Abstract

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

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Errata

Tom N. Cornsweet and John I. Yellott, "Intensity-dependent spatial summation: errata," J. Opt. Soc. Am. A 3, 165-165 (1986)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-3-1-165

References

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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]
  2. H. B. Barlow, “Temporal and spatial summation in human vision at different background intensities,” J. Physiol. (London) 141, 337–350 (1958).
  3. V. D. Glezer, “The receptive fields of the retina,” Vision Res. 5, 497–525 (1965).
    [CrossRef] [PubMed]
  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).
  6. D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
    [CrossRef]
  7. S. Schlaer, “The relation between visual acuity and illumination,” J. Gen. Physiol. 21, 165–188 (1937). [Schlaer’s data are shown in Fig. 3 of J. P. Thomas, “Spatial resolution and spatial interaction,” in Handbook of Perception, E. C. Carterette and M. P. Friedman, eds. (Academic Press, New York, 1975), Vol. V, Chap. 7.]
    [CrossRef]
  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.
  10. F. L. Van Ness, M. A. Bouman, “Spatial modulation transfer in the human eye,” J. Opt. Soc. Am. 57, 401–406 (1967).
    [CrossRef]
  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]

1980

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[CrossRef]

1979

H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
[CrossRef]

1972

D. H. Kelly, “Adaptation effects on spatio-temporal sine-wave thresholds,” Vision Res. 12, 89–101 (1972).
[CrossRef] [PubMed]

1971

B. Sakitt, “Configurational dependence of scotopic spatial summation,” J. Physiol. (London) 216, 513–529 (1971).

1967

1966

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.

1965

V. D. Glezer, “The receptive fields of the retina,” Vision Res. 5, 497–525 (1965).
[CrossRef] [PubMed]

1963

1958

H. B. Barlow, “Temporal and spatial summation in human vision at different background intensities,” J. Physiol. (London) 141, 337–350 (1958).

1957

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).

H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. (London)136, 469–488 (1957).

1954

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]

1948

1947

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]

1937

S. Schlaer, “The relation between visual acuity and illumination,” J. Gen. Physiol. 21, 165–188 (1937). [Schlaer’s data are shown in Fig. 3 of J. P. Thomas, “Spatial resolution and spatial interaction,” in Handbook of Perception, E. C. Carterette and M. P. Friedman, eds. (Academic Press, New York, 1975), Vol. V, Chap. 7.]
[CrossRef]

1877

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]

Aguilar, M.

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]

Barlow, H. B.

H. B. Barlow, “Temporal and spatial summation in human vision at different background intensities,” J. Physiol. (London) 141, 337–350 (1958).

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).

H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. (London)136, 469–488 (1957).

Bergen, J. R.

H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
[CrossRef]

Bouman, M. A.

Crawford, B. H.

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]

Enroth-Cugell, C.

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.

Fitzhugh, R.

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).

Glezer, V. D.

V. D. Glezer, “The receptive fields of the retina,” Vision Res. 5, 497–525 (1965).
[CrossRef] [PubMed]

Hildreth, E.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[CrossRef]

Kelly, D. H.

D. H. Kelly, “Adaptation effects on spatio-temporal sine-wave thresholds,” Vision Res. 12, 89–101 (1972).
[CrossRef] [PubMed]

Krauskopf, J.

Kuffler, S. W.

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).

Marr, D.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[CrossRef]

Ratliff, F.

F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, Calif., 1965).

Ricco, A.

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]

Robson, J. G.

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.

Rose, A.

Sakitt, B.

B. Sakitt, “Configurational dependence of scotopic spatial summation,” J. Physiol. (London) 216, 513–529 (1971).

Schlaer, S.

S. Schlaer, “The relation between visual acuity and illumination,” J. Gen. Physiol. 21, 165–188 (1937). [Schlaer’s data are shown in Fig. 3 of J. P. Thomas, “Spatial resolution and spatial interaction,” in Handbook of Perception, E. C. Carterette and M. P. Friedman, eds. (Academic Press, New York, 1975), Vol. V, Chap. 7.]
[CrossRef]

Stiles, W. S.

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]

Van Ness, F. L.

Wilson, H. R.

H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
[CrossRef]

Ann. Ottalmol.

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]

J. Gen. Physiol.

S. Schlaer, “The relation between visual acuity and illumination,” J. Gen. Physiol. 21, 165–188 (1937). [Schlaer’s data are shown in Fig. 3 of J. P. Thomas, “Spatial resolution and spatial interaction,” in Handbook of Perception, E. C. Carterette and M. P. Friedman, eds. (Academic Press, New York, 1975), Vol. V, Chap. 7.]
[CrossRef]

J. Opt. Soc. Am.

J. Physiol. (London)

B. Sakitt, “Configurational dependence of scotopic spatial summation,” J. Physiol. (London) 216, 513–529 (1971).

H. B. Barlow, “Increment thresholds at low intensities considered as signal/noise discriminations,” J. Physiol. (London)136, 469–488 (1957).

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).

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.

H. B. Barlow, “Temporal and spatial summation in human vision at different background intensities,” J. Physiol. (London) 141, 337–350 (1958).

Opt. Acta

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]

Proc. R. Soc. London Sec. B

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]

Proc. R. Soc. London Ser. B

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[CrossRef]

Vision Res.

V. D. Glezer, “The receptive fields of the retina,” Vision Res. 5, 497–525 (1965).
[CrossRef] [PubMed]

H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
[CrossRef]

D. H. Kelly, “Adaptation effects on spatio-temporal sine-wave thresholds,” Vision Res. 12, 89–101 (1972).
[CrossRef] [PubMed]

Other

F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, Calif., 1965).

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).

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Figures (15)

Fig. 1
Fig. 1

Schematic diagram of the IDS model. From top to bottom: input image profile (here, a sharp edge); photoreceptors; photoreceptor point-spread functions (for the Gaussian case of the model); output channels (arrows); output image profile (dots).

Fig. 2
Fig. 2

Point-spread functions of the Gaussian case of the IDS model shown for four input intensities.

Fig. 3
Fig. 3

Edge-response profiles. The input image was a step at zero from intensity I to I + D. I/D = 10 in all cases. Curve 1, I = 0.1; curve 2, I = 1; curve 3, I = 10; curve 4, I = 100.

Fig. 4
Fig. 4

Bar-response profiles for a narrow bar (top) and a wide bar (bottom) on a high-intensity background. Background intensity, I; bar intensity, I + ΔI. Bar widths are as indicated in the figure. Tick marks on the abscissa indicate a width of 1.0.

Fig. 5
Fig. 5

Response profiles for the same bars as in Fig. 4 when the background has low intensity.

Fig. 6
Fig. 6

Sinusoidal grating response profiles for high-contrast (90%) and low-contrast (20%) gratings.

Fig. 7
Fig. 7

MTF’s of the Gaussian IDS model for various input mean luminance levels.

Fig. 8
Fig. 8

Response profiles for square spots on a dark background. Spot area (A) times intensity (I) was held constant at 10. Curve 1, I = 1, A = 10; curve 2, I = 10, A = 1; curve 3, I = 100, A = 0.1; curve 4, I = 1000, A = 0.01.

Fig. 9
Fig. 9

Response profiles for square spots on nonzero backgrounds of various intensities. The input images were squares of intensity I + D surrounded by backgrounds of intensity I. The spot area (A) times its incremental intensity (D) was held constant (D × A = 10), and responses were computed for A = 0.01, 0.1, 1, and 10. Upper left, background intensity I = 0.01; upper right, I = 0.1; lower left, I = 1; lower right, I = 10.

Fig. 10
Fig. 10

Increment threshold as a function of test spot area for background fields of various intensities. The input images were square spots of area A and intensity I + D surrounded by uniform background fields of intensity I. Each curve shows, in log–log coordinates, the value of D required to produce a peak response of 1.15 as A increases over eight log units. Background intensities range from I = 1000 (top curve) to I = 0.001 (bottom curve). The diagonal straight line represents Ricco’s law; each curve follows this line up to some area value and then departs from it as shown.

Fig. 11
Fig. 11

TVI curves for test spots of different areas. These are replots of data from Fig. 10. Each curve shows the incremental intensity D required to produce a fixed peak-response value when the input is a square spot of area A and intensity I + D, surrounded by a background of intensity I. The three curves shown are for A = 0.01, A = 1, and A = 100. As background intensity increases, all curves eventually terminate in a diagonal straight line corresponding to Weber’s law.

Fig. 12
Fig. 12

Response profiles at threshold for a spot of fixed area on backgrounds of various intensities. Each curve shows the profile of the response to a square test spot of area A = 1 and intensity I + D surrounded by a uniform background of intensity I. The increment value D in each case is that required to produce a peak output value of 1.15. Top profile, background intensity I = 0.01; middle, I = 10; bottom, I = 1000.

Fig. 13
Fig. 13

Response profiles for a small square spot of fixed incremental intensity (D = 1000) superimposed upon a high-intensity (top curve, I = 100) or a low-intensity (bottom curve, I = 0.1) background. Spot width, 0.1.

Fig. 14
Fig. 14

Configurational effects within Ricco’s area. The top curve is the response profile for a single square spot with (area × intensity) = 10. The bottom curve is the response profile for a pair of square spots whose combined (area × intensity) value was also 10.

Fig. 15
Fig. 15

Component curves for the point-spread function of a constant-volume model that differs from both the IDS model and the LLI model. Here the point spread is the sum of two functions, one whose height increases proportionally with the input intensity I [here, a Gaussian of the form I × G1(x, y), where G1 has a fixed standard deviation σ1] and another whose height varies as cI, where c is a positive constant [here, (cI) × G2(x, y), where G2 has fixed standard deviation σ2 and σ2 > σ1]. Component curves for two values of I are shown on the left, and the corresponding composite point-spread functions are shown on the right.

Equations (46)

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I A > 10 / c 2 .
S { [ ( x p ) 2 + ( y q ) 2 ] , I } = I × S { I × [ ( x p ) 2 + ( y q ) 2 ] , 1 } .
I ( x , y ) × S { I ( x , y ) × [ ( x p ) 2 + ( y q ) 2 ] } ,
S ( p 2 + q 2 ) d p d q = 1 .
O [ I ( x , y ) ] ( p , q ) = I ( x , y ) × S { I ( x , y ) × [ ( x p ) 2 + ( y q ) 2 ] } d x d y .
S ( x 2 + y 2 ) = ( 1 / 2 π ) × exp [ ( 1 / 2 ) × ( x 2 + y 2 ) ]
[ I ( x , y ) / 2 π ] × exp { ( 1 / 2 ) × I ( x , y ) [ ( x p ) 2 + ( y q ) 2 ] } ,
I ( u j , υ k ) × S { I ( u j , υ k ) × [ ( u p ) 2 + ( υ q ) 2 ] } d u d υ ,
I ( u cos θ + υ sin θ , υ cos θ u sin θ ) × S { I ( u cos θ + υ sin θ , υ cos θ u sin θ ) × [ ( u p ) 2 + ( υ q ) 2 ] } d u d υ ,
O [ c I ( x , y ) ] ( p , q ) = O [ I ( x / c , y / c ) ] ( p c , q c ) .
I ( x / c , y / c ) S { I ( x / c , y / c ) × [ ( x p c ) 2 + ( y q c ) 2 ] } d x d y .
c I ( u , υ ) × S { c I ( u , υ ) × [ ( u p ) 2 + ( υ q ) 2 ] } d u d υ ,
O [ I ( x ) ] ( p , q ) = I ( x ) I ( x ) S { [ ( x p ) I ( x ) ] 2 + [ ( y q ) I ( x ) ] 2 } d y d x .
O [ I ( x ) ] ( p ) = I ( x ) $ { ( x p ) I ( x ) } d x ,
$ ( x ) = S ( x 2 + y 2 ) d y .
$ ( x ) = ( 1 / 2 π ) exp [ ( 1 / 2 ) ( x 2 + y 2 ) ] d y = ( 1 / 2 π ) exp [ ( 1 / 2 ) x 2 ] ,
O [ I ( x ) ] ( p ) = [ I ( x ) / 2 π ] × exp [ ( 1 / 2 ) I ( x ) ( x p ) 2 ] d x .
O ( p ) = N [ x ( I + D ) 1 / 2 ] + N [ x I ] ,
N [ z ] = z ( 1 / 2 π ) exp [ ( 1 / 2 ) x 2 ] d x .
O ( P max ) = N { [ ( 1 + I / D ) log ( 1 + D / I ) ] 1 / 2 } + N { [ ( I / D ) log ( 1 + D / I ) ] 1 / 2 } ,
O [ I ( x ) ] ( p ) = O [ I × V ( x ) ] ( p ) = O [ V ( x / I ) ] ( p I ) = O [ V ( x ) ] ( p I ) .
O ( 0 , 0 ) = 0 I × S { I [ x 2 + y 2 ] } d x d y + 0 ( I + D ) × S { ( I + D ) [ x 2 + y 2 ] } d x d y .
O ( p ) = N [ I × ( p W / 2 ) ] + N [ I × ( p + W / 2 ) ] + N [ ( I + D ) 1 / 2 × ( W / 2 p ) ] N [ ( I + D ) 1 / 2 × ( W / 2 + p ) ] .
O ( p ) = 1 + { 2 π 2 f 2 exp [ 2 π 2 f 2 ] } k cos 2 π f p .
G ( f , 1 ) = 2 π 2 f 2 exp ( 2 π 2 f 2 ) .
O [ L ( 1 + k cos 2 π f x ) ] ( p ) = O [ 1 + k cos 2 π f x / L ] ( p L ) = 1 + { 2 π 2 ( f / L ) 2 exp [ 2 π 2 ( f / L ) 2 ] } k cos 2 π f p .
G ( f , L ) = 2 π 2 ( f / L ) 2 exp [ 2 π 2 ( f / L ) 2 ] .
O [ L ( 1 + k cos 2 π f x ) ] ( p ) = O [ 1 + k cos 2 π f ( x / L ) ] ( p L ) = 1 + G ( f / L ) k cos 2 π ( f / L ) ( p L ) = 1 + G ( f / L ) k cos 2 π f p .
( 1 / π ) 2 ( 1 / σ 2 ) 2 [ 1 ( x 2 + y 2 ) / 2 σ 2 ] exp [ ( 1 / 2 ) ( x 2 + y 2 ) / σ 2 ] .
4 π 2 ( u 2 + υ 2 ) exp [ 2 π 2 σ 2 ( u 2 + υ 2 ) ] ,
4 π 2 f 2 exp ( 2 π 2 σ 2 f 2 ) .
O ( p , q ) = { N [ ( W / 2 p ) I ] N [ ( W / 2 + p ) I ] } × { N [ ( W / 2 q ) I ] N [ ( W / 2 + q ) I ] } .
O [ I ( x , y ) ] ( p , q ) = O ( c I ( x c , y c ) ] ( p , q ) = O [ I ( x , y ) ] ( p c , q c ) .
O ( p , q ) = 1 + ( { N [ A ( W / 2 p ) ] N [ A ( W / 2 + p ) ] } × { N [ A ( W / 2 q ) N [ A ( W / 2 + q ] } ) ( { N [ B ( W / 2 p ) ] N [ B ( W / 2 + p ) ] } × { N [ B ( W / 2 q ) ] N [ B ( W / 2 + q ) ] } ) ,
O [ p , q ] = 1 + ( { N [ A ( W / 2 q ) ] N [ A ( W / 2 q ) ] } × { N [ A ( S / 2 + W / 2 p ) ] N [ A ( S / 2 W / 2 p ) ] } ) + ( { N [ A ( W / 2 q ) ] N [ A ( W / 2 q ) ] } × { N [ A ( S / 2 + W / 2 p ) ] N [ A ( S / 2 W / 2 p ) ] } ) ( { N [ B ( W / 2 q ) ] N [ B ( W / 2 q ) ] } × { N [ B ( S / 2 + W / 2 p ) ] N [ B ( S / 2 W / 2 p ) ] } ) ( { N [ B ( W / 2 q ) ] N [ B ( W / 2 q ) ] } × { N [ B ( S / 2 + W / 2 p ) ] N [ B ( S / 2 W / 2 p ) ] } ) ,
[ I ( x , y ) ] n S { [ I ( x , y ) ] n [ ( x p ) 2 + ( y q ) 2 ] } ,
O [ I ( x , y ) ] ( p , q ) = [ I ( x , y ) ] n × S { [ I ( x , y ) ] n [ ( x p ) 2 + ( y q ) 2 ] } d y d x .
O [ c I ( x , y ) ] ( p , q ) = O [ I ( x / c n , y / c n ) ] ( p c n , q c n ) .
O ( p ) = [ ( 1 + k cos 2 π f x ) 1 / 2 / ( 2 π ) ] × exp [ ( 1 / 2 ) ( 1 + k cos 2 π f x ) ( x p ) 2 ] d x .
[ 1 + ( k / 2 ) cos 2 π f x ] 2 ( k 2 / 4 ) cos 2 2 π f x .
( 1 + k cos 2 π f x ) 1 + ( k / 2 ) cos 2 π f x ,
O ( p ) [ ( 1 + j cos 2 π f x ) / ( 2 π ) ] × exp [ ( 1 / 2 ) ( x p ) 2 ] × exp [ ( j ) ( cos 2 π f x ) ( x p ) 2 ] d x ,
O ( p ) [ ( 1 + j cos 2 π f x ) / ( 2 π ) ] × exp [ ( 1 / 2 ) ( x p ) 2 ] × [ 1 j ( x p ) 2 cos 2 π f x ] d x = 1 j [ 1 / ( 2 π ) ] ( x p ) 2 cos 2 π f x × exp [ ( 1 / 2 ) ( x p ) 2 ] d x + j [ 1 / ( 2 π ) ] cos 2 π f x exp [ ( 1 / 2 ) ( x p ) 2 ] d x j 2 [ 1 / ( 2 π ) ] ( x p ) 2 cos 2 π f x × exp [ ( 1 / 2 ) ( x p ) 2 ] d x .
O ( p ) 1 j [ 1 / ( 2 π ) ] υ 2 cos 2 π f ( υ + p ) × exp [ ( 1 / 2 ) υ 2 ] d υ + j [ 1 / ( 2 π ) ] × cos 2 π f ( υ + p ) exp [ ( 1 / 2 ) υ 2 ] d υ ,
O ( p ) 1 j cos 2 π f p [ 1 / ( 2 π ) ] υ 2 × cos 2 π f υ exp [ ( 1 / 2 ) υ 2 ] d υ + j cos 2 π f p [ 1 / ( 2 π ) ] cos 2 π f υ exp [ ( 1 / 2 ) υ 2 ] d υ .
O ( p ) 1 + [ 2 π 2 f 2 exp ( 2 π 2 f 2 ) ] k cos 2 π f p ,

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