space. performance is to consider each stimulus as a A line element model has been applied successpoint in n-dimensional space (usually Euclidean), fully to data on spatial frequency discrimination in which each dimension represents the by Wilson and Gelb (1984). Their model is based magnitude of response of one of n different on the assumption that there are six fundamental fundamental processes; criterion discrimination spatial frequency channels that encode size (or performance between two stimuli occurs when spatial frequency) at each point of the visual field their locations in this space are separated by a (Wilson, McFarlane & Phillips, 1983). These criterion distance. The crucial task in construct- channels are analogous to the three photoing such a model is, then, to postulate how receptor functions of color vision. Their model physical stimuli may be transformed into, or postulates how the responses of the channels produce, responses in the nervous system that vary as a function of the contrast of the stimuli, can be plotted in the theoretical space. and specifies rules for calculating distances in The most highly developed example of six-dimensional space. The model's theoretical such an approach is the account of wavelength spatial frequency discrimination functions could account for several sets of data (Hirsch & *Also at: Department of Psychology, University Of Hylton, 1982; Wilson & Gelb, 1979; Watson Connecticut, Storrs, Conn., U.S.A. & Robson, 1981). INTRODUCTION
1067
1068
DEAN YAGER and PATRICIAKRAMER
A second major theoretical approach to account for discrimination performance has a fundamentally different basis: by theoretically calculating the appearance of stimuli, one can then account for discrimination performance by relating it to the predicted rate of change of the appearance of stimuli as a function of the physical dimension in question. This class of model is more complete than the line element class, which accounts only for discrimination and matching performance; these models can also account for the supra-threshold phenomenal appearance of stimuli. As with line element models, this class of theories has been developed most extensively in color vision, to account for color appearance and color discriminations. Based on the earlier opponent-colors notions of Ewald Hering, Hurvich and Jameson developed quantitative expressions for relating perceived hue and saturation to the fundamental cone response functions (Hurvich & Jameson, 1955, 1957; Jameson & Hurvich, 1968; Hurvich, 1981). A spectral wavelength discrimination function was then generated by calculating the change in wavelength, for equally bright lights, that would produce a criterion sum of changes in the theoretical hue and saturation coefficient functions. Expressed differently, in regions of the spectrum where hue and/or saturation are changing rapidly as a function of wavelength (i.e. the absolute value of the first derivative of one or both of the functions is large), the predicted threshold difference in wavelength would be small, and vice versa. In this paper we present a set of assumptions and equations which allows us to compute an index of perceived spatial frequency, for simple spatial sine-wave stimuli at a constant contrast. Following the logic of the Hurvich/Jameson model for wavelength discrimination, we generate a predicted spatial frequency discrimination function by considering the slope of the function. (Index of perceived spatial frequency)= f(spatial frequency) in log-log cordinates. Thus, discrimination is determined by a proportionate change in the index.
MODEL FOR SPATIAL FREQUENCY DISCRIMINATION
Assumptions 1. Every point in visual space is processed by a limited number of spatial frequency band-pass filters, or spatial frequency channels (Graham, 1981, 1985). We have chosen to use the same six filters that Wilson and Gelb used in their line element model; the spatial frequency sensitivity profiles of these filters are the Fourier transforms of the difference of two or three spatial Gaussian functions (Wilson & Gelb, 1984). 2. The output of the ith filter is a product of its sensitivity (Si) to the hth spatial frequency (fh), and the contrast of the stimulus, Ch. The range of h is the entire visible spatial frequency spectrum, and the range of Ch and of Si is 0.0 to 1.0. 3. The spatial frequency channels have a compressive contrast transfer function (CTF) which relates the response of the channel, R,(fh), to the output of the linear filter
Ch"S~(L) Ri(fh)
:
C h • S i ( f h ) 7I-
B"
(1)
This is a simplified form of the CTF used by Wilson and Gelb (equation 5 in their paper). B is a positive constant that determines the steepness and maximum response of the filter; it is a free parameter in the model, although we have made the simplifying assumption that it has the same value, 0.5, for all channels.* The sensitivity factor, S, is determined by fitting the linear filters to the contrast sensitivity function: a contrast sensitivity function was measured at the retinal location where the discrimination function was measured, and the channels' sensitivity functions were fit to the contrast sensitivity data using an envelope model of sensitivity. 4. Only the element of a channel located at the optimum spatial position, with the largest response, contributes to detection and frequency discrimination. Thus, no spatial pooling is assumed. 5. The responses of the channels are labelled. This means that there is a unique perceptual correlate to activity in each of the channels. For the present, and for lack of any other non*Various other CTFs have been suggested (e.g. Wilson, arbitrary way of postulating labels, we assume 1980). At the contrast used in the presentexperimentand that a channel's label is the spatial frequency in the theoretical calculations, and with the channel at the peak of the sensitivity function of that sensitivitiesthat we used, it has been shownthat calculations of a perceivedfrequencyindex are quite insensitive channel, M~. This assumption is similar to the to the exact form of the CTF (Davis, Kramer & Yager, "width" labels for different receptive field sizes 1986). proposed by Georgeson (1980).
Model for spatial frequency discrimination 6. The responses of the channels are combined to produce an index of perceived spatial frequency, I(fh)
6
-ll/p
I£{R,(A)}'J
This expression is adapted from the width index of Georgeson (1980),* with the additional provision for a summation exponent, p, which could be any positive number. In the present version of the model, p = 1.0, which specifies equal weighting of the Ris. As p increases above 1.0, the weight of the m a x i m u m R~ becomes greater, and the calculated discrimination function (see below) becomes more and more "bumpy". 7. Since we will be modelling the discrimination of high-contrast stimuli only, we assumed that any random noise in the channels would be insignificant in comparison to the responses to the stimuli, and could be ignored. These seven assumptions are similar to those used to account for perceived spatial frequency and changes in perceived frequency as a function of stimulus contrast, by Gelb and Wilson (1983) and Davis, K r a m e r and Yager (1985), and in a preliminary report on a model for spatial frequency discrimination (Yager, K r a m e r & Richter, 1983).
1069
ence in spatial frequency that produces a criterion discrimination response, using one of the methods mentioned in the Introduction. That is, the change in spatial frequency (A f ) required for discrimination from a standard, fixed frequency ( f ) is divided by the standard frequency, and the resulting fraction. (A f / f ) plotted as a function of spatial frequency. The hypothesis that links the computed spatial frequency index function [equation (2), above] to the empirical spatial frequency discrimination functions is this: given a standard spatial frequencyfhl, a second spatial frequency, fh2, will be discriminated from fht at the criterion level if the absolute value of the expression I ( A I ) -- I(fh2) I(fhl )
(3)
is at a criterion value; that is, if there is a criterion proportional change in I ( f ) . The value of (A f / f ) may then be computed by determining the two spatial frequencies that are needed to generate the two values of the index of perceived spatial frequency in the above expression. (See Appendix for a sample calculation.) Thus, good discrimination (small values of A f / f ) is predicted where the index is changing rapidly (steep slope) as a function of spatial frequency, and vice versa.* The top panel of Fig. 1 is a schematic plot of equation (2), in log-log coordinates; regions
EXPERIMENTAL DATA AND THE LINKING HYPOTHESIS Spatial frequency discrimination thresholds are usually expressed as the proportional differ*Georgeson's (1980) expression for combining weighted labels from different channels was in terms of width. In equation (2), M = (1/W), where W is the width label from Georgeson. Thus, the denominator could have been written as [Rj(fh)][W], which is equivalent to the numerator of Georgeson's expression. We then took the reciprocal of this width index after weighting and summing the width labels, thus converting it to a spatial frequency index. See Davis et al. (1986) and Davis (1990) for a thorough discussion of different versions of the index. They showed that the computed index is quite insensitive to the exact form of the expression, for moderate levels of contrast. tThomas (1985) points out that spatial frequency discrimination cannot be based on the subjectiveindex alone in all circumstances: the subjectiveindex is contrast dependent, while discriminations sometimesare unaffected by random variations in contrast. His comment applies to low-contrast stimuli which are not detected on every trial, whereas we used stimuli that were always easily detected (contrast = 0.30).
!i v
| o~5 ,io ,~o ,io sio cycles/degree
Fig. 1. Top: theoretical index of perceived frequency as a function of physical frequency,calculated from equation (2). Bottom: theoretical spatial frequency discrimination function (see text, and Appendix).
1070
DEAN YAGER and PATRICIAKRAMER
homogeneous. Because of the increase in receptive field size and the shift of the peak sensitivities of spatial frequency channels toward lower spatial frequencies in the periphery (Wilson & Bergen, 1979), the linear filters in our model (assumption 1) have been scaled toward lower spatial frequencies from the filters used by Wilson and Gelb (1984). The six filters have peak sensitivities at about 0.4, 0.85, 1.4, 2.0, 4.0, and 8.0 c/deg for observers DY, ML and ER, and shifted by a further 20% for observer MW. This scaling of channels in the periphery relative to those in the central fovea is consistent with the estimates of Wilson and Bergen (1979). The responses of each channel were computed at 200 points on the spatial frequency axis, from 0.1 to 10.0c/deg, with C =0.30, at equal log unit intervals [Equation (1)]. I(f), the index of perceived spatial frequency, was then computed for each of the frequencies, by equation (2). The frequency index function was differentiated numerically, and the logarithm of the reciprocal of the slope was taken (see Appendix). This function was then scaled multiplicatively to fit, by eye, the psychophysical frequency discrimination function for each observer. At this stage in the development of the model, we did not feel that the effort entailed in finding a statistical best-fit was justified; rather, the main point of this paper is to demonstrate that a
of relatively high and low slope alternate. A predicted Af/f function is shown in the bottom panel of Fig. 1, which was computed by the method described in the previous paragraph and in the Appendix. FITTING THEORY TO DATA
Since many visual functions change rapidly within a few degrees of the center of the visual field, a large stimulus such as that used by Hirsch and Hylton (1982) and Richter and Yager (1984) in their empirical studies of spatial frequency discrimination will, if it is centered on the fovea, fall onto a variety of different sets of visual receptive fields that have different ranges of spatial frequency sensitivity. One consequence of this condition could be that variations in spatial frequency discrimination across the spectrum may be obscured, since several sets of spatial frequency channels could be active, and discrimination could be mediated by the most sensitive set of channels for each part of the spectrum, as suggested by Richter and Yager (1984) and Woodward, Ettinger and Yager (1985). Thus, we have chosen to model data on frequency discrimination that had been obtained with a 3 by 4 deg rectangular stimulus centered at 10deg above the fovea (Richter & Yager, 1984), where the visual field is much more
L/
DY .12 .10
.08,
0042-6989/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press pie
A MODEL FOR PERCEIVED SPATIAL FREQUENCY SPATIAL FREQUENCY DISCRIMINATION
AND
DEAN YAGER Department of Vision Sciences, State University of New York, College of Optometry, 100 E. 24th Street, New York, NY 10010, U.S.A. PATRICIA KRAMER* Department of Psychology, College of the Holy Cross, Worcester, MA 01610, U.S.A. (Received 25 August 1985; in revised form 26 September 1989) Abstract--The responses of labelled spatial frequency channels are combined to generate an index of perceived spatial frequency. Spatial frequency discrimination thresholds are shown to be inversely related to the slope of the function of the index vs spatial frequency, when it is plotted on log-log co-ordinates. Spatial frequency
Theory
Channels
Discrimination thresholds
discrimination by the "line-element" model introduced by Helmholtz, and developed by Two visual stimuli must be different on at least Schr6dinger, Stiles, MacAdam, and others. one perceptual dimension if an observer can (Thorough reviews of this approach are given discriminate between them. The discrimination in Wyszecki & Stiles, 1967, and Bouman & task may have many different forms. A few of Walraven, 1972.) In general, these models the possibilities are these: a criterion d' with a postulate that the outputs of three types of "same/different" judgment; criterion proportion photorec~ptors are represented as points in correct responses in ordering the stimuli along three-dimensional space, and discrimination a physical dimension; and criterion proportion threshold is reached when the difference in correct responses in a two-alternative forced- receptor activities caused by two different color choice matching procedure where a third stimuli result in the two stimuli being separated stimulus is identical to one of the alternatives. by a criterion distance in the receptor-response One theoretical approach to discrimination space. performance is to consider each stimulus as a A line element model has been applied successpoint in n-dimensional space (usually Euclidean), fully to data on spatial frequency discrimination in which each dimension represents the by Wilson and Gelb (1984). Their model is based magnitude of response of one of n different on the assumption that there are six fundamental fundamental processes; criterion discrimination spatial frequency channels that encode size (or performance between two stimuli occurs when spatial frequency) at each point of the visual field their locations in this space are separated by a (Wilson, McFarlane & Phillips, 1983). These criterion distance. The crucial task in construct- channels are analogous to the three photoing such a model is, then, to postulate how receptor functions of color vision. Their model physical stimuli may be transformed into, or postulates how the responses of the channels produce, responses in the nervous system that vary as a function of the contrast of the stimuli, can be plotted in the theoretical space. and specifies rules for calculating distances in The most highly developed example of six-dimensional space. The model's theoretical such an approach is the account of wavelength spatial frequency discrimination functions could account for several sets of data (Hirsch & *Also at: Department of Psychology, University Of Hylton, 1982; Wilson & Gelb, 1979; Watson Connecticut, Storrs, Conn., U.S.A. & Robson, 1981). INTRODUCTION
1067
1068
DEAN YAGER and PATRICIAKRAMER
A second major theoretical approach to account for discrimination performance has a fundamentally different basis: by theoretically calculating the appearance of stimuli, one can then account for discrimination performance by relating it to the predicted rate of change of the appearance of stimuli as a function of the physical dimension in question. This class of model is more complete than the line element class, which accounts only for discrimination and matching performance; these models can also account for the supra-threshold phenomenal appearance of stimuli. As with line element models, this class of theories has been developed most extensively in color vision, to account for color appearance and color discriminations. Based on the earlier opponent-colors notions of Ewald Hering, Hurvich and Jameson developed quantitative expressions for relating perceived hue and saturation to the fundamental cone response functions (Hurvich & Jameson, 1955, 1957; Jameson & Hurvich, 1968; Hurvich, 1981). A spectral wavelength discrimination function was then generated by calculating the change in wavelength, for equally bright lights, that would produce a criterion sum of changes in the theoretical hue and saturation coefficient functions. Expressed differently, in regions of the spectrum where hue and/or saturation are changing rapidly as a function of wavelength (i.e. the absolute value of the first derivative of one or both of the functions is large), the predicted threshold difference in wavelength would be small, and vice versa. In this paper we present a set of assumptions and equations which allows us to compute an index of perceived spatial frequency, for simple spatial sine-wave stimuli at a constant contrast. Following the logic of the Hurvich/Jameson model for wavelength discrimination, we generate a predicted spatial frequency discrimination function by considering the slope of the function. (Index of perceived spatial frequency)= f(spatial frequency) in log-log cordinates. Thus, discrimination is determined by a proportionate change in the index.
MODEL FOR SPATIAL FREQUENCY DISCRIMINATION
Assumptions 1. Every point in visual space is processed by a limited number of spatial frequency band-pass filters, or spatial frequency channels (Graham, 1981, 1985). We have chosen to use the same six filters that Wilson and Gelb used in their line element model; the spatial frequency sensitivity profiles of these filters are the Fourier transforms of the difference of two or three spatial Gaussian functions (Wilson & Gelb, 1984). 2. The output of the ith filter is a product of its sensitivity (Si) to the hth spatial frequency (fh), and the contrast of the stimulus, Ch. The range of h is the entire visible spatial frequency spectrum, and the range of Ch and of Si is 0.0 to 1.0. 3. The spatial frequency channels have a compressive contrast transfer function (CTF) which relates the response of the channel, R,(fh), to the output of the linear filter
Ch"S~(L) Ri(fh)
:
C h • S i ( f h ) 7I-
B"
(1)
This is a simplified form of the CTF used by Wilson and Gelb (equation 5 in their paper). B is a positive constant that determines the steepness and maximum response of the filter; it is a free parameter in the model, although we have made the simplifying assumption that it has the same value, 0.5, for all channels.* The sensitivity factor, S, is determined by fitting the linear filters to the contrast sensitivity function: a contrast sensitivity function was measured at the retinal location where the discrimination function was measured, and the channels' sensitivity functions were fit to the contrast sensitivity data using an envelope model of sensitivity. 4. Only the element of a channel located at the optimum spatial position, with the largest response, contributes to detection and frequency discrimination. Thus, no spatial pooling is assumed. 5. The responses of the channels are labelled. This means that there is a unique perceptual correlate to activity in each of the channels. For the present, and for lack of any other non*Various other CTFs have been suggested (e.g. Wilson, arbitrary way of postulating labels, we assume 1980). At the contrast used in the presentexperimentand that a channel's label is the spatial frequency in the theoretical calculations, and with the channel at the peak of the sensitivity function of that sensitivitiesthat we used, it has been shownthat calculations of a perceivedfrequencyindex are quite insensitive channel, M~. This assumption is similar to the to the exact form of the CTF (Davis, Kramer & Yager, "width" labels for different receptive field sizes 1986). proposed by Georgeson (1980).
Model for spatial frequency discrimination 6. The responses of the channels are combined to produce an index of perceived spatial frequency, I(fh)
6
-ll/p
I£{R,(A)}'J
This expression is adapted from the width index of Georgeson (1980),* with the additional provision for a summation exponent, p, which could be any positive number. In the present version of the model, p = 1.0, which specifies equal weighting of the Ris. As p increases above 1.0, the weight of the m a x i m u m R~ becomes greater, and the calculated discrimination function (see below) becomes more and more "bumpy". 7. Since we will be modelling the discrimination of high-contrast stimuli only, we assumed that any random noise in the channels would be insignificant in comparison to the responses to the stimuli, and could be ignored. These seven assumptions are similar to those used to account for perceived spatial frequency and changes in perceived frequency as a function of stimulus contrast, by Gelb and Wilson (1983) and Davis, K r a m e r and Yager (1985), and in a preliminary report on a model for spatial frequency discrimination (Yager, K r a m e r & Richter, 1983).
1069
ence in spatial frequency that produces a criterion discrimination response, using one of the methods mentioned in the Introduction. That is, the change in spatial frequency (A f ) required for discrimination from a standard, fixed frequency ( f ) is divided by the standard frequency, and the resulting fraction. (A f / f ) plotted as a function of spatial frequency. The hypothesis that links the computed spatial frequency index function [equation (2), above] to the empirical spatial frequency discrimination functions is this: given a standard spatial frequencyfhl, a second spatial frequency, fh2, will be discriminated from fht at the criterion level if the absolute value of the expression I ( A I ) -- I(fh2) I(fhl )
(3)
is at a criterion value; that is, if there is a criterion proportional change in I ( f ) . The value of (A f / f ) may then be computed by determining the two spatial frequencies that are needed to generate the two values of the index of perceived spatial frequency in the above expression. (See Appendix for a sample calculation.) Thus, good discrimination (small values of A f / f ) is predicted where the index is changing rapidly (steep slope) as a function of spatial frequency, and vice versa.* The top panel of Fig. 1 is a schematic plot of equation (2), in log-log coordinates; regions
EXPERIMENTAL DATA AND THE LINKING HYPOTHESIS Spatial frequency discrimination thresholds are usually expressed as the proportional differ*Georgeson's (1980) expression for combining weighted labels from different channels was in terms of width. In equation (2), M = (1/W), where W is the width label from Georgeson. Thus, the denominator could have been written as [Rj(fh)][W], which is equivalent to the numerator of Georgeson's expression. We then took the reciprocal of this width index after weighting and summing the width labels, thus converting it to a spatial frequency index. See Davis et al. (1986) and Davis (1990) for a thorough discussion of different versions of the index. They showed that the computed index is quite insensitive to the exact form of the expression, for moderate levels of contrast. tThomas (1985) points out that spatial frequency discrimination cannot be based on the subjectiveindex alone in all circumstances: the subjectiveindex is contrast dependent, while discriminations sometimesare unaffected by random variations in contrast. His comment applies to low-contrast stimuli which are not detected on every trial, whereas we used stimuli that were always easily detected (contrast = 0.30).
!i v
| o~5 ,io ,~o ,io sio cycles/degree
Fig. 1. Top: theoretical index of perceived frequency as a function of physical frequency,calculated from equation (2). Bottom: theoretical spatial frequency discrimination function (see text, and Appendix).
1070
DEAN YAGER and PATRICIAKRAMER
homogeneous. Because of the increase in receptive field size and the shift of the peak sensitivities of spatial frequency channels toward lower spatial frequencies in the periphery (Wilson & Bergen, 1979), the linear filters in our model (assumption 1) have been scaled toward lower spatial frequencies from the filters used by Wilson and Gelb (1984). The six filters have peak sensitivities at about 0.4, 0.85, 1.4, 2.0, 4.0, and 8.0 c/deg for observers DY, ML and ER, and shifted by a further 20% for observer MW. This scaling of channels in the periphery relative to those in the central fovea is consistent with the estimates of Wilson and Bergen (1979). The responses of each channel were computed at 200 points on the spatial frequency axis, from 0.1 to 10.0c/deg, with C =0.30, at equal log unit intervals [Equation (1)]. I(f), the index of perceived spatial frequency, was then computed for each of the frequencies, by equation (2). The frequency index function was differentiated numerically, and the logarithm of the reciprocal of the slope was taken (see Appendix). This function was then scaled multiplicatively to fit, by eye, the psychophysical frequency discrimination function for each observer. At this stage in the development of the model, we did not feel that the effort entailed in finding a statistical best-fit was justified; rather, the main point of this paper is to demonstrate that a
of relatively high and low slope alternate. A predicted Af/f function is shown in the bottom panel of Fig. 1, which was computed by the method described in the previous paragraph and in the Appendix. FITTING THEORY TO DATA
Since many visual functions change rapidly within a few degrees of the center of the visual field, a large stimulus such as that used by Hirsch and Hylton (1982) and Richter and Yager (1984) in their empirical studies of spatial frequency discrimination will, if it is centered on the fovea, fall onto a variety of different sets of visual receptive fields that have different ranges of spatial frequency sensitivity. One consequence of this condition could be that variations in spatial frequency discrimination across the spectrum may be obscured, since several sets of spatial frequency channels could be active, and discrimination could be mediated by the most sensitive set of channels for each part of the spectrum, as suggested by Richter and Yager (1984) and Woodward, Ettinger and Yager (1985). Thus, we have chosen to model data on frequency discrimination that had been obtained with a 3 by 4 deg rectangular stimulus centered at 10deg above the fovea (Richter & Yager, 1984), where the visual field is much more
L/
DY .12 .10
.08,
