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OCR for page 135
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
OCR for page 136
136
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DAVID R WILLL4MS
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. .
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FIGURE 1 A psychophysicist's view of the foveal cone mosaic, characterized by the
spacing between rows of cones, r, and cone diameter, d.
OCR for page 137
THE INVISIBLE CONE MOSAIC
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,\ \
/~\
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Central fovea
Cone
aperture
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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
OCR for page 138
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
OCR for page 139
THE INVISIBLE CONE MOSAIC
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139
of\ PRE- SAMPL I NG I
\ \ F I LTER I
\ \ / t:YQUIST
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POST-SAMPLING \
FI LEER
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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
OCR for page 140
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.
OCR for page 141
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
OCR for page 142
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
OCR for page 143
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
OCR for page 144
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.
OCR for page 145
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
OCR for page 146
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.
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1986 Models for the arrangement of long and medium wavelength cones in the
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OCR for page 147
THE INVISIBLE CONE MOSAIC
147
Bergmann, C.
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fir rationelle Medicine 2:83-108.
Boynton, R.M.
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1975 Contrast sensitivity of individual colour mechanisms of human vision. Journal
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Cicerone, C.M., and J.L. Nerger
1989 The relative numbers of long-wavelength-sensitive to middle-wavelength-
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DAVID R WILLL4MS
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Representative terms from entire chapter:
nyquist frequency
