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Receptive-field-like functions inferred from large-area psychophysical measurements
Vision Res. Vol. 25. No. 12, pp. 1895-1900. 1985 Printed in Great Britain. All n&s resencd
Copynghr
0042-6989 85 53 013+ 0.00 C= 1985 Pergamon Press Ltd
RECEPTIVE-FIELD-LIKE FUNCTIONS INFERRED FROM LARGE-AREA PSYCHOPHYSICAL MEASUREMENTS D. H. KELLY Visual Sciences Program, SRI International,
333 Ravenswood Avenue, Menlo Park, CA 94025, U.S.A.
(Received 19 October 1984; in revised form I July 1985) means of quasi-sinusoidal, circular stimulus patterns. frequency modulated to correct for the radial component of retinal inhomogeneity. we attempt to make all eccentricities within a l6-deg field contribute equally to the threshold response. Wide-field contrast-sensitivity data obtained with these frequency-modulated stimuli are modeled using a canonical form of local contrast sensitivity function (CSF), scaled linearly with eccentricity. Calculating Fourier transforms of the constant-velocity, local CSF. we obtain line-spread and point-spread functions that can also be interpreted as receptive-field responses at various eccentricities. These results are compared to other data on local spatial processing in the retina. Abstract-By
Contrast sensitivity inhomogeneity
Receptive fields
Frequency-modulated
INTRODUCTION
measurements of centerfields in the vertebrate retina have motivated many types of psychophysical experiments that search for quantitative relations between the psychophysics and the physiology of local spatial processing in the human visual system. Well-known examples of such psychophysical studies include the sensitization-disk experiments of Westheimer (1967) and others, the fine-line experiments of Limb and Rubinstein (1977) and of Wilson (1978) and his collaborators (e.g. Hines, 1976), and the thin-annulus experiments of Blommaert and Roufs (1981). The techniques used in these studies usually involve two parameters, an independent and a dependent variable, which yield a threshold function that is more or less localized in the visual field. Another technique for inferring the nature of spatial processing depends on the measurement of thresholds for detecting sinewave gratings. This technique also provides two parameters (contrast vs spatial frequency). However, sinewave measurements cannot be accurately localized in the visual field, particularly if they are extended to the lowest spatial frequencies for which the visual process has a significant response (Kelly, 1965, 1977). If spatial processing were uniform across the retina, then the distinction between large-area and local responses might be less important, but clearly that is not the case. The structure and function of the retina (e.g. the sizes of receptive fields) change smoothly but drastically within the first few degrees from the fovea (Fischer, 1973; Wilson and Sherman, 1976). However, a fovea-centered stimulus field only a few degrees in diameter is still too small to study the low-frequency range of interest. Electrophysiological surround receptive
targets
Optic flow
Retinal
Nevertheless, the possibility of deriving local properties from large-area sinewave thresholds has remained a tantalizing one, partly because the Fourier transform of the sinewave contrast-sensitivity function resembles, at least qualitatively, the profile of a center/surround retinal receptive field. This was apparent in the early results of Pate1 (1966), who first calculated line-spread functions from sinewave thresholds. The notion that certain properties of the retina (spatial, temporal, and chromatic) may control the threshold in some simple detection tasks does not conflict with the more elaborate forms of visual processing revealed by more complex paradigms such as pattern adaptation (Blakemore and Campbell, 1969) or masking (Stromeyer and Julesz, 1972). These higher visual processes may be irrelevant to the unmasked, unadapted, detection thresholds considered here, so that the form of the threshold function for the detection of sinewave stimuli depends on properties of the retina; i.e. the rest of the visual system acts as a null detector in these experiments (Kelly and Burbeck, 1984). However, the systematic increase in receptive-field size with eccentricity has been an intractable problem in trying to interpret contrast sensitivity functions in terms of local spatial processing. Here we present a new method of coping with this aspect of retinal inhomogeneity. Using stimulus patterns that compensate over a wide field for the eccentricitydependent component of retinal structure, we obtain a canonical form for retinal receptive fields that seems to be at least as accurate as the results of the line-spread and point-spread studies mentioned above (Westheimer, 1967; Hines, 1976; Limb and Rubinstein, 1977; Wilson, 1978; Blommaert and Roufs, 198 I).
1895
D. H.
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The simplest interpretation of our results would be that the sinewave thresholds are governed by a single type of receptive field (e.g. an X-like ganglion crli) that changes size only with eccentricity. Although the presence of multiple retinaf mechanisms (e.g. X. Y and W ceils in the cat) serving the same, or nearly the same. retinal locus is well established. there is no reason to suppose that they all have identical thresholds. It seems possible that an X-like mechanism is dominant at threshold, although a Y-like mechanism may be more sensitive at suprathreshold levels (Shapiey and Victor, 1978; Burbeck. 1981: Burbeck and Kelly. 1981). The contrast-sensitivity data presented here difter from those of our previous study with radially scaled stimuli (Kelly, 1985b) in two important ways. First, the scale factor of the present stimuli was chosen to match the estimated radial inhomogeneity of the author’s right eye, so that the resulting sensitivity data can be directly transformed into the canonical spread function of the present mode!. (In the previous study, severat different stimulus scale factors were used to test this model but none of them exactly matched the subject’s retinal scale factor). Second, in collecting the present data, the temporal frequency of this s~atiafly-optimized stimulus was not hetd constant (as in Kelly, 1985b). but was varied with spatial frequency in order to hold the retinal t&city constant. We have shown elsewhere (Kelly, i979, 19g5a) that this constant-velocity technique is the only form of controlled temporal variation that can match the contrast-sensitivity data obtained with natural eye movements. Before calculating the main results of the present study, we therefore compared the scaled-target data obtained with this constantvelocity technique to scaled-target data obtained at a constant temporal frequency. Fortunately, we found that both types of data can be fitted by the model used previously (Kelly, 1985b), with a modest change in the value of one parameter.
KELLY
hence the o~t~murn optic-Row gratings) for a given subject has been described in detail elsewhere (Kelly, 1985b). When the scale factor of the target matches the scale factor of the subject‘s retina. the model predicts that the measured contrast sensitivity function (CSF) will assume the same form as the local CSF at any point on the retina. This standard, canonical form is given by SUllt,,,)
= V‘ifLY exp( - P/‘L/“,,,)
(1)
The constant p controls the ~dndwidth of this function, which depends on the temporal properties of the stimufus. The locaf normalizing spatial frequency, / maxiis given by fm.,, = _/,i( 1 f Kr )
(2)
where f, is the frequency of maximum sensitivity at the fovea, r is eccentricity in degrees, and K is the retinal scale factor (Kelly, 1985b). Thus the local ~~x~rnurn shifts by one octave at the eccentricity, f = K-1. (For the author’s right eye, fL.= 3.5 cyclesideg and K-’ = 2.3 deg. in reasonably good agreement with other inhomogeneity measures.) RESULTS {A )
Constant ceiociry I‘S consranl temporal fkequency
Figure I shows two sets of measurements of contrast sensitivity as a function of (fovea]) spatial frequency, taken with circular FM stimuli matching the subject’s estimated retinal scale factor (and hence approximating the local CSF just described). Both are plotted as functions of the local spatial frequency at the fovea [where f,,, =f, in equation (?)I. For the
METHODS
We controlled the effects of eye movements and other potential sources of unwanted temporal variation by stabilizing the retinal image, thereby keeping our circular targets always concentric with the subject’s point of fixation. The desired temporal variation was introduced at the distal stimulus in the form of radial motion (i.e. optic flow), which induced little or no optokinetic nystagmus. The local temporal frequency was held constant over the entire I6-deg field. Any one of our circularly symmetric stimulus patterns is locally sinusoidaf, but its period increases linearly with eccentricity. With the appropriate scale factor for this radial variation, such a frequencymodulated (FM) pattern may provide nearly equal stimulation at all eccentricities. A circular-retina model that permits us to derive this scale factor {and
1*
/
a.2
05 WATIAL
I
1
2
1
/
5
10
FAEQUENCYsy/deg
Fig. I. Contrast sensitivity as a function of (fovea0 spatial frequency under two temporal conditions. Squares: constant temporal frequency (0.S Hz). Circles: temporal frequency varied with spatial frequency to keep v&city constant (0.19deg/sec at fovea). Circufar, linearly scaled patterns were used. as illustrated elsewhere (Kelly, 1985b). Temporal frequency did nat vary with radius in either case; hence veiocity increased linearly with radius in both cases (as in “optic Row’“). Solid and dashed curves are both plots of equation (I). with different values of parameter p.
Receptive-field function
1897
Fig. 2. Fovea1 version of the local contrast sensitivity function (at 0.19 deg/sec), according to the present model. This surface represents the same data as the dashed curve of Fig. I, but plotted in linear coordinates as a figure of revolution in two-dimensional frequency space.
open squares, temporal frequency was held constant at 0.5 Hz as in our previous study (Kelly, 1985b). (However, these are new data with K-’ = 2.3 deg.) For the open circles, temporal frequency was varied with spatial frequency to hold the radial velocity constant at any given eccentricity; at the fovea, this local velocity was always 0.19 deg/sec (in the range of the subject’s natural drift motions). Both curves in Fig. 1 are theoretical, being plots of equation (1) with the appropriate parameter values; the fit is very good in both cases. The decrease in the bandwidth of the constant-velocity curve was obtained by increasing the parameter p, from I to 1.4. The experimental and theoretical results shown in Fig. 1 confirm that the author’s model of retinal inhomogeneity (Kelly, 1985b) fits the constantvelocity data for scaled circular targets as well as the constant-temporal-frequency data from which the model was derived. (B) Line -spread and point -spread ~~~cti~ffs We have calculated both line-spread and pointspread functions from the inhomogeneity-corrected, constant-velocity (dashed) curve in Fig. I. The linespread function is calculated as the one-dimensional Fourier transform of the CSF, scaled for any desired eccentricity by equation (2). Results in this form are
useful for comparison with data on local subthreshold summation between fine lines (Hines, 1976; Limb and Rubinstein, 1977; Wilson, 1978). The point-spread function is calculated from our essentially one-dimensional data by treating the local CSF [equation (I) and Fig. I] as the radial profile of a circularly symmetric function in two-dimensional frequency space. Figure 2 shows a perspective view of that function, plotted in linear frequency units (which compress the CSF at low frequencies and expand it at high frequencies, relative to the log units of Fig. I). The frequency scales refer to the local CSF at the fovea (where f,,, =f=). For both spread-function calculations, we used the scaled-target CSF at a fovea1 velocity of 0.19 deg/sec at the fovea. With unscaled, rectilinear gratings, such a slow drift rate gives results that are very similar to the CSF obtained by viewing a stationary grating with natural eye movements (Kelly, 1979). Hence, the constant-velocity data (dashed curve in Fig. 1) should more nearly represent the way receptive fields function with natural eye movements. Figure 3(a) shows three line-spread functions calculated as one-dimensional Fourier transforms of equation (I) and scaled in accord with equation (2) to represent eccentricities of 0, 3, and 6 deg. The positive and negative areas of these functions are
D. H.
KELLY
0"
(b)
Fig. 3. (a) Retinal line-spread functions at eccentricities of 0, 3 and 6 deg. according to the present model. These functions were calculated as one-dimensional Fourier transforms of equation (I) with p = 1.4. and scaled in terms of equation (2) with l/K = 2.3 deg. (b) Retinal point-spread function at the fovea, calculated as the two-dimensional Fourier transform of the surface shown in Fig. 2. invariant with eccentricity and they integrate to zero; both properties follow from the form of equation (I). (If the local sensitivity is significantly anisotropic, then these results represent line-spread only in the radial direction, because our circular targets have no tangential component.) A standard criterion for comparing the widths of sombrero-shaped line-spread functions is the distance between zero-crossings; i.e. the width of the excitatory zone. For the calculated functions shown in Fig. 3(a), these widths are 4.4 min arc at the fovea, 10.7 min at an eccentricity of 3 deg, and 16.8 min at 6deg. Because these data were compensated for inhomogeneity, the fovea1 width is significantly smaller than the value we obtained earlier by transforming the CSF for uniform, homogeneous gratings (Kelly, 1977). However, it seems to be in good agreement with fovea1 line-spread functions inferred by an entirely different method; namely, subthreshold summation between adjacent line stimuli. With slow temporal modulation, Wilson (1978) obtained a foveai width of 5.6 min. Comparable results of
several such studies vary from a maximum of 6.6 min (Hines, 1976) to a minimum of 3.9 min (subject JOL of Limb and Rubinstein, 1977). The other line-spread functions of Fig. 3(a), at eccentricities of 3” and 6’, are also in good agreement with comparable subthreshold summation data. For example, at 2.5 deg eccentricity, Hines (1976) obtained a zero-crossing width of 8.7 min; at 3.3 deg, the width of Limb and Rubinstein’s line-spread function was about 10min for both their subjects. Figure 3(b) shows the point-spread function calculated as the two-dimensional Fourier transform of the local CSF shown in Fig. 2. Like the CSF of our isotropic model, the point-spread function must also be a figure of rotation. The distance scales represent the foveai version of this canonical function; its change of scale with eccentricity can be inferred from the line-spread functions shown in the same figure. DISCUSSION
Although the point-spread function contains no information that is not also implicit in the line-spread
function {given the assumption of a linear, isotropic mechanism). this two-dimensional response seems easier to compare with the spatial responses of retinal ganglion cells, which are often expressed in a similar form. Even the relatively broad. flat shape of the negative surround illustrated in Fig. 3(b) has been reported in X-type ganglion cells (Shapiey, 1985). To evaluate how well our wide-field results with scaled stimuli actually represent local properties. the most useful comparisons are with local measurements of spatial processing in the human retina that are presumably not distorted by retinal inhomogeneity. Thus, the agreement between our line-spread functions and the subthreshofd summation results described above provide corroboration for the corresponding point-spread function as well. Here we
consider some related measurements with local, circular targets. Blommaert and Roufs (1981) attempted to carry out the two-dimensional, circular analog of the parallel-line experiments. With one subject, they obtained a fovea1 point-spread function that was narrower than would be expected from most of the parallel-line results. However, it agreed fairly well with their calculated prediction from Limb and Rubinstein’s (1977) subject JOL. Westheimer’s (1967) sensitization experiments also used circularly symmetric stimuli, providing other results that can be compared with our point-spread functions. In this case, the critical diameter of his sensitizing disk, where the test threshold is maximums should equal the zero-crossing diameter of the pointspread function (under our standard assumptions). Note that these diameters must be greater than the line-spread widths of Fig. 3(a). At eccentricities of 0, 2.5, and 5deg, Westheimer (1967) obtained critical diameters of 5, 14.2, and 2 I .3 min arc (his Fig. 4), compared to zero-crossing diameters of 6.3, 14.6, and 22.9min for the pointspread functions calculated here at the same eccentricities; his doubling eccentricity was between I and 2 deg, compared to our vaiue of 2.3” for K-r. Thus a perfect fit to Westheimer’s data would require slightly smaller values off, and K than those used in Fig. 3(a). Averaged over a 16-deg field, however, these two changes tend to compensate each other, resulting in only a second-order effect in fitting data like those of Fig. I. The CSF measured with our scaled, inhomogeneous stimuli differs in both shape and bandwidth from the CSF measured with ordinary, homogeneous gratings over the same area. The magnitude and direction of these differences can be explained by the hypothesis (Van Doorn ef al., 1972) that the inhomogeneity-corrected thresholds are affected by only one size of receptive field at each eccentricity, Although that post&ate can be relaxed somewhat, the form of the scaled CSF implies a strict limit on the size distribution of the receptive fields that affect the threshold at any given eccentricity.
This local distribution is evidently much more homogeneous than the distribution of receptive-field
sizes over the 16’ field used in our experiments. Since our inferred spread-functions are dependent on a model of retinal inhomogeneity (Kelly. 1985b). and the local-target measurements are not. agreement between these two types of results tends to confirm that our technique satisFactoriIy compensates for the major component of the inhomogeneity. as the model predicts. Like most psychophysical models, this one is necessarily an oversimplified version of what is known about the relevant visual physiology. Nevertheless, it represents a significant step toward remedying two important defects (spatial uniformity and temporal control) of previous attempts to infer receptive-~eld~l~ke properties from sinewave thresholds. We expect these results to be useful in computational modeling of the visual process. Ackno,rlerlgemenls-This work was supported by National Institutes of Health Grant EY 01 IX?. The computer graphics were done by John Peters. REFERENCES
Bfaketnore C. and Campbell F. W. (1969) On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images. 1. PhpioL, Lond. 203, 237-260. Blommaert E. J. J. and Roufs J. A. J. (1981) The fovea1 point spread function as a determinant for detail vision. Vision Res. 21, 1223-1233. Burbeck C. A. (1981) Criterion-free pattern and flicker thresholds. f. opt. SW. Rm. 7f, t343-1350. Burbeck C. A. and Kelly D. H. (1981) Contrast gain
measurements and the trans~ent~sust~ineddichotomy. J. opr. sot. Am. If, t335-t342. Fischer B. (1973) Overlap of receptive field centers and representation of the visual field in the cat’s optic tract. Vision Res. 13, 2113-2 120. Hines M. (1976) Line spread function variation near the fovea. Vision Res. f6, 567-572. Kelly D. Ii. (1965) Spatial frequency, bandwidth and resolution. Ap$. Upf. 4% 435-437. Kelly D, H. (1977) Visual contrast sensitivity. Upiira ~crrt 24, llx-129. Kelly D. H. (1919) Motion and vi&n. II Stabilized spatiotemporal threshold surface. J. apt. Sm. Am. 69, 1340-1349. Kelly D. H. (1985a) Visual processing of moving stimuli. J. opt. Sot. Am. A 2, 216-225. Kelly D. H. (1985b) Retinal inhomogeneity. III Circularretina theory. J. opt. Sot. Am. A 2, 810-8 19. Kelly D. H. and Burbeck C. A. (1984) Critical problems in spatial vision. Crit. Rec. Binned. Engng ?fE, 125~t77. Limb f. 0. and Rubinstein 6. B. (1977) A mode1 of threshold vision ~~corpo~t~~g inhomogene~ty of the visual field. Vision Rex 17, 571-584. Pate1 A. S. (1966) Spatial resolution by the human visual system. The effect of mean retinal illuminance. J. opt. Sot. Am. 56, 689-694. Shapley R. M. (1985) Parallel channels from retina to brain. Scienc. Am. In press. Shapley R. M. and Victor J. D. (1978) The et&t of contrast on the transfer properties of cat retinal ganglion cells. 1. Pfzysiol.. tonbe 285, 275-293. Stromeyet C. F. IfI and Julesz B. (1972) Spatial-frequency masking in vision: critical bands and spread of masking. f. opt, Sm. Am. 62, i221-1232. Van Down A. J., Koenderink J. J. and Bouman IM. A.
D. H. (1972) The influence of retinal inhomogeneity on the perception of spatial patterns. Kxhernelik 10, 223-230. Westheimer G. (1967) Spatial interaction in human cone vision. J. Physiol. 190, 139-153. Wilson H. R. (1978) Quantitative characterization of two
KELLY
types of line-spread function near the fovea. Visian Rex. 18. 971-981. Wilson J. R. and Sherman S. IV. (1976) Receptive-field characteristics of neurons in cat striate cortex; changes with visual field eccentricity. J. !\‘europh,vsiol. 39.517-533.
Copynghr
0042-6989 85 53 013+ 0.00 C= 1985 Pergamon Press Ltd
RECEPTIVE-FIELD-LIKE FUNCTIONS INFERRED FROM LARGE-AREA PSYCHOPHYSICAL MEASUREMENTS D. H. KELLY Visual Sciences Program, SRI International,
333 Ravenswood Avenue, Menlo Park, CA 94025, U.S.A.
(Received 19 October 1984; in revised form I July 1985) means of quasi-sinusoidal, circular stimulus patterns. frequency modulated to correct for the radial component of retinal inhomogeneity. we attempt to make all eccentricities within a l6-deg field contribute equally to the threshold response. Wide-field contrast-sensitivity data obtained with these frequency-modulated stimuli are modeled using a canonical form of local contrast sensitivity function (CSF), scaled linearly with eccentricity. Calculating Fourier transforms of the constant-velocity, local CSF. we obtain line-spread and point-spread functions that can also be interpreted as receptive-field responses at various eccentricities. These results are compared to other data on local spatial processing in the retina. Abstract-By
Contrast sensitivity inhomogeneity
Receptive fields
Frequency-modulated
INTRODUCTION
measurements of centerfields in the vertebrate retina have motivated many types of psychophysical experiments that search for quantitative relations between the psychophysics and the physiology of local spatial processing in the human visual system. Well-known examples of such psychophysical studies include the sensitization-disk experiments of Westheimer (1967) and others, the fine-line experiments of Limb and Rubinstein (1977) and of Wilson (1978) and his collaborators (e.g. Hines, 1976), and the thin-annulus experiments of Blommaert and Roufs (1981). The techniques used in these studies usually involve two parameters, an independent and a dependent variable, which yield a threshold function that is more or less localized in the visual field. Another technique for inferring the nature of spatial processing depends on the measurement of thresholds for detecting sinewave gratings. This technique also provides two parameters (contrast vs spatial frequency). However, sinewave measurements cannot be accurately localized in the visual field, particularly if they are extended to the lowest spatial frequencies for which the visual process has a significant response (Kelly, 1965, 1977). If spatial processing were uniform across the retina, then the distinction between large-area and local responses might be less important, but clearly that is not the case. The structure and function of the retina (e.g. the sizes of receptive fields) change smoothly but drastically within the first few degrees from the fovea (Fischer, 1973; Wilson and Sherman, 1976). However, a fovea-centered stimulus field only a few degrees in diameter is still too small to study the low-frequency range of interest. Electrophysiological surround receptive
targets
Optic flow
Retinal
Nevertheless, the possibility of deriving local properties from large-area sinewave thresholds has remained a tantalizing one, partly because the Fourier transform of the sinewave contrast-sensitivity function resembles, at least qualitatively, the profile of a center/surround retinal receptive field. This was apparent in the early results of Pate1 (1966), who first calculated line-spread functions from sinewave thresholds. The notion that certain properties of the retina (spatial, temporal, and chromatic) may control the threshold in some simple detection tasks does not conflict with the more elaborate forms of visual processing revealed by more complex paradigms such as pattern adaptation (Blakemore and Campbell, 1969) or masking (Stromeyer and Julesz, 1972). These higher visual processes may be irrelevant to the unmasked, unadapted, detection thresholds considered here, so that the form of the threshold function for the detection of sinewave stimuli depends on properties of the retina; i.e. the rest of the visual system acts as a null detector in these experiments (Kelly and Burbeck, 1984). However, the systematic increase in receptive-field size with eccentricity has been an intractable problem in trying to interpret contrast sensitivity functions in terms of local spatial processing. Here we present a new method of coping with this aspect of retinal inhomogeneity. Using stimulus patterns that compensate over a wide field for the eccentricitydependent component of retinal structure, we obtain a canonical form for retinal receptive fields that seems to be at least as accurate as the results of the line-spread and point-spread studies mentioned above (Westheimer, 1967; Hines, 1976; Limb and Rubinstein, 1977; Wilson, 1978; Blommaert and Roufs, 198 I).
1895
D. H.
IY96
The simplest interpretation of our results would be that the sinewave thresholds are governed by a single type of receptive field (e.g. an X-like ganglion crli) that changes size only with eccentricity. Although the presence of multiple retinaf mechanisms (e.g. X. Y and W ceils in the cat) serving the same, or nearly the same. retinal locus is well established. there is no reason to suppose that they all have identical thresholds. It seems possible that an X-like mechanism is dominant at threshold, although a Y-like mechanism may be more sensitive at suprathreshold levels (Shapiey and Victor, 1978; Burbeck. 1981: Burbeck and Kelly. 1981). The contrast-sensitivity data presented here difter from those of our previous study with radially scaled stimuli (Kelly, 1985b) in two important ways. First, the scale factor of the present stimuli was chosen to match the estimated radial inhomogeneity of the author’s right eye, so that the resulting sensitivity data can be directly transformed into the canonical spread function of the present mode!. (In the previous study, severat different stimulus scale factors were used to test this model but none of them exactly matched the subject’s retinal scale factor). Second, in collecting the present data, the temporal frequency of this s~atiafly-optimized stimulus was not hetd constant (as in Kelly, 1985b). but was varied with spatial frequency in order to hold the retinal t&city constant. We have shown elsewhere (Kelly, i979, 19g5a) that this constant-velocity technique is the only form of controlled temporal variation that can match the contrast-sensitivity data obtained with natural eye movements. Before calculating the main results of the present study, we therefore compared the scaled-target data obtained with this constantvelocity technique to scaled-target data obtained at a constant temporal frequency. Fortunately, we found that both types of data can be fitted by the model used previously (Kelly, 1985b), with a modest change in the value of one parameter.
KELLY
hence the o~t~murn optic-Row gratings) for a given subject has been described in detail elsewhere (Kelly, 1985b). When the scale factor of the target matches the scale factor of the subject‘s retina. the model predicts that the measured contrast sensitivity function (CSF) will assume the same form as the local CSF at any point on the retina. This standard, canonical form is given by SUllt,,,)
= V‘ifLY exp( - P/‘L/“,,,)
(1)
The constant p controls the ~dndwidth of this function, which depends on the temporal properties of the stimufus. The locaf normalizing spatial frequency, / maxiis given by fm.,, = _/,i( 1 f Kr )
(2)
where f, is the frequency of maximum sensitivity at the fovea, r is eccentricity in degrees, and K is the retinal scale factor (Kelly, 1985b). Thus the local ~~x~rnurn shifts by one octave at the eccentricity, f = K-1. (For the author’s right eye, fL.= 3.5 cyclesideg and K-’ = 2.3 deg. in reasonably good agreement with other inhomogeneity measures.) RESULTS {A )
Constant ceiociry I‘S consranl temporal fkequency
Figure I shows two sets of measurements of contrast sensitivity as a function of (fovea]) spatial frequency, taken with circular FM stimuli matching the subject’s estimated retinal scale factor (and hence approximating the local CSF just described). Both are plotted as functions of the local spatial frequency at the fovea [where f,,, =f, in equation (?)I. For the
METHODS
We controlled the effects of eye movements and other potential sources of unwanted temporal variation by stabilizing the retinal image, thereby keeping our circular targets always concentric with the subject’s point of fixation. The desired temporal variation was introduced at the distal stimulus in the form of radial motion (i.e. optic flow), which induced little or no optokinetic nystagmus. The local temporal frequency was held constant over the entire I6-deg field. Any one of our circularly symmetric stimulus patterns is locally sinusoidaf, but its period increases linearly with eccentricity. With the appropriate scale factor for this radial variation, such a frequencymodulated (FM) pattern may provide nearly equal stimulation at all eccentricities. A circular-retina model that permits us to derive this scale factor {and
1*
/
a.2
05 WATIAL
I
1
2
1
/
5
10
FAEQUENCYsy/deg
Fig. I. Contrast sensitivity as a function of (fovea0 spatial frequency under two temporal conditions. Squares: constant temporal frequency (0.S Hz). Circles: temporal frequency varied with spatial frequency to keep v&city constant (0.19deg/sec at fovea). Circufar, linearly scaled patterns were used. as illustrated elsewhere (Kelly, 1985b). Temporal frequency did nat vary with radius in either case; hence veiocity increased linearly with radius in both cases (as in “optic Row’“). Solid and dashed curves are both plots of equation (I). with different values of parameter p.
Receptive-field function
1897
Fig. 2. Fovea1 version of the local contrast sensitivity function (at 0.19 deg/sec), according to the present model. This surface represents the same data as the dashed curve of Fig. I, but plotted in linear coordinates as a figure of revolution in two-dimensional frequency space.
open squares, temporal frequency was held constant at 0.5 Hz as in our previous study (Kelly, 1985b). (However, these are new data with K-’ = 2.3 deg.) For the open circles, temporal frequency was varied with spatial frequency to hold the radial velocity constant at any given eccentricity; at the fovea, this local velocity was always 0.19 deg/sec (in the range of the subject’s natural drift motions). Both curves in Fig. 1 are theoretical, being plots of equation (1) with the appropriate parameter values; the fit is very good in both cases. The decrease in the bandwidth of the constant-velocity curve was obtained by increasing the parameter p, from I to 1.4. The experimental and theoretical results shown in Fig. 1 confirm that the author’s model of retinal inhomogeneity (Kelly, 1985b) fits the constantvelocity data for scaled circular targets as well as the constant-temporal-frequency data from which the model was derived. (B) Line -spread and point -spread ~~~cti~ffs We have calculated both line-spread and pointspread functions from the inhomogeneity-corrected, constant-velocity (dashed) curve in Fig. I. The linespread function is calculated as the one-dimensional Fourier transform of the CSF, scaled for any desired eccentricity by equation (2). Results in this form are
useful for comparison with data on local subthreshold summation between fine lines (Hines, 1976; Limb and Rubinstein, 1977; Wilson, 1978). The point-spread function is calculated from our essentially one-dimensional data by treating the local CSF [equation (I) and Fig. I] as the radial profile of a circularly symmetric function in two-dimensional frequency space. Figure 2 shows a perspective view of that function, plotted in linear frequency units (which compress the CSF at low frequencies and expand it at high frequencies, relative to the log units of Fig. I). The frequency scales refer to the local CSF at the fovea (where f,,, =f=). For both spread-function calculations, we used the scaled-target CSF at a fovea1 velocity of 0.19 deg/sec at the fovea. With unscaled, rectilinear gratings, such a slow drift rate gives results that are very similar to the CSF obtained by viewing a stationary grating with natural eye movements (Kelly, 1979). Hence, the constant-velocity data (dashed curve in Fig. 1) should more nearly represent the way receptive fields function with natural eye movements. Figure 3(a) shows three line-spread functions calculated as one-dimensional Fourier transforms of equation (I) and scaled in accord with equation (2) to represent eccentricities of 0, 3, and 6 deg. The positive and negative areas of these functions are
D. H.
KELLY
0"
(b)
Fig. 3. (a) Retinal line-spread functions at eccentricities of 0, 3 and 6 deg. according to the present model. These functions were calculated as one-dimensional Fourier transforms of equation (I) with p = 1.4. and scaled in terms of equation (2) with l/K = 2.3 deg. (b) Retinal point-spread function at the fovea, calculated as the two-dimensional Fourier transform of the surface shown in Fig. 2. invariant with eccentricity and they integrate to zero; both properties follow from the form of equation (I). (If the local sensitivity is significantly anisotropic, then these results represent line-spread only in the radial direction, because our circular targets have no tangential component.) A standard criterion for comparing the widths of sombrero-shaped line-spread functions is the distance between zero-crossings; i.e. the width of the excitatory zone. For the calculated functions shown in Fig. 3(a), these widths are 4.4 min arc at the fovea, 10.7 min at an eccentricity of 3 deg, and 16.8 min at 6deg. Because these data were compensated for inhomogeneity, the fovea1 width is significantly smaller than the value we obtained earlier by transforming the CSF for uniform, homogeneous gratings (Kelly, 1977). However, it seems to be in good agreement with fovea1 line-spread functions inferred by an entirely different method; namely, subthreshold summation between adjacent line stimuli. With slow temporal modulation, Wilson (1978) obtained a foveai width of 5.6 min. Comparable results of
several such studies vary from a maximum of 6.6 min (Hines, 1976) to a minimum of 3.9 min (subject JOL of Limb and Rubinstein, 1977). The other line-spread functions of Fig. 3(a), at eccentricities of 3” and 6’, are also in good agreement with comparable subthreshold summation data. For example, at 2.5 deg eccentricity, Hines (1976) obtained a zero-crossing width of 8.7 min; at 3.3 deg, the width of Limb and Rubinstein’s line-spread function was about 10min for both their subjects. Figure 3(b) shows the point-spread function calculated as the two-dimensional Fourier transform of the local CSF shown in Fig. 2. Like the CSF of our isotropic model, the point-spread function must also be a figure of rotation. The distance scales represent the foveai version of this canonical function; its change of scale with eccentricity can be inferred from the line-spread functions shown in the same figure. DISCUSSION
Although the point-spread function contains no information that is not also implicit in the line-spread
function {given the assumption of a linear, isotropic mechanism). this two-dimensional response seems easier to compare with the spatial responses of retinal ganglion cells, which are often expressed in a similar form. Even the relatively broad. flat shape of the negative surround illustrated in Fig. 3(b) has been reported in X-type ganglion cells (Shapiey, 1985). To evaluate how well our wide-field results with scaled stimuli actually represent local properties. the most useful comparisons are with local measurements of spatial processing in the human retina that are presumably not distorted by retinal inhomogeneity. Thus, the agreement between our line-spread functions and the subthreshofd summation results described above provide corroboration for the corresponding point-spread function as well. Here we
consider some related measurements with local, circular targets. Blommaert and Roufs (1981) attempted to carry out the two-dimensional, circular analog of the parallel-line experiments. With one subject, they obtained a fovea1 point-spread function that was narrower than would be expected from most of the parallel-line results. However, it agreed fairly well with their calculated prediction from Limb and Rubinstein’s (1977) subject JOL. Westheimer’s (1967) sensitization experiments also used circularly symmetric stimuli, providing other results that can be compared with our point-spread functions. In this case, the critical diameter of his sensitizing disk, where the test threshold is maximums should equal the zero-crossing diameter of the pointspread function (under our standard assumptions). Note that these diameters must be greater than the line-spread widths of Fig. 3(a). At eccentricities of 0, 2.5, and 5deg, Westheimer (1967) obtained critical diameters of 5, 14.2, and 2 I .3 min arc (his Fig. 4), compared to zero-crossing diameters of 6.3, 14.6, and 22.9min for the pointspread functions calculated here at the same eccentricities; his doubling eccentricity was between I and 2 deg, compared to our vaiue of 2.3” for K-r. Thus a perfect fit to Westheimer’s data would require slightly smaller values off, and K than those used in Fig. 3(a). Averaged over a 16-deg field, however, these two changes tend to compensate each other, resulting in only a second-order effect in fitting data like those of Fig. I. The CSF measured with our scaled, inhomogeneous stimuli differs in both shape and bandwidth from the CSF measured with ordinary, homogeneous gratings over the same area. The magnitude and direction of these differences can be explained by the hypothesis (Van Doorn ef al., 1972) that the inhomogeneity-corrected thresholds are affected by only one size of receptive field at each eccentricity, Although that post&ate can be relaxed somewhat, the form of the scaled CSF implies a strict limit on the size distribution of the receptive fields that affect the threshold at any given eccentricity.
This local distribution is evidently much more homogeneous than the distribution of receptive-field
sizes over the 16’ field used in our experiments. Since our inferred spread-functions are dependent on a model of retinal inhomogeneity (Kelly. 1985b). and the local-target measurements are not. agreement between these two types of results tends to confirm that our technique satisFactoriIy compensates for the major component of the inhomogeneity. as the model predicts. Like most psychophysical models, this one is necessarily an oversimplified version of what is known about the relevant visual physiology. Nevertheless, it represents a significant step toward remedying two important defects (spatial uniformity and temporal control) of previous attempts to infer receptive-~eld~l~ke properties from sinewave thresholds. We expect these results to be useful in computational modeling of the visual process. Ackno,rlerlgemenls-This work was supported by National Institutes of Health Grant EY 01 IX?. The computer graphics were done by John Peters. REFERENCES
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