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The :Invisible Cone Mosaic DAVID R. WIL LIA M S The last 5 years have seen a resurgence in interest among theoreticians, anatomists, and psychophysicists in the imaging properties of the cone mosaic. This recent activity might lead one to the conclusion that the cone mosaic represents a critical and important bottleneck for spatial vision. However, I argue that the cone mosaic is spatially "transparent" for most natural images we might encounter and does not pose a primary limitation on spatial vision. This may sound heretical at a meeting devoted to photoreceptors, but I do not mean to imply that the recent work on cone sampling has been for naught. On the contrary, this work is interesting because it has shown us how evolution has smoothly incorporated the cone mosaic into the visual system, leaving almost no perceptual trace. By considering the mosaic in the context of the visual stages that precede and follow it, the reasons for its invisibility have become apparent. The geometry of the cone array gives rise to at least two optical properties that can, under the right laboratory conditions, affect spatial vision. These two properties are illustrated in Figure 1. The first has to do with the spacing between receptors, r, which can lead to the ambiguity known as aliasing. The second has to do with the diameter of the cone aperture, d, which causes them to act as low-pass spatial filters. These two factors are potential sources of information loss that are quite different in nature, and I will discuss them in turn, explaining why neither has much impact on ordinary spatial vision. 135
136 I.., ,.,. DAVID R WILLL4MS ~ d . . , . >% of. . . . . · ~ ; .. ~ ~ 'i . . ~ . N.; · ~ . .^ )\ -2.\ ... ~ FIGURE 1 A psychophysicist's view of the foveal cone mosaic, characterized by the spacing between rows of cones, r, and cone diameter, d.
THE INVISIBLE CONE MOSAIC ~ o.~o U) ().'o .H ().10 _ ().~: 137 1.()() ~ ,\ \ /~\ Optics \ _ t ~ SO 40 i\ fx Neural Optics \ \ Central fovea Cone aperture 1 ~1 1 I,l I - - - - 60 SO 10() 190 1 4(, Spatial Frequency ( c/deg) FIGURE 2 Modulation transfer functions for the foveal cone aperture and the optics of the eye (2-mm pupil). The foveal cone Nyquist frequency, fN is shown as a vertical dotted line. The curve labeled "Neural" is a contrast sensitivity function arbitrarily normalized to unit at its peak. It represents foveal contrast sensitivity to interference fringes (Williams, 1985~. The dotted portion of this curse indicates spatial frequencies that produce aliasing. CONE SPACING It has been known for some time that under normal viewing conditions blurring by the optics of the eye protects the fovea from aliasing. The ver- tical dotted line in Figure 2 shows the Nyquist frequency at the very center of the fovea, 56 cycles/degree. This estimate comes from psychophysical observations of abasing (Williams, 1985a, 19~, 1989) and is in reason- able agreement with anatomical estimates (Osterberg, 1935; Miller, 1979; Curcio et al., 1987~. Spatial frequencies below the Nyquist frequency are represented veridically by the visual system, while those above it are subject to aliasing. The modulation transfer function (MTF) of the eye's optics chosen here is that of Campbell and Gubisch (1966) obtained with a 2-mm pupil. The diffraction limit dictated by this pupil size is 63 cycles/degree at 555 nm. Slightly larger pupils could produce slightly more contrast at higher spatial frequencies, but in general there is precious little modulation in the retinal image in the aliasing range. The situation is considerably different in the parafovea and periphery, where the MTF of the eye probably exceeds the cone Nyquist frequency. Unfortunately, we have surprisingly little information about the off-axis image quality of the human eye. I am not aware of any measurements of the MTF obtained with a normal daylight pupil size. Jennings and Charman
138 DAVID R WILLIAMS (1981) are frequently cited for their claim that optical quality is relatively constant across the central 25 deg of the retina. However, their data were obtained with a fully dilated 7.5-mm pupil, so their line-spread functions, even at the fovea, are roughly four times broader than those of Campbell and Gubisch (1966), obtained with an optimum pupil size. Though it remains to be seen how good the off-axis optical quality of the eye is under optimal conditions, there is no doubt that the retinal image there can be modulated at spatial frequencies above the cone Nyquist frequency. Bergman (1858) described effects observed near the grating resolution limit that may be attributable to aliasing under normal viewing conditions. Smith and Cass (1987) and Thibos and Still (1988) have confirmed this, also without the use of high-contrast interference fringes. Some observers can see extrafoveal aliasing by viewing a unity con- trast grating with the eccentric retina. A Ronchi ruling placed on a light table works reasonably well. It helps to orient the grating parallel to the retinal meridian in which the grating lies since ocular aberrations, such as lateral chromatic aberration, will reduce contrast less. Gratings with a spatial frequency above the cone Nyquist frequency (e.g., above roughly 15 cycles/degree at 10-de" eccentricity in the nasal retina) can take on the appearance of dynamic two-dimensional noise. Under these conditions there is little or no perceptual evidence for the original grating: the stripes cannot be resolved and their orientation cannot be discerned. However, the speckled appearance of the grating distinguishes it from a uniform field of the same luminance (such as a neutral density filter matched to the average transmittance of the Ronchi ruling). Nonetheless, there are a number of reasons why cone aliasing is not a severe problem in the extrafovea under normal viewing conditions (Williams, 1986~. Off-axis aberrations and perhaps increased retinal scatter help to some extent. The power spectra of natural scenes typically decline with increasing frequency. The high spatial frequency, high-contrast, si- nusoidal gratings used in the laboratory are rare events in natural scenes. Natural scenes are typically complex, so that weak aliasing effects may be masked by the predominant spatial frequencies in the image that lie below the Nyquist frequency. Furthermore, small refractive errors can produce dramatic losses in contrast at high frequencies, and it seems likely that the peripheral retina is typically less well refracted than the fovea. Even with laser interference fringes, the most visible aliasing effects we have been able to produce are never more than about five times contrast threshold. Yellott (1982) has suggested that disorder in the cone mosaic plays a major role in eliminating aliasing in the periphery, since it smears aliasing energy into a broad range of spatial frequencies and orientations. However, contrast sensitivity for detecting aliasing in the extrafoveal retina where the mosaic is disordered is not very different from that for detecting aliasing
THE INVISIBLE CONE MOSAIC U) A o o £ 139 of\ PRE- SAMPL I NG I \ \ F I LTER I \ \ / t:YQUIST \ \/ ~ LACY \ \ 1 POST-SAMPLING \ FI LEER L \ ~ 0 - 5 fN f N 1.5 f N SPATIAL E~REQU=CY FIGURE 3 Hypothetical scheme, qualitatively similar to that found in peripheral retina, that allows low-pass filtering following cone sampling to eliminate aliasing under nonnal viewing conditions. with the more regular foveal mosaic. To be sure, disorder in the mosaic prevents regular patterns from being aliased into regular moire patterns. But such regular patterns are rare in nature, and it seems more likely that the difference in regularity of the cone mosaic in fovea and periphery is a consequence of some disruptive effect of rods outside the foveal center. Hirsch and Hylton (1984) have pointed out that neural filtering fol- lowing the sampling process can reduce aliasing distortion, though at the expense of overall spatial bandwidth. Consider the hypothetical example illustrated in Figure 3. Suppose the optical attenuation preceding sam- pling, labeled "pre-sampling filter" in the figure, has a cutoff at 1.5 times the Nyquist frequency, f N. Aliasing corresponds to a reflection of signals above fN to corresponding spatial frequencies below, as shown by the hatched area. However, if a postsampling filter, which might correspond, for example, to the centers of ganglion cell receptive fields, had a cutoff frequency as high as only 0.5 f N. the system as a whole would be protected from aliasing. Disorder in the cone mosaic would reduce the effectiveness of this scheme somewhat, but it could still greatly reduce the visibility of aliasing. A situation similar to this exists in the peripheral retina. Even with interference fringes, contrast sensitivity is clearly limited by postrecep- toral factors in the peripheral retina, and acuity falls far short of the cone Nyquist frequency. So the scheme suggested by Hirsch and Hylton may
40 DAVID R W7LLL4MS operate there. There are a large number of factors that mitigate against aliasing distortion outside the fovea, so that the granularity of the cone mosaic does not intrude in everyday vision. CONE DIAMETER The mosaic as a whole can never resolve a pattern that cannot be re- solved by its individual elements. The cones in the human eye are typically smaller than the point-spread function of the eye's optics, so that the con- trast attenuation produced by individual cones is negligible. Figure 2 shows a psychophysical estimate of the aperture of foveal cones from MacLeod et al. (1985~. Their estimate was obtained by measuring the contrast de- modulation in the visual system prior to an intensive nonlinearity. This estimate should be considered as a lower bound on the spatial bandwidth of receptors because of uncertainty about the effect of additional sources of demodulation besides the receptor aperture, such as retinal scatter. The true curves might actually be shallower than this but are not likely to be steeper. This estimate is slightly shallower than that of Miller and Bernard (1983), which was based on anatomical measurements of inner-segment diameter in the primate. MacLeod, Williams, and Makous (1985) calculate an upper bound on the effective foveal cone diameter of about 1.8 microns, compared with their estimate of 2.4. But what is the relationship between anatomical inner-segment diameter and the functional light-gathering aper- ture of cones? How does the presence of neighboring cones affect the waveguide aperture? Our present knowledge of cone wave~uide r)ror)erties is insufficient to answer these questions. -my- - - r--r- This leaves us with another puzzle, since cones everywhere could benefit from being somewhat larger, which would allow them to catch more photons without producing appreciable contrast loss in the "visible" range of spatial frequencies. Why then are cones so small for a given spacing between them? In the fovea perhaps there are physical constraints that prevent cones from occupying more of the image plane. Is their size in the periphery limited by a requirement to make room for quantum-hung~y rods? Or are there other factors that limit their size? In any case it is clear that neither the cone aperture nor the spacing between cones has an important impact on spatial vision (Williams, 1985b), the possible exception being a modest amount of aliasing in extrafoveal vision under ordinary viewing conditions.
TlIE INVISIBLE CONE MOSAIC SAMPLING BY THE THREE CONE MOSAICS 141 So far I have been treating the cone mosaic as a homogeneous pop- ulation, ignoring the three submosaics of short (S), middle (M), and long wavelength-sensitive (L) cones that comprise it. The spatial sampling rate of each submosaic is of course lower than the sampling rate of the mosaic as a whole. One might suppose that this would make each submosaic susceptible to aliasing distortion. But as with the mosaic as a whole, the visual system has succeeded in hiding the three submosaics from subjective experience. The S cone mosaic runs the greatest risk of undersampling because it accounts for less than 10 percent of the cone population. However, as Yellott et al. (1984) have argued, it is protected in part by axial chromatic aberration in the eye, which is most severe at the short wavelengths to which the B cones are sensitive. Under most conditions the short wavelengths to which the S cones are sensitive produce such a blurred image that the array of S cones is adequate to represent it. Williams and Collier (1983) (see also Williams et al., 1983) showed that aliasing with the S cone mosaic can be seen, but it required careful correction for chromatic aberration, and even then the phenomenon was difficult to detect. Are the M and L cone mosaics susceptible to aliasing? One view would hold that it would be: if foveal resolution is limited by cone sampling, then visual acuity for a grating that modulates only a single cone type should be reduced as a result of an effective reduction in spatial sampling rate. For example, assume for simplicity that the loss of one cone type through chromatic adaptation reduces the number of effective sampling elements underlying the fringe by a factor of 2. (The actual value will be somewhat more or less depending on the ratio of M to L cones.) If the remaining cones were arranged in a perfectly regular lattice, the Nyquist frequency of the submosaic would be 1// or 71 percent of its value when M and L cones operate together. This simplistic application of sampling theory would then predict that visual resolution would be reduced correspondingly. However, Green (1968) and Cavonious and Estevez (1975) measured contrast sensitivity at high spatial frequencies under conditions designed to isolate M or L cones. Neither study found a difference in performance at high spatial frequencies when only one submosaic mediated detection compared with both cone classes operating together. Nancy Coletta and I recently took another look at this issue, making measurements of visual acuity with laser interference fringes superimposed on chromatic back- grounds. We hoped the use of these high-contrast stimuli would improve our chances of seeing a difference. But so far we also have failed to find a difference between M cone acuity, L cone acuity, and M plus L cone acuity. Chromatic adaptation was used to attempt to isolate M or L cones
142 DAVID R WILLIAMS in the manner of Green (1968~. Measurements were made for three kinds of interference fringe stimuli: (1) those of equal contrast for M and L cones produced by superimposing either a 488 nm or a 633 nm fringe on a background of the same wavelength, (2) fringes that favored M cone grating detection in which a unity contrast 488 nm grating was superimposed on a 660 nm uniform adapting field, and (3) fringes that favored L cone grating detection in which a unity contrast 633 nm grating was superimposed on a 460 nm background. The observer made two settings for each of these conditions of chromatic adaptation: (1) he adjusted the spatial frequency of the fringe to his resolution limit, defined as the highest spatial frequency at which he could perceive fine stripes at the appropriate orientation, and (2) he adjusted the spatial frequency of the interference fringe to produce the coarsest zebra stripes at the foveal center. That is, he identified a "moire zero" (Williams, 1988) in which the period of the grating matches the spacing between receptors. The mean settings for both these tasks are plotted in Figure 4. The abscissa is an index of the cone isolation achieved by the three adaptation conditions, where O indicates perfect M cone isolation, 1 indicates perfect L cone isolation, and 0.5 indicates equal contrasts in both M and L cones. The contrasts in the M and L cones for each of these stimuli were calculated using Smith and Pokorny fundamentals (Boynton, 1979~. These conditions were chosen, based on the results of contrast sensitivity measurements, so that both fringes near the resolution limit and zebra stripes would be above contrast threshold for the favored cone mechanism but well below threshold for the unfavored cone mechanism. The results show that neither resolution nor the moire zero depends on the ratio of contrasts in the two cone types, even when the ratio approaches a factor of 10. Furthermore, the observer could detect no obvious subjective difference between the zebra stripes seen when both cone types operated together and when one or the other cone type was strongly favored. Why should foveal resolution not decrease when only one submosaic is operating? The logic that leads to the prediction of a resolution loss under chromatic adaptation rests on the assumption that the remaining submosaic is perfectly regular. This is unlikely to be true. Williams and Coletta (1987) have shown that under some conditions observers can reliably identify the orientation of a grating even when its spatial frequency exceeds the Nyquist limit by 50 percent. That is, the invariance of visual resolution with chromatic adaptation is consistent with the supra-Nyquist visual resolution previously observed in the pare fovea. Despite the silencing of sampling elements by chromatic adaptation, there are apparently sufficient samples remaining to provide the observer with enough information to recognize a grating. It seems that the visual system can sustain substantial loss of sampling elements and still retain high visual acuity. This observation is
THE INVISIBLE CONE MOSAIC 120 100 ~0 g 60 40 u, 20 143 . O O O COARSEST ZEBRA STRI PES AD _ RESOLUTION _1 I ~ ~ I I ~ I 1 1 0.0 0.5 i.0 G CONE R CONTRAST R CONE ISOLATION R CONTRAST + G CONTRAST ISOLATION FIGURE 4 Measurements of interference fringe acuity and the coarsest aliasing patterns obtained under venous conditions of chromatic adaptation. relevant to the clinical assessment of retinal damage since visual acuity measures are not likely to be sensitive to random losses of large numbers of neural elements across the visual field. Why does the moire zero not change when only one submosaic is operating? The moire zero should not change because the silencing of sampling elements from a mosaic, in either a random or regular fashion, does not eliminate the periodicity in the mosaic that is responsible for the moire zero. This fact can be understood by considering Figure 5. The left half of the image shows a regular triangular lattice sampling a grating whose period nearly equals the sample spacing. That is, the grating frequency is near a moire zero for the mosaic. The low-frequency diagonal grating is the resulting moire or alias. On the right, two-thirds of the samples from the original array have been removed in random fashion. Clearly, the moire is largely unaffected. Figure 6 shows that this is true even if the removal of cones is nonrandom. In this case, on the right, only every third row of
44 DAVID R WILLIAMS FIGURE ~ Square wave grating sampled by a regular triangular lattice on the left and the same lattice with two-thirds of the elements removed at random on the right (a submosaic). The grating frequency nearly equals twice the Nyquist frequency of the complete mosaic, producing an alias (low-~equengy diagonal grating) that is conspicuous with the submosaic as well. receptors has been retained, yet the moire is the same. So there is in fact no theoretical expectation that the spatial frequency yielding the coarsest zebra stripes should change when one submosaic is effectively removed. This does not imply that there are no differences in the aliasing effects predicted for a complete mosaic and the submosaics that comprise them. The submosaic in Figure 5 introduces aliasing noise at all input spatial frequencies, unlike the complete mosaic. More interesting effects arise when the submosaic has a regularity over and above that of the mosaic as a whole. For example, the submosaic in Figure 6 has a Nyquist frequency for horizontal gratings that is only a third that of the complete mosaic. This implies that there will be a second moire zero for the submosaic at one-third the frequency of the moire zero shared by both mosaics. That is, regular submosaics should show additional moire effects at low frequencies over and above those seen with the complete sampling array.
THE INVISIBLE CONE MOSAIC 145 FIGURE 6 Square wave grating sampled by a regular triangular lattice on the left and the same lattice with two-thirds of the rows of cones removed on the right (a submosaic). The grating frequency nearly equals twice the Nyquist Sequence of the complete mosaic, producing an alias (low-~equengy diagonal grating) that is conspicuous with the submosaic as well. Sekiguchi, Williams, and Packer (in press) have simulated the aliasing effects predicted by a variety of packing arrangements of the M and L cones. These simulated effects can be quite striking and yield "chromatic aliases." These aliases should vary in chromaticity as well as brightness even when viewing a single monochromatic interference fringe. We are presently engaged in a search for chromatic aliasing, since the spatial frequencies and orientations of the gratings that produce them would specify the packing arrangement of M and L cones. We have identified a red-green zebra stripe pattern when viewing interference fringes near the foveal Nyquist frequency (Sekiguchi et al., in press), but unfortunately this effect does not seem to be generated by chromatic abasing. Indeed, it is interesting that chromatic aliasing is not more obvious, and this may indicate that the assignment rule for the M and L cones is not particularly regular. Completely random submosaics could produce two-dimensional
146 DAVID R W7LLL4MS aliasing noise but no regular chromatic zebra stripes (Ahumada, 1986~. Alternatively, it may be that the low spatiotemporal bandwidth of chromatic mechanisms makes it impossible to see chromatic aliasing. Our failure to observe chromatic aliasing with gratings so far is still something of a puzzle, however, since a quite different set of observations suggest that it should be possible to see it. The observation that the color appearance of foveal point sources fluctuates from flash to flash has been attributed to the sampling effects of the M and L cone mosaic (Krauskopf, 1964, 1978; Krauskopf and Srebro, 1965; Cicerone and Nerger, 1989; Vimal et al., 1989~. In any case, the M and L cone submosaics, like the mosaic as a whole, are apparently well protected from aliasing distortion under normal viewing conditions. The extreme ratios of M to L cone stimulation that can be produced with appropriate choices of monochromatic light in the laboratory are a far cry from the effects of the smooth and continuous reflectance spectra of real scenes. These submosaics are presumably less susceptible to undersampling both because of their high densities~ompared with the S cones, for example and because the overlap in their spectral sensitivity curves ensures that the images in each of the two submosaics are-highly correlated. This is not to say that effects cannot be found in the laboratory, and we are in the process of trying to develop more sensitive tests to reveal them. But the stimuli we have already used are likely to be far more selective than those encountered in ordinary viewing. Since this meeting, evidence for aliasing by the M and L cone submo- saics has been found. Williams, Sekiguchi, and Packer (1990) have argued that the red and green subjective colors seen when viewing fine achromatic gratings are caused by M and L cone aliasing. But what about arrays of cells deeper in the visual system? Do they produce aliasing? A likely candidate would be the array of ganglion cells in the peripheral retina, whose density is SLY to nine times lower than that of cones. Coletta and Williams (1987a, 1987b, 1987c) have shown that the ability to detect interference fringes above the cone Nyquist frequency must be due in part to cone aliasing. But is it possible that some of this aliases originates at sites deeper in the visual system? Or has the postreceptoral visual system protected itself from aliasing? One simple way of doing this Is to make receptive field centers large enough that they demodulate spatial frequencies above the array's Nyquist frequency. Precisely how well the visual system succeeds at this is not yet known. REFERENCES Ahumada, A., Jr. 1986 Models for the arrangement of long and medium wavelength cones in the central fovea. Joumal of the Optical Society of America A 3:P92.
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148 DAVID R WILLL4MS Sekiguchi, N., D.R. Williams, and O. Packer in press Distortion of Nyquist frequency gratings in foveal vision. Vision Research. Smith, R.A., and P.F. Cass 1987 Aliasing in the parafovea with incoherent light. Joumal of the Optical Society of America A 4:1530-1534. Thibos, UN., and D.L Still 1988 What limits visual resolution in peripheral vision? Investigative Ophthabnology and Vision Science 29(Suppl.~:138. Vimal, R.LP., J. Pokorny, V.C. Smith, and S.K. Shevell 1989 Foveal cone thresholds. Vision Research 29:61-78. Williams, D.R. 1985 Aliasing in human foveal vision. Vision Research 25:195-205. 1986 Seeing through the photoreceptor mosaic. Trends in Neuroscience 9:193-198. 1988 Topography of the foveal cone mosaic in the living human eye. Vision Research 28:433 454. in press Photoreceptor sampling and aliasing in human vision. In Tutorials in Optics, Duncan Moore, ed. Number 1, 1987 OSA Annual Meeting. Joumal of the Optical Society of America Williams, D.R., and N.J. Coletta 1987 Cone spacing and the visual resolution limit. Journal of the Optical Society of America A 4:151~1523. Williams, D.R., and R.J. Collier 1983 Consequences of spatial sampling by a human photoreceptor mosaic. Science 221:385-387. Williams, D.R., R.J. Collier, and B.J. Thompson 1983 Spatial resolution of the short-wavelength mechanism. Pp. 487-508 in Colour Vision: Physiology and Psychophysics, J.D. Mollon and L^l: Sharpe, eds. London: Academic Press. Williams, D.R., N. Sekiguchi, and O. Packer 1990 Spatial aliasing by chromatic mechanisms. Investigative Ophthalmology and Vision Science (Supp.) 31~4):494. Yellott, J.I., Jr. 1982 Spectral analysis of spatial sampling by photoreceptors: topological disorder prevents aliasing. Vision Research 22:1205-1210. Yellott, J.I., Jr., B.A. Wandell, and T.N. Cornsweet 1984 The nervous system. Pp. 257-316 in Handbook of Physiology. Bethesda, Md.: American Physiological Society.
