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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 2, Iss. 10 — Oct. 1, 1985
  • pp: 1769–1786

Intensity-dependent spatial summation

Tom N. Cornsweet and John I. Yellott, Jr.  »View Author Affiliations

JOSA A, Vol. 2, Issue 10, pp. 1769-1786 (1985)

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Psychophysical evidence indicates that, in the human retina, the size of the spatial-summation area decreases as illuminance increases. Such a relationship would be beneficial for the detection of spatial contrast in the presence of photon noise. We analyze an image-processing mechanism in which the area of a strictly positive point-spread function varies inversely with local illuminance while its volume remains constant. In addition to its expected effect of improving spatial resolution as illuminance increases, this mechanism also yields center-surround antagonism and all other manifestations of bandpass filtering and accounts for Ricco’s law and Weber’s law—including the failures of both laws as a function of test conditions. The relationship between this mechanism and lateral inhibition is analyzed.

© 1985 Optical Society of America

Original Manuscript: November 13, 1984
Manuscript Accepted: April 16, 1985
Published: October 1, 1985

Tom N. Cornsweet and John I. Yellott, "Intensity-dependent spatial summation," J. Opt. Soc. Am. A 2, 1769-1786 (1985)

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  1. Psychophysical demonstrations of spatial summation begin with A. Ricco, “Relazione fra il minimo angolo visuale e l’intensita luminoso,” Ann. Ottalmol. 6, 373–479 (1877);the literature is reviewed by P. E. Hallett, “Spatial summation,” Vision Res. 3, 9–24 (1963)and by B. Sakitt, “Configuration dependence of scotopic spatial summation,” J. Physiol. (London) 216, 513–529 (1971);Physiological demonstrations of spatial summation in the vertebrate retina begin with H. K. Hartline, “The receptive fields of optic nerve fibers,” Am. J. Physiol. 130, 690–699 (1940);a recent review of that literature is P. O. Bishop, “Processing of visual information within the retinostriate system,” in Volume III of the Handbook of Physiology: The Nervous System, I. Darian-Smith, ed. (American Physiological Society, Bethesda, Md., 1984);spatial summation at the photoreceptor level (receptor coupling) was first reported by D. A. Baylor, M. G. F. Fourtes, P. M. O’Bryan, “Receptive fields of cones in the retina of the turtle,” J. Physiol. (London) 214, 265–294 (1971);many studies are described in H. B. Barlow, P. Fatt, eds., Vertebrate Photoreception (Academic, New York, 1977). [CrossRef]
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  4. When the illuminance is I, the quantum catch is a Poisson random variable with mean and variance IA, and when the illuminance rises to I+ cI, the catch is Poisson with mean and variance (1 + c)IA. Taking the probability of detecting the change to be the probability that the catch for I+ cI exceeds the catch for I and using the normal approximation to the Poisson, it follows that the detection probability is the probability that a normal random variable with mean cIA and variance IA(2 + c) is greater than zero. To make this probability greater than 0.999, cIA/[IA(2 + c)]1/2 must be greater than 3. The order-of-magnitude value 10/c2 underestimates the actual required value of IA [i.e., (9/c2)(2 + c)] by a factor ranging from 0.55 (when c = 0.01) to 0.37 (when c = 1).
  5. A. Rose, “The sensitivity performance of the human eye on an absolute scale,” J. Opt. Soc. Am. 49, 645–663 (1948);Vision: Human and Electronic (Plenum, New York, 1974).
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  8. H. B. Barlow, R. Fitzhugh, S. W. Kuffler, “Change of organization in the receptive fields of the cat’s retina during dark adaption,” J. Physiol. (London) 137, 338–354 (1957).
  9. C. Enroth-Cugell, J. G. Robson, “The contrast sensitivity of the ganglion cells of the cat,” J. Physiol. (London) 187, 517–552 (1966);A. M. Derrington, P. Lennie, “The influence of temporal frequency and adaptation level on receptive field organization of retinal ganglion cells in cat,” J. Physiol. (London) 333, 343–366 (1982).These experiments measured spatial CSF’s for individual X cells over a wide range of mean luminance levels and fit them with modulation transfer functions implied by a linear difference-of-Gaussians receptive field model. In both cases the X-cell CSF changed from bandpass to low pass as mean luminance fell from photopic levels to near absolute threshold, indicating a loss of lateral inhibitory effects. The parameters of the best fitting MTF’s implied that this change was due almost entirely to changes in the relative sensitivities of the center and surround mechanisms: The spatial areas of the center and surround apparently changed very little with mean luminance. Analyzing these data from a signal-detection standpoint, we find that that interpretation implies a very large decrease in the quantum efficiency of the cat retina with light adaptation: for Derrington and Lennie’s X-cell 25-J (their Fig. 9) quantum efficiency apparently fell by around 4 log units as mean luminance increased from 3.8 × 10−5 to 200 cd/m2. Psychophysical evidence indicates that human quantum efficiency falls by only a factor of 10 over the same range (Ref. 5). Comparative visual-acuity measurements show that as mean luminance rises from 10−5 to 10 cd/m2, human visual acuity improves by a factor of 30, whereas cat acuity rises by only a factor of 3. [T. Pasternak and W. H. Merigan, “The luminance dependence of spatial vision in the cat,” Vision Res. 21, 1333–1339 (1981)]. Taken altogether, these results suggest that cat and human retinas respond quite differently to changes in the light level. We are not aware of any study measuring spatial CSF’s for primate retinal ganglion cells as a function of mean luminance, but we would expect substantial changes in the apparent size of receptive fields.
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  11. D. H. Kelly, “Adaptation effects on spatio-temporal sine-wave thresholds,” Vision Res. 12, 89–101 (1972). [CrossRef] [PubMed]
  12. H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. (London)136, 469–488 (1957).
  13. M. Aguilar, W. S. Stiles, “Saturation of the rod mechanism of the retina at high levels of stimulation,” Opt. Acta 1, 59–65 (1954). [CrossRef]
  14. H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979). [CrossRef]
  15. B. Sakitt, “Configurational dependence of scotopic spatial summation,” J. Physiol. (London) 216, 513–529 (1971).
  16. B. H. Crawford, “Visual adaptation in relation to brief conditioning stimuli,” Proc. R. Soc. London Sec. B 134, 283–302 (1947). [Data shown as Fig. 3.10 in H. Ripps and R. A. Weale, “Visual adaptation,” in The Eye, 2nd ed., H. Davson, ed. (Academic, New York, 1976), Vol. 2A.] [CrossRef]
  17. F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, Calif., 1965).
  18. J. Krauskopf, “The effect of retinal image stabilization on the appearance of heterochromatic targets,” J. Opt. Soc. Am. 53, 741–744 (1963). [CrossRef] [PubMed]

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