Advances in Photoreception Proceedings of a Symposium on Frontiers of Visual Science (1990) / Chapter Skim
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The Photoreceptor Mosaic as an Image Sampling Device
The Photoreceptor Mosaic as an Image Sampling Device
Pages 117-134
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From page 117... ...
Once this point was settled, the visual consequences of receptor sampling seem to have attracted surprisingly little attention for quite a long time. But in this century the general topic of signal sampling has assumed great practical importance, and mathematicians and engineers have developed sophisticated tools for analyzing the information transmission properties of arbitrary sampling schemes.
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From page 118... ...
, but instead follows a different blueprint based on some other approach to recovering continuous images from discrete samples. Part of this paper deals with image recovery algorithms that do not require the kind of precise architecture assumed by Shannon's theorem and therefore seem better suited to the kind of image sampling found in the primate retina.
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From page 119... ...
Shannon's theorem deals strictly with bandlimited signals and infinite sampling arrays. In the real world, of course, we never encounter either one.
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From page 120... ...
Shannon's theorem generalizes readily to two dimensions in other words, to image sampling (e.g., Goodman, 1968~. There it assumes that the sampling array is a perfect lattice and tells us that any image can be reconstructed from its sample values provided it contains no spatial frequencies higher than a cutoff a Nyquist limit that depend on the spacing of the sample points.
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From page 121... ...
SAMPLING PARAMETERS OF THE FOVEAL CONE MOSAIC Now we turn to the parameters of retinal image sampling by the cones. I will deal first with the fovea and then with the rest of the retina.
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From page 122... ...
after sampling consists of multiple replicas of G one centered at each point in the spectrum of the sampling array.
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From page 123... ...
In the fovea then the spatial bandwidth of the retinal image matches the Nyquist limit implied by overall cone density, and the cone mosaic is locally regular enough to create narrowband aliasing of super-Nyquist .
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From page 124... ...
s . FIGURE 4 Spatial regularity of the central foveal receptor lattice.
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From page 125... ...
But the mosaic overall certainly does not have the kind of perfect lattice structure assumed by Shannon's theorem. IMAGE RECOVERY FROM IRREGULAR SAMPLES: YEN'S THEOREM Does this lack of spatial regularity imply that foveal receptor sampling necessarily loses information?
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From page 126... ...
(C) Yen reconstruction of a sinusoid at one-half the nominal Nyquist frequency.
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From page 127... ...
But instead of using the same sine function for every point, Yen's algorithm requires each point to have its own idiosyncratic interpolation function one that depends on the distances between that point and all the others. Panel B shows the interpolation functions for three of the points from panel ~ One can see that Yen interpolation functions look like distorted sine functions drawn by a careless draftsman: they are often distinctly asymmetric, and no two are exactly alike.
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From page 128... ...
EXTRAFOVEAL IMAGE SAMPLING Yen's theorem allows us to say that if the visual system can learn the actual positions of its foveal cones, then despite spatial disorder in the cone mosaic, receptor sampling of normal foveal images need not impose any loss of information. The critical point is that in the fovea the average sampling rate matches the spatial frequency bandwidth imposed by the optics of the eye.
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From page 129... ...
As cone density decreases beyond that point, the radius of the desert island decreases, but it always has a value that is approximately twice the nominal Nyquist frequency implied by the local density. [A desert island spectrum indicates that the extrafoveal cones are packed essentially at random but are subject to a constraint on the minimum cone-cone distance (Yellott, 1983b)
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From page 130... ...
. The horizontal bar across the desert island portion of the spectn~m marks a distance equal to twice the Nyquist frequency implied by local cone density.
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From page 131... ...
Bottom right: postsampling spectrum when F > R/2. Aliased energy in the form of broadband noise is widely scattered over the Nyquist region, with no concentration at any single sub-Nyquist frequency, and sharp spikes remain at the input frequency points OF.
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From page 132... ...
and the construction of those functions requires knowledge of the actual sampling position. But here again, once the interpolation functions have been created, they are good for life, and the reconstruction of any given input image is computationally just as simple as it would be for spatially regular sampling below the Nyquist limit.
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From page 133... ...
1858 Anatomisches und physiologisches uber die netzhaut des auges. Zeitschnit fir rationelle Medicine 2:83-108.
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From page 134... ...
1983b Nonhomogeneous Poisson disks model the photoreceptor mosaic. Investigative Ophthalmology and Visual Science 24 (Suppl.~:147 (abstract)
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