Viston Res. Vol. 16. pp. 56: to j-2.
Pergamon Press 1976. Prtnted in Circa Bntam.
LINE SPREAD FUNCTION VARIATION NEAR THE FOVEA’
Department
of Physics and Department of Biophysics and Theoretical Biology, The University of Chicago, Chicago, IL 60637, U.S.A. (Receiced 4 April 1975. in
recked
firm
11 September
1975)
Abstract-Line spread functions (LSFs) for thin lines at five eccentricities near the fixation point were measured psychophysically by subthreshold addition of flanking lines. With increasing eccentricity, the LSFs decrease in height and increase in excitatory center and inhibitory trough width. The LSF data are used to defme a model consisting of an inhomogeneous linear filter followed by a peak output detector. This model is consistent with the threshold data for any shape bar whenever the filter output for that bar has only one peak. Probability summation between independent events is adequate to account for thresholds for some multipeak patterns.
INTRODUCTION Extensive anatomical and physiological evidence exists for the increase of average receptive field size with increasing distance from the visual fixation point (Boycott and W&&e, 1974; Fisher, 1973; Hubel and Wiesel, 1960). Psychophysically this is borne out by measurements of acuity (Aulhom and Harms, 1972; Sloan, 1951), summation areas (Westheimer, 1967), Mach band locations (Shipley and Wier, 1972), and sinusoid modulation transfer functions (MTFs) (Hilz and Cavonius, 1974). Similarly, there is some evidence for a distribution of receptive fiefd sizes at a single point. Different patterns seem to be detected by different receptive field populations (Kulikowski and KingSmith, 1973). Determination of the relative importance of these two sources of variation is difficult when patterns are used which subtend a visual angle larger than the fovea. For such patterns, especially wide field gratings, the effects of both variations are interchangeable. However, if patterns are used which are detected by a portion of the visual system serving a small area of the visual field, separation of the two effects will occur. Thresholds for analogous patterns at different eccentricities will then reflect more purely the effects of eccentricity. This paper presents measurements of line spread functions (LSFs) and sensitivities to bars of various shapes and widths at several eccentricities near the fovea to aid in separating the increase in mean field size with eccentricity from a possible wide range of field sizes at a single eccentricity. %lETHODS The apparatus is similar to that of Shapley and Tolhurst (1973). A 1 MHz triangle and 122 Hz triggered sawtooth are applied to the Y- and X- axes of a 5 in. oscilloscope (P31 phosphor) to produce a uniform blue-green raster ’ Presented in partial fulfillment of the requirements for the Ph.D. degree in the Department of Physics, The University of Chicago. YR.166-t
with luminance of 30cd,im’. Vertical patterns are generated by modulating the Z-axis in synchrony with the sawtooth. Contrast and Z-axis input voltage are proportional when the pattern’s spatial frequencies are less than 20 cycles/in. and the pattern’s maximum contrast is less than 2@“$ A PDP 8/f computer generates the spatial pattern by reading 12-bit words from a 512-word list into a digital to analog converter (DAC) at a rate of 13.3 psec/word. Thus the pattern consists of 512 bars, each independently set to one of 4096 values. Another DAC is updated every beam retrace and the two outputs are continuously analog multiplied together to provide temporal modulation. The computer’s relay drivers set the contraSt by means of a 64 position resistor network calibrated in 1,/Soctave steps. Figure 1 shows some of the spatial luminance distributions used in this paper. As the photograph shows, the step nature of the patterns is not visible and the sampled functions appear continuous. A subject views the screen monocularly from a distance of 36 in. with his head in a chin rest. The screen is masked to 8” horizontal x 1.6’ vertical by a large illuminated cardboard surround similar to the screen in hue and brightness. The horizontal dimension of the mask was chosen so that patterns at 2.5’ would be available. The vertical dimension was chosen as a compromise between possible edge effects at the top and bottom of the pattern and effects of recep tive field size variation in the vertical direction. Two small marks on the cardboard above and below the center aid in fixation and accommodation. The computer randomly selects the pattern to be displayed and sets the contrast 05-1.5 octaves below the last contrast threshold of that pattern. The subject, after satisfying himself that his gaze is centered on the screen, initiates a @5-set long presentation of the pattern by pressing a button. The temporal presentation used, in which the pattern smoothly appears and then fades ous is a Gaussian of the form e- [(r - 025 set)/@2 set]‘. The choice of this temporal presentation was a compromise between several conflicting considerations. First, slow drift eye movements having a velocity of about 6’/sec and microsaccadic eye movements with a mean amplitude of 56’ dictate a fast presentation (Alpem, 1972). Second this study is concerned with spatial properties of the visual system. A very fast or an on-off presentation would allow
567
temporal transients to afYect the data. Smce transient effects die out after less than 0.1 set (Kelly. 1971). a dowI> appearing longer presentation is necessary for the spatial form of the line spread function to become apparent. Each press of the button initiates the same pattern with a contras: increase of 1 S octa\e. IVhen the subject finally sees a change from mean luminance (usually after 110 presses), he pushes a second button to signal the computer to record the contrast and randomly select another pattern. In a 1-hr session. subjects were able to complete 360 threshold settings. For each line pattern. 2-t threshold settings were made at each eccentricity. For bar patterns, 12 settings were made. Standard errors within a day’s session were usually between 0.2 and 0.3 octaves. Averaging data from several days’ experiments did not decrease the standard error but did smooth the data considerably. Data were taken on three subjects with corrected vision, one of whom did not know the overall purpose of the experiments. Data on a fourth naive subject was thrown out because of standard errors each day of 0.5 octaves and inconsistency between days.
the dimensions of sensitivity deg. Thus. ,*’ S = max ( LSF& - s).f‘(z~d.u. - - 7,
With equation (1) the empirical line spread function can easily be determined through a knowledge of the sensitivity to various combinations of thin lines. For a single line having a width of 1’. we approximatef(x) by a Dirac delta function, f;,,,(x) = 4.x - e). 1 minute. Then we have from equation (1). LSF,(O) = S,,,,,, ,lne. 1 minute- ‘.
Since the LSF data will be used to predict thresholds for a variety of patterns, several assumptions about pattern detection must be made. The simplest assumptions are a linear spatial filter followed by a peak output detector. The detection task is to discriminate between the luminance patterns &,,,,, and &,,,,,(I + m.f‘W.s(r)).
Here. L,,,, is the mean luminance andf(x).g(t) is a spatiotemporal pattern suitably normalized so that its maximum value is 1. The modulation, m. is defined as (.L,,, - &,,,,,)/ L mran At threshold, l/m is termed the sensitivity, S. These patterns are passed through a linear spatial filter defined by the convolution of a weighting function or receptive field. x, at eccentricity, E, and .the stimulus. The filter produces a response
(2)
We now
calculate the response to a thin center line at eccentricity e, flanked by two half contrast lines at distance a away from the center line (see Fig. 1). Now 1; = (6(.x - E) f ‘d(x l - E+$5(x
- E + dt
I
a)
minute
For LSFs with a strong excitatory center. the maximum response still occurs at E. Thus S, = [LSF,(O) + jLSF,((r) + iLSF,( -a)].
THEORI
(1)
I minute.
Assuming a symmetrical LSF and using equation (2), we have LSF,(n) = [S, - Srlnste,J
.minute- ‘.
(3)
Consequently. by using patterns with a variety of spacing the entire LSF can be mapped. Unfortunately, there is a major difficulty in using the part of the LSF determined by equation (3) to predict the sensitivity to wide bars through equation (1). The problem is that an extremely small error in Ssinglctine introduces an extremely large error in wide bar threshold prediction. The error is equal to the product of the error in .&te tint and the area of the stimulus within an LSF width. For this reason the area of the LSF should be determined by the sensitivity of the LSF to a uniform field. This problem is discussed more fully in the linear filter test section of this paper.
RESULTS
In all experiments g(r) is the same and so we have neglected the time course of the response. Temporal effects are contamed implicitly in the weighting function as are the effects of the mean luminance. We assume that detection occurs when for some threshold modulation, m = l/S, the maximum change in the filtered response is 0, where:
This can be written in terms of S. S = max
1
J(x) dx.
We define the quantity in brackets as the empirical line spread function at eccentricity E, LSF,. This quantity has * Prior to these measurements, a control experiment with center line-single flanking line combinations was performed to check for asymmetry within receptive fields that mediate line detection. Since no significant deviation from symmetry could be found, we felt justified in using half contrast flanking lines on both sides of a center line for LSF measurements.
LSFs were determined by measuring the change in single line sensitivity due to the presence of two flanking half contrast lines at varying distance from a center line at eccentricity 6’. At the small eccentricities used there was no significant difference between nasal and temporal visual field field sensitivity for any pattern. Therefore contrast sensitivities for patterns measured at corresponding nasal and temporal points were averaged together. Figures 2 and 3 show LSFs for two subjects. Figure 2 makes explicit the symmetry of the LSFs and facilitates comparison between 0” and 25’. Figure 3 shows the progressive change in LSF size for 0’. 1.25” and 25”. All data has been analyzed as discussed in the theory section. The qualitative forms of the LSFs are similar for all subjects and eccentricities and consist of a strong excitatory center with a less strong inhibitory surround tapering off to zero. For center-flanking line separations greater than 16’ the subthreshold flanking lines do not affect the sensitivity to the center fine. For ail subjects, a change in eccentricity of 2.5’ is accompanied by a decline in LSF height. The ratio of positive peak at 0’ to positive peak at 2.5’ is 15-1.6. Similarly, there is a clear decrease in inhibitory trough magnitude. Again. excitatory center width
Lmax
Lmean
Fig.
1. The 5
Line spread function variation near the fovea
569
Table 1. The Gaussian parameters that fit the LSF data. The ratio b/r is 1.5 for all subjects and eccentricities. The area of the LSFs. A-B. are constant for each subject. These Gaussian fits are used in equation (1) to predict the sensitivities to arbitrary patterns.
-200 -16
-12
-8
0
4
DISTANCE
-4
(MINI
8
I2
16
Fig. 2. LSFs at 0” (closed circles) and 2.5’ (open circles) for subject MH. The LSFs are superimposed to facilitate comparison. Fits to the data by the difference of two Gaussians are shown as continuous lines and make explicit the symmetry of the LSFs. Parameters that fit the data are gathered in Table 1.
but not so rapidly as the decrease in single line sensitivity. The center width ratio between 2.5” and 0” is about 1.3. Finally, the inhibitory trough shows consjderable broadening. Each curve fitting the data in Figs. 2 and 3 is the difference of two Gaussians, increases,
gie -(x-&2z’
_
I3
e-(x -d’l2B’
,hip
.
For each eccentricity, the LSFs were fit by eye by adjusting the height and center width of the function. The area of the function corresponds to the sensitivity to a uniform field incremental stimulus and is invariant with eccentricity (see Linear Filter Test section). The ratio, p/z, was found to be constant at I.5 for all subjects and eccentricities. Table 1 shows the parameters used for three subjects.
LINEAR
FILTER TEST
To check this linear filter hypothesis, sensitivity measurements were made for vertical bars with
various luminance profiles over a large range of bar widths at the eccentricities 0” and 2.5’. Figures 4, 5 and 6 show the results on one subject for cosine bars, cosine edges. and square bars (see Fig. 1). Cosine bars and cosine edges of width W are defined as cos(rr(x - 6)/W’), Ix - E/ < W/2 and +[I + cos(7c(.u- e)I’W)], 0 < (x - E) < W respectively. Results for the other two subjects are similar. In all cases the lower curve of a pair is for 2.5” and the upper is for 0’ eccentricity. No pattern was found whose sensitivity was greater in the periphery than in the center. Generally, sensitivities for thin bars are substantially lower in the periphery and the width required to reach maximum sensitivity is larger. The sensitivities for wide cosine bars at 0” and 2.5” approach the same value. This implies that the area under the LSFs is invariant with eccentricity, if we assume that the LSFs mediate wide cosine bar detection. Response of the narrow LSF to a uniform stimulus is within 5% of its response to a 1’ wide cosine bar. These wide bar sensitivities were thus used to define the invariant area of the Gaussian fits to the LSF data. The continuous lines in Fig. 4. 5 and 6 were obtained by using the Gaussian fits to the LSFs of Fig. 4 in equation (1). The predicted sensitivity to the narrowest cosine bars is close to the measured sensitivity. This is expected since the physical pattern which consists of only three lines, two of which have a magnitude 0.7 of the third, is similar to the situation of half contrast lines immediately flanking a center line. It is clear that the measured LSFs are consistent with the sensitivities to cosine bars of all widths. If wide cosine bars measure the area of an LSF, wide cosine edges measure the area of the LSF center
IO_
I* BAR
Fig. 3. LSFs at 0”. 1.25” and 2.5” for subject HW. For economy of space, the redundant left half of each LSF is omitted. Parameters that fit the data are gathered in Table 1.
2.
WIDTH (DEG)
Fig. 4. Contrast sensitivity to cosine bars of various widths for subject HW. The closed circles are for cosine bars centered at 0”. The open circles are for 25’. The continuous lines are the linear filter-peak detector model prediction using the Gaussian fits to the 0” and 2.5’ data of Fig. 3.
MICH.AEL HISES
0 4.5 cycles
IO'
Fig. 5. Same as Fi_e.4 except the patterns used are cosine edges. The discontinuity at the edge is centered at 0’ and 2.3’ respectively. and one inhibitory flank. This is because the LSF location that gives the maximum convolution response to an edge has its excitatory region. inhibitory region boundary at the edge. Figure 5 shows that the fit is good. In the extension to square bars the whole procedure fails in a way that is consistent for all subjects and eccentricities. Narrow bars are consistent with the LSFs; however at that bar width where the convolution response reaches a peak, the measured sensitivity maintains its peak value and only slowly decreases to a plateau. It is interesting to note that at this bar width, edge enhancement becomes visible and the convolution response breaks up into two peaks which locate the edges of the square. Also, according to Figs. 3 and 6, subjects are more sensitive to a 2” wide square bar than a wide cosine edge though the sensitivity to an edge 1’ from the center is less than the sensitivity to an edge located at the center. DISCWSIO~
Acuity, measured in terms of ability to resolve high contrast detail, falls off by at least a factor of 2 in 2’ (Millodot, 1972: Weymouth et al., 1928). It is natural to relate the acuity to line spread function center sizes and expect a similar fall off. There may be several reasons why the LSFs presented in this paper do not demonstrate such a large variation with eccentricity. First, the height of the lines used to obtain the LSFs is almost 2’ long. If we assume good fixation and a radial organization of the retina in which receptive field sizes are the same at a constant distance from the fixation point, then the LSF at 0’ contains contributions from receptive fields at l”, the LSF at 1.25’ contains 1.6” contributions, and the 2.5’ LSF contains 2.7’ contributions. Thus, the 0” LSF may be broadened to a greater degree than the 23’ LSF. Second, stability of fixation was not determined 90 >
In
BAR
WIDTH
COEG)
Fig. 6. Same as Fig. 4 except the patterns used are square bars centered at 0’ and 2.5’.
!
1
1
1
1
.5
1
2
4
8
frequency
Icycfe/deg 1
Fig. 7. Effect of number of cvcles of a cosine grating on the modulation transfer fun&ion. Data is the log mean of three subjects. The peak detector prediction for 1.3 cycle gratings is shown as the solid line. The peak to trough detector prediction for 1.5 cycle gratings is shown as the dotted line.
and any departure from fLvation on the center of the screen would tend to lessen the effect of eccentricity (see Methods). Finally, acuity may not be mediated by the same receptive fields that are maximally sensitive to a single thin line. However, the intent of the line experiments was not to obtain data useful for acuity discussions and the prediction of high spatial frequency properties of the retina, but to assess the relevance of eccentricity in the detection of distributed patterns containing intermediate and low spatial frequencies. Clearly a theory of threshold detection must take into account the fact that high frequencies in the periphery are less important than low frequencies. Unfortunately the domain of l-dimensional patterns that can be predicted using the l-dimensional LSFs and the linear model is bounded in three separate areas. Two areas represent local problems to the theory in the sense that eccentricity variation is not important and one area represents a global problem that involves eccentricity. The first class of patterns whose thresholds cannot be accounted for with a single LSF is represented by square bars. A single channel model does not permit such a wide peak in square bar sensitivity YS width. One correction to the model would be to allow for a distribution of receptive field sizes at each eccentricity. The optimum square bar for stimulation of a receptive field with center and surround has a width equal to the receptive field center size. This is the basis for the size tuned mechanisms of Thomas (1970). Receptive fields with wide but weak centers would not be stimulated by lines; their response could only be elicited by patterns that afforded a large amount of summation. Of course, permitting different size receptive fields at a single point obviates the need for the LSF to fit wide bar cosine patterns. The only criterion that the newly postulated receptive field must obey is that its response be consistent with thresholds of other patterns. If receptive fields having a dip in the excitatory region are implausible, consistency requires that the sensitivity ratio of square and cosine bars cannot be greater than the ratio between square and cosine bar areas, ~~2. This condition is satisfied for the data in Fig. 4 and 6. From Fig. 6. the maximum in square bar sensitivities at 0’ and 2.5’ occurs at a width about twice that expected from the convolution response of the corresponding LSF
Line spread function variation near the fovea
571
Thus, if such a distribution of center sizes exists, it tinct peaks in the convolution response. ft should be is clear from the figure that at each eccentricity the noted that with high frequency gratings and extremely largest needed is at least twice as wide as the smallest. narrow bars, Campbell et nl. (1969) showed that a An alternative to such a distribution is to assume linear filter, peak to trough detector worked very well, that for wide bars a subject is detecting one of two although even in this range there was a certain ambiedges. Following King-Smith and Kulikowski (19751, guity in the detector with one cycle sine wave patwe assume that we are dealing with probabi~i~ sum- terns. mation between two independent events. Generally, The third class, patterns with several features each frequency of seeing curves are necessary data for a separated by more than one LSF width, also focuses proper analysis within a probability summation attention on the detector. Effects of receptive field size framework, hence the following analysis is meant to distribution at a single point and simple assessments be indicative only. We assume that the logarithm of of convolution response magnitude differences are the sensitivity measurements for a single edge are nor- clearty inadequate in accounting for sensitivities. mally distributed with parameters corresponding to Severat investigators have looked at the effect of the measured mean and S.D. Based on experimental number of cycles on grating sensitivity [Hoekstra er results, we shall take the mean as unity and the SD. nl., 1974). Grating sensitivity increases with number as 0.09 log units (@3 octaves). If P,(V) is the probof cycles until saturation at about 8 cycles. Magnuski ability of seeing an edge at contrast x then the (1973), using cosine gratings with 1-set counterphase probability of seeing at least one edge of a wide flicker, triangular modulation in the vertical direction bar at contrast .Y is P,(X) = t - (1 - PI(s)) = and the method of adjustment, found a sensitivity Y,(s) - Pi(x). By hypothe& the measured threshold ratio between 8.5 and 05 cycle gratings of about 1.7. Under conditions of pattern generation used in this would occur when P&x) = 05. This occurs when paper the ratio between 8.5 and 1.5 cycle gratings PI(u) = 0.293. For the standard normal distribution this occurs at a contrast of -0545 SD. from the is greater than a factor of 2 (Fig. 7). Similarly, we mean. For a S.D. of 0.3 octaves, this increases the have found that, for thin line patterns in which the sensitivity to a square bar by iY,i. As can be seen lines are separated by 0.5’ and the number of lines from Fig. 6. this increase is just what is needed to is varied, the sensitivity vs number of lines in the pattern increases up to a factor of 1.7 over the sensifit the wide square bar data. The presentation of results has been restricted to tivity to a single thin line alone. It seems reasonable positive going localized patterns in an attempt to to suppose that the statistical nature of the particular minimize the effect of details of the detector. A detection process becomes important with such patterns. second class of patterns which are still relatively localized but have dark flanks brings the inadequacy of the simple peak detector to the fore. Consider an ex- ~c~~o~~~e~g~effts-I wish to thank Professors J. Cowan periment similar to the one with cosine bars but this and H. R. Wilson for many useful discussions and much time with cosine bars of opposite polarity flanking encouragement. This work was supported by N.I.H. training grant GM 2037-06. the center one, a 1.5 cycle grating. The open squares in Fig. 7 show an average MTF for such gratings. At low frequencies such that the center cosine bar is wider than the measured LSFs, the convolution REFERENCES of grating and LSF yields a peak response (solid line of Fig. 7) that is the same as the response to the Alpetn M. (1972) Eye movements. Handbook of Sensory cosine bar alone. Experimentally, however, one is Physiolog,v, VII’-%,pp. 303-330. Springer, Berlin. more sensitive to the grating than to the cosine bar Aulhorn E. and Harms H. (1972) Visual perimetry. Handat these frequencies. Since this difference is substanbook of’Sensory Physiology, VII,:4, pp. 102-145. Springer, tial, a peak to trough detector may be indicated. Note Berlin. that, with a strict peak to trough detector, the centerBoycott B. 8. and W&sfe H. (1974) The morphoiogical types of ganglion ceils of the domestic cat’s retina. J. flanking line experiments, instead of providing a diPhysic&, Lond. 240, 397-419. rect LSF, should producz secondary excitatory surrounds after the inhibitory region. This is because the Campbell F. W., Carpenter R. H. S. and Levinson J. Z. (1969) Visibility of aperiodic patterns compared with inhibitory regions of the center and flanking line add that of sinusoidal gratings. J. Physiol., Land. 204, so that the extra trough depth increases the peak to 283-298. trough response. These excitatory regions, which Fisher B. (1973) Overlap of receptive field cent&s and should be about half as strong as the inhibitor representation of the visual field in the cat’s optic tract. regions, are not apparent in the line data. MaintainVision Res. 13, 21132120. ing a peak to trough detector in this case would Hilz R. and Cavonius C. R. (1974) Functional organization of the peripheral retina: sensitivity to periodic stimuli. require replacement of the linear part of the model Vision Res. 14, 1333-1337. with some type of nonlinear rectifying filter. At any Hoekstra J., van der Goot D. P. J., van den Brink G. rate, the resulting receptive field, peak to trough and B&en F. A. (1974) The influence of the number detector combination fits the 0.5 cycfe but is unable of cycles upon the visual contrast threshold for spatial to account for I.5 cycle grating sensitivities. The sine wave patterns. Vision Res. 14, 365-368. theoreticai response for 1.5 cycle gratings is shown Hubel D. H. and Wiesei T. N. (1960) Receptive fields of as the dotted line in Fig. 7. Surprisingly, it fits very optic nerve fibres in the spider monkey. J. Phrsiol., Land. well the 2.5 cycle MTF. Again, complications might 1% 572-580. enter due to the possibility of different receptive field Kelly D. H. (1971) Theory of flicker and transient responses, II. Counterphase gratings. J. opr. Sot. &I. 61, 632-640. sizes at a single point and the presence of three dis-
King-Smith P. E. and Kulikowski J. J. (19753 The detection of gratings by independent activation of line detectors. J. Physiol., Lond. 217. 237-271. Kulikowski J. J. and King-Smith P. E. (1973) Spatial arrangement of line. edge. and grating detectors revealed by subthreshoid summation. Vision Res. 13, 1455-1478. Magnuski H. S. (1973) Visual system analysis through use of finite field sine gratings. Ph.D. thesis. MIT. Mitlodot M. (19721Variation of visual acuity in the central region of the retina. Br. J. physiol. Opr. 27, S-28. Shapley R. M. and Tolhurst D. J. (1973) Edge detectors in human vision. J. Phwiol.. Land. 229, 16C183.
Shipley T. and Wier C. (1972). Asymmetries in the lMach band phenomena. K.vbernetik 10. 181-189. Sloan L. L. (1951) Measurement of visual acuity: a critical review. Archs Ophthal.. Chicago 45, 704-725. Thomas J. P. (1970) Model of the function of receptive fields in human vision. Psychol. Reo. 77, 121-134. Westheimer G. (1967) Spatial interaction in human cone vision. J. Physiol.. Land. 190, 139-154. Weymouth F. W., Hines D. C.. Acres L. H.. Raaf J. E. and Wheeler M. C. (1928) Visual acuity within the area centralis and the relation to eye movements and fixation. Am. J. Ophthal. 11. 947-960.
Pergamon Press 1976. Prtnted in Circa Bntam.
LINE SPREAD FUNCTION VARIATION NEAR THE FOVEA’
Department
of Physics and Department of Biophysics and Theoretical Biology, The University of Chicago, Chicago, IL 60637, U.S.A. (Receiced 4 April 1975. in
recked
firm
11 September
1975)
Abstract-Line spread functions (LSFs) for thin lines at five eccentricities near the fixation point were measured psychophysically by subthreshold addition of flanking lines. With increasing eccentricity, the LSFs decrease in height and increase in excitatory center and inhibitory trough width. The LSF data are used to defme a model consisting of an inhomogeneous linear filter followed by a peak output detector. This model is consistent with the threshold data for any shape bar whenever the filter output for that bar has only one peak. Probability summation between independent events is adequate to account for thresholds for some multipeak patterns.
INTRODUCTION Extensive anatomical and physiological evidence exists for the increase of average receptive field size with increasing distance from the visual fixation point (Boycott and W&&e, 1974; Fisher, 1973; Hubel and Wiesel, 1960). Psychophysically this is borne out by measurements of acuity (Aulhom and Harms, 1972; Sloan, 1951), summation areas (Westheimer, 1967), Mach band locations (Shipley and Wier, 1972), and sinusoid modulation transfer functions (MTFs) (Hilz and Cavonius, 1974). Similarly, there is some evidence for a distribution of receptive fiefd sizes at a single point. Different patterns seem to be detected by different receptive field populations (Kulikowski and KingSmith, 1973). Determination of the relative importance of these two sources of variation is difficult when patterns are used which subtend a visual angle larger than the fovea. For such patterns, especially wide field gratings, the effects of both variations are interchangeable. However, if patterns are used which are detected by a portion of the visual system serving a small area of the visual field, separation of the two effects will occur. Thresholds for analogous patterns at different eccentricities will then reflect more purely the effects of eccentricity. This paper presents measurements of line spread functions (LSFs) and sensitivities to bars of various shapes and widths at several eccentricities near the fovea to aid in separating the increase in mean field size with eccentricity from a possible wide range of field sizes at a single eccentricity. %lETHODS The apparatus is similar to that of Shapley and Tolhurst (1973). A 1 MHz triangle and 122 Hz triggered sawtooth are applied to the Y- and X- axes of a 5 in. oscilloscope (P31 phosphor) to produce a uniform blue-green raster ’ Presented in partial fulfillment of the requirements for the Ph.D. degree in the Department of Physics, The University of Chicago. YR.166-t
with luminance of 30cd,im’. Vertical patterns are generated by modulating the Z-axis in synchrony with the sawtooth. Contrast and Z-axis input voltage are proportional when the pattern’s spatial frequencies are less than 20 cycles/in. and the pattern’s maximum contrast is less than 2@“$ A PDP 8/f computer generates the spatial pattern by reading 12-bit words from a 512-word list into a digital to analog converter (DAC) at a rate of 13.3 psec/word. Thus the pattern consists of 512 bars, each independently set to one of 4096 values. Another DAC is updated every beam retrace and the two outputs are continuously analog multiplied together to provide temporal modulation. The computer’s relay drivers set the contraSt by means of a 64 position resistor network calibrated in 1,/Soctave steps. Figure 1 shows some of the spatial luminance distributions used in this paper. As the photograph shows, the step nature of the patterns is not visible and the sampled functions appear continuous. A subject views the screen monocularly from a distance of 36 in. with his head in a chin rest. The screen is masked to 8” horizontal x 1.6’ vertical by a large illuminated cardboard surround similar to the screen in hue and brightness. The horizontal dimension of the mask was chosen so that patterns at 2.5’ would be available. The vertical dimension was chosen as a compromise between possible edge effects at the top and bottom of the pattern and effects of recep tive field size variation in the vertical direction. Two small marks on the cardboard above and below the center aid in fixation and accommodation. The computer randomly selects the pattern to be displayed and sets the contrast 05-1.5 octaves below the last contrast threshold of that pattern. The subject, after satisfying himself that his gaze is centered on the screen, initiates a @5-set long presentation of the pattern by pressing a button. The temporal presentation used, in which the pattern smoothly appears and then fades ous is a Gaussian of the form e- [(r - 025 set)/@2 set]‘. The choice of this temporal presentation was a compromise between several conflicting considerations. First, slow drift eye movements having a velocity of about 6’/sec and microsaccadic eye movements with a mean amplitude of 56’ dictate a fast presentation (Alpem, 1972). Second this study is concerned with spatial properties of the visual system. A very fast or an on-off presentation would allow
567
temporal transients to afYect the data. Smce transient effects die out after less than 0.1 set (Kelly. 1971). a dowI> appearing longer presentation is necessary for the spatial form of the line spread function to become apparent. Each press of the button initiates the same pattern with a contras: increase of 1 S octa\e. IVhen the subject finally sees a change from mean luminance (usually after 110 presses), he pushes a second button to signal the computer to record the contrast and randomly select another pattern. In a 1-hr session. subjects were able to complete 360 threshold settings. For each line pattern. 2-t threshold settings were made at each eccentricity. For bar patterns, 12 settings were made. Standard errors within a day’s session were usually between 0.2 and 0.3 octaves. Averaging data from several days’ experiments did not decrease the standard error but did smooth the data considerably. Data were taken on three subjects with corrected vision, one of whom did not know the overall purpose of the experiments. Data on a fourth naive subject was thrown out because of standard errors each day of 0.5 octaves and inconsistency between days.
the dimensions of sensitivity deg. Thus. ,*’ S = max ( LSF& - s).f‘(z~d.u. - - 7,
With equation (1) the empirical line spread function can easily be determined through a knowledge of the sensitivity to various combinations of thin lines. For a single line having a width of 1’. we approximatef(x) by a Dirac delta function, f;,,,(x) = 4.x - e). 1 minute. Then we have from equation (1). LSF,(O) = S,,,,,, ,lne. 1 minute- ‘.
Since the LSF data will be used to predict thresholds for a variety of patterns, several assumptions about pattern detection must be made. The simplest assumptions are a linear spatial filter followed by a peak output detector. The detection task is to discriminate between the luminance patterns &,,,,, and &,,,,,(I + m.f‘W.s(r)).
Here. L,,,, is the mean luminance andf(x).g(t) is a spatiotemporal pattern suitably normalized so that its maximum value is 1. The modulation, m. is defined as (.L,,, - &,,,,,)/ L mran At threshold, l/m is termed the sensitivity, S. These patterns are passed through a linear spatial filter defined by the convolution of a weighting function or receptive field. x, at eccentricity, E, and .the stimulus. The filter produces a response
(2)
We now
calculate the response to a thin center line at eccentricity e, flanked by two half contrast lines at distance a away from the center line (see Fig. 1). Now 1; = (6(.x - E) f ‘d(x l - E+$5(x
- E + dt
I
a)
minute
For LSFs with a strong excitatory center. the maximum response still occurs at E. Thus S, = [LSF,(O) + jLSF,((r) + iLSF,( -a)].
THEORI
(1)
I minute.
Assuming a symmetrical LSF and using equation (2), we have LSF,(n) = [S, - Srlnste,J
.minute- ‘.
(3)
Consequently. by using patterns with a variety of spacing the entire LSF can be mapped. Unfortunately, there is a major difficulty in using the part of the LSF determined by equation (3) to predict the sensitivity to wide bars through equation (1). The problem is that an extremely small error in Ssinglctine introduces an extremely large error in wide bar threshold prediction. The error is equal to the product of the error in .&te tint and the area of the stimulus within an LSF width. For this reason the area of the LSF should be determined by the sensitivity of the LSF to a uniform field. This problem is discussed more fully in the linear filter test section of this paper.
RESULTS
In all experiments g(r) is the same and so we have neglected the time course of the response. Temporal effects are contamed implicitly in the weighting function as are the effects of the mean luminance. We assume that detection occurs when for some threshold modulation, m = l/S, the maximum change in the filtered response is 0, where:
This can be written in terms of S. S = max
1
J(x) dx.
We define the quantity in brackets as the empirical line spread function at eccentricity E, LSF,. This quantity has * Prior to these measurements, a control experiment with center line-single flanking line combinations was performed to check for asymmetry within receptive fields that mediate line detection. Since no significant deviation from symmetry could be found, we felt justified in using half contrast flanking lines on both sides of a center line for LSF measurements.
LSFs were determined by measuring the change in single line sensitivity due to the presence of two flanking half contrast lines at varying distance from a center line at eccentricity 6’. At the small eccentricities used there was no significant difference between nasal and temporal visual field field sensitivity for any pattern. Therefore contrast sensitivities for patterns measured at corresponding nasal and temporal points were averaged together. Figures 2 and 3 show LSFs for two subjects. Figure 2 makes explicit the symmetry of the LSFs and facilitates comparison between 0” and 25’. Figure 3 shows the progressive change in LSF size for 0’. 1.25” and 25”. All data has been analyzed as discussed in the theory section. The qualitative forms of the LSFs are similar for all subjects and eccentricities and consist of a strong excitatory center with a less strong inhibitory surround tapering off to zero. For center-flanking line separations greater than 16’ the subthreshold flanking lines do not affect the sensitivity to the center fine. For ail subjects, a change in eccentricity of 2.5’ is accompanied by a decline in LSF height. The ratio of positive peak at 0’ to positive peak at 2.5’ is 15-1.6. Similarly, there is a clear decrease in inhibitory trough magnitude. Again. excitatory center width
Lmax
Lmean
Fig.
1. The 5
Line spread function variation near the fovea
569
Table 1. The Gaussian parameters that fit the LSF data. The ratio b/r is 1.5 for all subjects and eccentricities. The area of the LSFs. A-B. are constant for each subject. These Gaussian fits are used in equation (1) to predict the sensitivities to arbitrary patterns.
-200 -16
-12
-8
0
4
DISTANCE
-4
(MINI
8
I2
16
Fig. 2. LSFs at 0” (closed circles) and 2.5’ (open circles) for subject MH. The LSFs are superimposed to facilitate comparison. Fits to the data by the difference of two Gaussians are shown as continuous lines and make explicit the symmetry of the LSFs. Parameters that fit the data are gathered in Table 1.
but not so rapidly as the decrease in single line sensitivity. The center width ratio between 2.5” and 0” is about 1.3. Finally, the inhibitory trough shows consjderable broadening. Each curve fitting the data in Figs. 2 and 3 is the difference of two Gaussians, increases,
gie -(x-&2z’
_
I3
e-(x -d’l2B’
,hip
.
For each eccentricity, the LSFs were fit by eye by adjusting the height and center width of the function. The area of the function corresponds to the sensitivity to a uniform field incremental stimulus and is invariant with eccentricity (see Linear Filter Test section). The ratio, p/z, was found to be constant at I.5 for all subjects and eccentricities. Table 1 shows the parameters used for three subjects.
LINEAR
FILTER TEST
To check this linear filter hypothesis, sensitivity measurements were made for vertical bars with
various luminance profiles over a large range of bar widths at the eccentricities 0” and 2.5’. Figures 4, 5 and 6 show the results on one subject for cosine bars, cosine edges. and square bars (see Fig. 1). Cosine bars and cosine edges of width W are defined as cos(rr(x - 6)/W’), Ix - E/ < W/2 and +[I + cos(7c(.u- e)I’W)], 0 < (x - E) < W respectively. Results for the other two subjects are similar. In all cases the lower curve of a pair is for 2.5” and the upper is for 0’ eccentricity. No pattern was found whose sensitivity was greater in the periphery than in the center. Generally, sensitivities for thin bars are substantially lower in the periphery and the width required to reach maximum sensitivity is larger. The sensitivities for wide cosine bars at 0” and 2.5” approach the same value. This implies that the area under the LSFs is invariant with eccentricity, if we assume that the LSFs mediate wide cosine bar detection. Response of the narrow LSF to a uniform stimulus is within 5% of its response to a 1’ wide cosine bar. These wide bar sensitivities were thus used to define the invariant area of the Gaussian fits to the LSF data. The continuous lines in Fig. 4. 5 and 6 were obtained by using the Gaussian fits to the LSFs of Fig. 4 in equation (1). The predicted sensitivity to the narrowest cosine bars is close to the measured sensitivity. This is expected since the physical pattern which consists of only three lines, two of which have a magnitude 0.7 of the third, is similar to the situation of half contrast lines immediately flanking a center line. It is clear that the measured LSFs are consistent with the sensitivities to cosine bars of all widths. If wide cosine bars measure the area of an LSF, wide cosine edges measure the area of the LSF center
IO_
I* BAR
Fig. 3. LSFs at 0”. 1.25” and 2.5” for subject HW. For economy of space, the redundant left half of each LSF is omitted. Parameters that fit the data are gathered in Table 1.
2.
WIDTH (DEG)
Fig. 4. Contrast sensitivity to cosine bars of various widths for subject HW. The closed circles are for cosine bars centered at 0”. The open circles are for 25’. The continuous lines are the linear filter-peak detector model prediction using the Gaussian fits to the 0” and 2.5’ data of Fig. 3.
MICH.AEL HISES
0 4.5 cycles
IO'
Fig. 5. Same as Fi_e.4 except the patterns used are cosine edges. The discontinuity at the edge is centered at 0’ and 2.3’ respectively. and one inhibitory flank. This is because the LSF location that gives the maximum convolution response to an edge has its excitatory region. inhibitory region boundary at the edge. Figure 5 shows that the fit is good. In the extension to square bars the whole procedure fails in a way that is consistent for all subjects and eccentricities. Narrow bars are consistent with the LSFs; however at that bar width where the convolution response reaches a peak, the measured sensitivity maintains its peak value and only slowly decreases to a plateau. It is interesting to note that at this bar width, edge enhancement becomes visible and the convolution response breaks up into two peaks which locate the edges of the square. Also, according to Figs. 3 and 6, subjects are more sensitive to a 2” wide square bar than a wide cosine edge though the sensitivity to an edge 1’ from the center is less than the sensitivity to an edge located at the center. DISCWSIO~
Acuity, measured in terms of ability to resolve high contrast detail, falls off by at least a factor of 2 in 2’ (Millodot, 1972: Weymouth et al., 1928). It is natural to relate the acuity to line spread function center sizes and expect a similar fall off. There may be several reasons why the LSFs presented in this paper do not demonstrate such a large variation with eccentricity. First, the height of the lines used to obtain the LSFs is almost 2’ long. If we assume good fixation and a radial organization of the retina in which receptive field sizes are the same at a constant distance from the fixation point, then the LSF at 0’ contains contributions from receptive fields at l”, the LSF at 1.25’ contains 1.6” contributions, and the 2.5’ LSF contains 2.7’ contributions. Thus, the 0” LSF may be broadened to a greater degree than the 23’ LSF. Second, stability of fixation was not determined 90 >
In
BAR
WIDTH
COEG)
Fig. 6. Same as Fig. 4 except the patterns used are square bars centered at 0’ and 2.5’.
!
1
1
1
1
.5
1
2
4
8
frequency
Icycfe/deg 1
Fig. 7. Effect of number of cvcles of a cosine grating on the modulation transfer fun&ion. Data is the log mean of three subjects. The peak detector prediction for 1.3 cycle gratings is shown as the solid line. The peak to trough detector prediction for 1.5 cycle gratings is shown as the dotted line.
and any departure from fLvation on the center of the screen would tend to lessen the effect of eccentricity (see Methods). Finally, acuity may not be mediated by the same receptive fields that are maximally sensitive to a single thin line. However, the intent of the line experiments was not to obtain data useful for acuity discussions and the prediction of high spatial frequency properties of the retina, but to assess the relevance of eccentricity in the detection of distributed patterns containing intermediate and low spatial frequencies. Clearly a theory of threshold detection must take into account the fact that high frequencies in the periphery are less important than low frequencies. Unfortunately the domain of l-dimensional patterns that can be predicted using the l-dimensional LSFs and the linear model is bounded in three separate areas. Two areas represent local problems to the theory in the sense that eccentricity variation is not important and one area represents a global problem that involves eccentricity. The first class of patterns whose thresholds cannot be accounted for with a single LSF is represented by square bars. A single channel model does not permit such a wide peak in square bar sensitivity YS width. One correction to the model would be to allow for a distribution of receptive field sizes at each eccentricity. The optimum square bar for stimulation of a receptive field with center and surround has a width equal to the receptive field center size. This is the basis for the size tuned mechanisms of Thomas (1970). Receptive fields with wide but weak centers would not be stimulated by lines; their response could only be elicited by patterns that afforded a large amount of summation. Of course, permitting different size receptive fields at a single point obviates the need for the LSF to fit wide bar cosine patterns. The only criterion that the newly postulated receptive field must obey is that its response be consistent with thresholds of other patterns. If receptive fields having a dip in the excitatory region are implausible, consistency requires that the sensitivity ratio of square and cosine bars cannot be greater than the ratio between square and cosine bar areas, ~~2. This condition is satisfied for the data in Fig. 4 and 6. From Fig. 6. the maximum in square bar sensitivities at 0’ and 2.5’ occurs at a width about twice that expected from the convolution response of the corresponding LSF
Line spread function variation near the fovea
571
Thus, if such a distribution of center sizes exists, it tinct peaks in the convolution response. ft should be is clear from the figure that at each eccentricity the noted that with high frequency gratings and extremely largest needed is at least twice as wide as the smallest. narrow bars, Campbell et nl. (1969) showed that a An alternative to such a distribution is to assume linear filter, peak to trough detector worked very well, that for wide bars a subject is detecting one of two although even in this range there was a certain ambiedges. Following King-Smith and Kulikowski (19751, guity in the detector with one cycle sine wave patwe assume that we are dealing with probabi~i~ sum- terns. mation between two independent events. Generally, The third class, patterns with several features each frequency of seeing curves are necessary data for a separated by more than one LSF width, also focuses proper analysis within a probability summation attention on the detector. Effects of receptive field size framework, hence the following analysis is meant to distribution at a single point and simple assessments be indicative only. We assume that the logarithm of of convolution response magnitude differences are the sensitivity measurements for a single edge are nor- clearty inadequate in accounting for sensitivities. mally distributed with parameters corresponding to Severat investigators have looked at the effect of the measured mean and S.D. Based on experimental number of cycles on grating sensitivity [Hoekstra er results, we shall take the mean as unity and the SD. nl., 1974). Grating sensitivity increases with number as 0.09 log units (@3 octaves). If P,(V) is the probof cycles until saturation at about 8 cycles. Magnuski ability of seeing an edge at contrast x then the (1973), using cosine gratings with 1-set counterphase probability of seeing at least one edge of a wide flicker, triangular modulation in the vertical direction bar at contrast .Y is P,(X) = t - (1 - PI(s)) = and the method of adjustment, found a sensitivity Y,(s) - Pi(x). By hypothe& the measured threshold ratio between 8.5 and 05 cycle gratings of about 1.7. Under conditions of pattern generation used in this would occur when P&x) = 05. This occurs when paper the ratio between 8.5 and 1.5 cycle gratings PI(u) = 0.293. For the standard normal distribution this occurs at a contrast of -0545 SD. from the is greater than a factor of 2 (Fig. 7). Similarly, we mean. For a S.D. of 0.3 octaves, this increases the have found that, for thin line patterns in which the sensitivity to a square bar by iY,i. As can be seen lines are separated by 0.5’ and the number of lines from Fig. 6. this increase is just what is needed to is varied, the sensitivity vs number of lines in the pattern increases up to a factor of 1.7 over the sensifit the wide square bar data. The presentation of results has been restricted to tivity to a single thin line alone. It seems reasonable positive going localized patterns in an attempt to to suppose that the statistical nature of the particular minimize the effect of details of the detector. A detection process becomes important with such patterns. second class of patterns which are still relatively localized but have dark flanks brings the inadequacy of the simple peak detector to the fore. Consider an ex- ~c~~o~~~e~g~effts-I wish to thank Professors J. Cowan periment similar to the one with cosine bars but this and H. R. Wilson for many useful discussions and much time with cosine bars of opposite polarity flanking encouragement. This work was supported by N.I.H. training grant GM 2037-06. the center one, a 1.5 cycle grating. The open squares in Fig. 7 show an average MTF for such gratings. At low frequencies such that the center cosine bar is wider than the measured LSFs, the convolution REFERENCES of grating and LSF yields a peak response (solid line of Fig. 7) that is the same as the response to the Alpetn M. (1972) Eye movements. Handbook of Sensory cosine bar alone. Experimentally, however, one is Physiolog,v, VII’-%,pp. 303-330. Springer, Berlin. more sensitive to the grating than to the cosine bar Aulhorn E. and Harms H. (1972) Visual perimetry. Handat these frequencies. Since this difference is substanbook of’Sensory Physiology, VII,:4, pp. 102-145. Springer, tial, a peak to trough detector may be indicated. Note Berlin. that, with a strict peak to trough detector, the centerBoycott B. 8. and W&sfe H. (1974) The morphoiogical types of ganglion ceils of the domestic cat’s retina. J. flanking line experiments, instead of providing a diPhysic&, Lond. 240, 397-419. rect LSF, should producz secondary excitatory surrounds after the inhibitory region. This is because the Campbell F. W., Carpenter R. H. S. and Levinson J. Z. (1969) Visibility of aperiodic patterns compared with inhibitory regions of the center and flanking line add that of sinusoidal gratings. J. Physiol., Land. 204, so that the extra trough depth increases the peak to 283-298. trough response. These excitatory regions, which Fisher B. (1973) Overlap of receptive field cent&s and should be about half as strong as the inhibitor representation of the visual field in the cat’s optic tract. regions, are not apparent in the line data. MaintainVision Res. 13, 21132120. ing a peak to trough detector in this case would Hilz R. and Cavonius C. R. 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