Abstract

We have carried out a series of frequency-of-seeing experiments similar to those performed by Hecht, Shlaer, and Pirenne [J. Gen. Physiol. <b>25</b>, 819–840 (1942)], using an Ar<sup>+</sup> laser operated at 514.5 nm as the source of light. In certain blocks of trials, our subjects were encouraged to report as seen those trials in which the stimulus might have been present. It was determined that sensitivity and reliability were traded against each other over a broad range: for our four subjects, the detection of 147 photons at the cornea with 60% frequency of seeing entailed, on the average, a 1% false-positive rate (FPR), whereas the detection of 34 photons at the cornea with 60% frequency of seeing was accompanied by a 33% FPR. A new neural-counting model has been developed in the framework of signal-detection theory. It combines Poisson stimulus fluctuations with additive and multiplicative neural noise, both of which are known to be present in the visual system at threshold. The resulting probability-of-detection curves, derived from the Neyman Type-A counting distribution, are in good accord with our experimental frequency-of-seeing data for sensible values of the model parameters. We deduce that, on the average, our four subjects are able to detect a single photon at the retina with 60% frequency of seeing, at the expense of a 55% FPR. In Part 2 of this set of papers [P. R. Prucnal and M. C. Teich, Biol. Cybern. <b>43</b>, 87–96 (1982)], we use the normalizing transform, together with probit analysis, to provide improved estimates of threshold parameters, whereas in Part 3 [M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, and W. J. McGill, submitted to Biol. Cybern.], we consider the effects of non-Poisson quantum fluctuations.

© 1982 Optical Society of America

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  39. The Poisson nature of the detected photons follows directly from the small value of the degeneracy parameter of the light, δ = 〈m〉τc/T, where τc, is the coherence time of the light [see L. Mandel, Proc. Phys. Soc. 74, 233–242 (1959); B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978)]. For the source used by HSP, 〈m〉 ≃ 10, T ≃ 10-3 sec, and τc ≃ 10-13 sec, whereas for our multimode Ar+ laser source τc 10-9 sec [see G. Vannucci and M. C. Teich, Appl. Opt. 19, 548–553 (1980)]. In both cases δ « 1. We shall see in Part 3 of this series32 that certain rather common sources of visible radiation, such as television and oscilloscope images produced by cathodoluminescence, as well as image-intensified light [see A. van Meeteren, Vision Res. 18, 257–267 (1978)], in general, generate non-Poisson photon statistics [see Refs. 43 and 47]. Their use in vision experiments therefore requires exceptional care.
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  44. H. Grimm, "Tafeln der Neyman-Verteilung Typ-A," Biom. Z. 6, 10–23 (1964).
  45. M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, and W. J. McGill, "Non-Poisson nature of the effective noise in the visual system near threshold," J. Opt. Soc. Am. 68, 1454(A) (1978).
  46. Several of our stipulations on the format of detection are, no doubt, unnecessarily restrictive. For example, the assumptions of a single counting focus and a fixed integration time could likely be relaxed without seriously affecting the outcome.
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  54. P. R. Prucnal and M. C. Teich, "An increment threshold law for stimuli of arbitrary statistics," J. Math. Psychol. 21, 168–177 (1980).
  55. H. B. Barlow, W. R. Levick, and M. Yooh, "Responses to single quanta of light in retinal ganglion cells of the cat," Vision Res. 11, Suppl. 3, 87–101 (1971).
  56. H. B. Barlow and W. R. Levick, "Threshold setting by the surround of cat retinal ganglion cells," J. Physiol. 259, 737–757 (1976).
  57. P. Lennie, "Scotopic increment thresholds in retinal ganglion cells," Vision Res. 19, 425–430 (1979).
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  70. S. Yeandle and J. B. Spiegler, "Light evoked and spontaneous discrete waves in the ventral nerve photoreceptor of Limulus," J. Gen. Physiol. 61, 552–571 (1973).
  71. We are grateful to H. B. Barlow and W. R. Levick for providing us with experimental data.
  72. W. R. Levick, "Form and function of cat retinal ganglion cells," Nature 254, 659–662 (1975).
  73. P. O. Bishop, W. R. Levick, and W. O. Williams, "Statistical analysis of the dark discharge of lateral geniculate neurones," J. Physiol. 170, 598–612 (1964).
  74. H. Nakahama, N. Ishii, M. Yamamoto, and H. Saito, "Stochastic properties of spontaneous impulse activity in central single neurons," Tohoku J. Exp. Med. 104, 373–409 (1971).
  75. If we assume that the counting time is very large in comparison with the decay times τp, for all k stages, and that the multiplication parameter of each stage is given by αj, the variance-to-mean ratio is49 [equation]. This is the origin of the quantity ρ used in Part 2 of this series of papers.31 If the multiplication parameters of all stages are identical, and equal to α, then and R = 1 + α[(1 - αk-1)/(1 - α)] and ρ = α[(1 - αk-1)/(1 - α)]. For k = 1 and k = 2 we recover the usual expressions for the Poisson and Neyman Type-A distributions, respectively. Random deletion from an SNDP also maintains the cluster property of this process, although with a reduction in the multiplication parameter α.47
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  78. D. R. Cox and W. L. Smith, "On the superposition of renewal processes," Biometrika 41, 91–99 (1954).
  79. G. Østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 1–103 (1935).
  80. H. R. Wilson and J. R. Bergen, "A four mechanism model for threshold spatial vision," Vision Res. 19, 19–32 (1979).

1982 (1)

P.R. Prucnal and M. C. Teich, "Multiplication noise in the human visual system at threshold: 2. Probit estimation of parameters," Biol. Cybern. 43, 87–96 (1982).

1981 (5)

P. G. Lillywhite, "Multiplicative intrinsic noise and the limits to visual performance," Vision Res. 21, 291–296 (1981).

M. C. Teich, "Role of the doubly stochastic Neyman Type-A and Thomas counting distributions in photon detection," Appl. Opt. 20, 2457–2467 (1981).

M. C. Teich and B. E. A. Saleh, "Fluctuation properties of multiplied-Poisson light: Measurement of the photon-counting distribution for radioluminescence radiation from glass," Phys. Rev. A 24, 1651–1654 (1981).

M. C. Teich and B. E. A. Saleh, "Interevent-time statistics for shot-noise-driven self-exciting point processes in photon detection," J. Opt. Soc. Am. 71, 771–776 (1981).

B. E. A. Saleh, J. Tavolacci, and M. C. Teich, "Discrimination of shot-noise-driven Poisson processes by external dead time: application to radioluminescence from glass," IEEE J. Quantum Electron. QE-17, 2341–2350 (1981).

1980 (4)

M. C. Teich and P. Diament, "Relative refractoriness in visual information processing," Biol. Cybern. 38, 187–191 (1980).

D. A. Baylor, G. Matthews, and K.-W. Yau, "Two components of electrical dark noise in toad retinal rod outer segments," J. Physiol. 309, 591–621 (1980).

P. R. Prucnal and M. C. Teich, "An increment threshold law for stimuli of arbitrary statistics," J. Math. Psychol. 21, 168–177 (1980).

H. B. Barlow, "The absolute efficiency of perceptual decisions," Phil. Trans. R. Soc. Lond. Ser. B 290, 71–82 (1980); A. F. Burgess, R. F. Wagner, R. J. Jennings, and H. B. Barlow "Efficiency of human visual signal discrimination," Science 214, 93–94 (1981).

1979 (5)

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton and W. J. McGill, "Role of quantum fluctuations and the Neyman Type-A distribution in human vision," J. Opt. Soc. Am. 69, 1469(A) (1979).

P. Lennie, "Scotopic increment thresholds in retinal ganglion cells," Vision Res. 19, 425–430 (1979).

D. A. Baylor, T. D. Lamb, and K.-W. Yau, "Responses of retinal rods to single photons," J. Physiol. 288, 613–634 (1979).

K.-W. Yau, G. Matthews, and D. A. Baylor, "Thermal activation of the visual transduction mechanism in retinal rods," Nature 279, 785–786 (1979).

H. R. Wilson and J. R. Bergen, "A four mechanism model for threshold spatial vision," Vision Res. 19, 19–32 (1979).

1978 (4)

1977 (3)

M. C. Teich, P. R. Pruenal, and G. Vannucci, "Optimum photon detection with a simple counting processor," Opt. Lett. 1, 208–210 (1977).

M. E. Breton, M. C. Teich, and L. Matin, "Intensity fluctuations produced by multimode lasers in combination with dielectric beamsplitters," Behavioral Res. Methods Instrum. 9, 324–325 (1977).

K.-W. Yau, T. D. Lamb, and D. A. Baylor, "Light-induced fluctuations in membrane current of single toad rod outer segments," Nature 269, 78–80 (1977).

1976 (1)

H. B. Barlow and W. R. Levick, "Threshold setting by the surround of cat retinal ganglion cells," J. Physiol. 259, 737–757 (1976).

1975 (1)

W. R. Levick, "Form and function of cat retinal ganglion cells," Nature 254, 659–662 (1975).

1974 (3)

B. Sakitt, "Canonical ratings," Percept. Psychophys. 6, 478–488 (1974).

T. E. Cohn, "A new hypothesis to explain why the increment threshold exceeds the decrement threshold," Vision Res. 14, 1277–1279 (1974).

T. E. Cohn, L. N. Thibos, and R. N. Kleinstein, "Detection of a luminance increment," J. Opt. Soc. Am. 64, 1321–1327 (1974); T. E. Cohn and D. J. Lasley, "Detectability of a luminance increment: Effect of spatial uncertainty," J. Opt. Soc. Am. 64, 1715–1719 (1974).

1973 (1)

S. Yeandle and J. B. Spiegler, "Light evoked and spontaneous discrete waves in the ventral nerve photoreceptor of Limulus," J. Gen. Physiol. 61, 552–571 (1973).

1972 (1)

B. Sakitt, "Counting every quantum," J. Physiol. 223, 131–150 (1972).

1971 (2)

H. Nakahama, N. Ishii, M. Yamamoto, and H. Saito, "Stochastic properties of spontaneous impulse activity in central single neurons," Tohoku J. Exp. Med. 104, 373–409 (1971).

H. B. Barlow, W. R. Levick, and M. Yooh, "Responses to single quanta of light in retinal ganglion cells of the cat," Vision Res. 11, Suppl. 3, 87–101 (1971).

1970 (2)

W. R. Levick and J. L. Zacks, "Responses of cat retinal ganglion cells to brief flashes of light," J. Physiol. 206, 677–700 (1970).

J. Nachmias and E. C. Kocher, "Detection and discrimination of luminance increments," J. Opt. Soc. Am. 60, 382–389 (1970).

1967 (1)

W. J. McGill, "Neural counting mechanisms and energy detection in audition," J. Math. Psychol. 4, 351–376 (1967).

1966 (1)

M. Treisman, "A statistical decision model for sensory discrimination which predicts Weber's law and other sensory laws: some results of a computer simulation," Percept. Psychophys. 1, 203–230 (1966).

1964 (3)

M. Treisman, "Noise and Weber's law: the discrimination of brightness and other dimensions," Psychol. Rev. 71, 314–330 (1964).

H. Grimm, "Tafeln der Neyman-Verteilung Typ-A," Biom. Z. 6, 10–23 (1964).

P. O. Bishop, W. R. Levick, and W. O. Williams, "Statistical analysis of the dark discharge of lateral geniculate neurones," J. Physiol. 170, 598–612 (1964).

1963 (3)

H. R. Blackwell, "Neural theories of simple visual discriminations," J. Opt. Soc. Am. 53, 129–160 (1963).

G. S. Brindley, "The relation of frequency of detection to intensity of a stimulus for a system of many independent detectors each of which is stimulated by a m-quantum coincidence," J. Physiol. 169, 412–415 (1963).

J. Nachmias and R. M. Steinman, "Study of absolute visual detection by the rating-scale method," J. Opt. Soc. Am. 53, 1206–1213 (1963).

1957 (1)

S. W. Kuffler, R. FitzHugh, and H. B. Barlow, "Maintained activity in the cat's retina in light and darkness," J. Gen. Physiol. 40, 683–702 (1957).

1956 (1)

1954 (5)

E. J. Denton and M. H. Pirenne, "The absolute sensitivity and functional stability of the human eye," J. Physiol. 123, 417–442 (1954).

W. W. Peterson, T. G. Birdsall, and W. C. Fox, "The theory of signal detectability," IRE Trans. Prof. Group Inf. Theory PGIT-4, 171–212 (1954).

W. P. Tanner, Jr., and J. A. Swets, "The human use of information—I. Signal detection for the case of the signal known exactly," IRE Trans. Prof. Group Inf. Theory PGIT-4, 213–221 (1954).

W. P. Tanner, Jr., and J. A. Swets, "A decision-making theory of visual detection," Psychol. Rev. 61, 401–409 (1954).

D. R. Cox and W. L. Smith, "On the superposition of renewal processes," Biometrika 41, 91–99 (1954).

1953 (1)

D. R. Cox and W. L. Smith, "The superposition of several strictly periodic sequences of events," Biometrika 40, 1–11 (1953).

1950 (1)

E. Baumgardt, "The quantic and statistical basis of visual excitation," J. Gen. Physiol. 31, 269–290 (1950).

1949 (1)

H. A. van der Velden, "Quanteuse verschijnselen bij het zien," Ned. Tijdschr. Natuurkd. 15, 147–151 (1949).

1947 (1)

1944 (1)

H. A. van der Velden, "Over het aantal lichtquanta, dat nodig is voor een lichtprikkel bij het menselijk oog," Physica 11, 179–189 (1944); "The number of quanta necessary for the perception of light of the human eye," Ophthalmologica 111, 321–331 (1946) [translation].

1943 (1)

HI. de Vries, "The quantum character of light and its bearing upon the threshold of vision, the differential sensitivity and acuity of the eye," Physica 10, 553–564 (1943).

1942 (1)

S. Hecht, S. Shlaer, and M. H. Pirenne, "Energy, quanta, and vision," J. Gen. Physiol. 25, 819–840 (1942).

1939 (1)

J. Neyman, "On a new class of 'contagious' distributions, applicable in entomology and bacteriology," Ann. Math. Stat. 10, 35–57 (1939).

1936 (1)

W. J. Crozier, "On the sensory discrimination of intensities," Proc. Nat. Acad. Sci. U.S.A. 22, 412–416 (1936).

1935 (1)

G. Østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 1–103 (1935).

1889 (1)

S. P. Langley, "Energy and vision," Philos. Mag. Ser. 5, 27, 1–23 (1889).

Barlow, H. B.

H. B. Barlow, "The absolute efficiency of perceptual decisions," Phil. Trans. R. Soc. Lond. Ser. B 290, 71–82 (1980); A. F. Burgess, R. F. Wagner, R. J. Jennings, and H. B. Barlow "Efficiency of human visual signal discrimination," Science 214, 93–94 (1981).

H. B. Barlow and W. R. Levick, "Threshold setting by the surround of cat retinal ganglion cells," J. Physiol. 259, 737–757 (1976).

H. B. Barlow, W. R. Levick, and M. Yooh, "Responses to single quanta of light in retinal ganglion cells of the cat," Vision Res. 11, Suppl. 3, 87–101 (1971).

S. W. Kuffler, R. FitzHugh, and H. B. Barlow, "Maintained activity in the cat's retina in light and darkness," J. Gen. Physiol. 40, 683–702 (1957).

H. B. Barlow, "Retinal noise and absolute threshold," J. Opt. Soc. Am. 46, 634–639 (1956).

H. B. Barlow, "Retinal and central factors in human vision limited by noise," in Photoreception in Vertebrates, H. B. Barlow and P. Fatt, eds. (Academic, New York, 1977), Chap. 19, pp. 337–358.

H. B. Barlow, T. E. Cohn (School of Optometry; University of California, Berkeley; Calif. 94720), W. R. Levick, and L. N. Thibos, "Low light level performance of cat retinal ganglion cells" (personal communication).

We are grateful to H. B. Barlow and W. R. Levick for providing us with experimental data.

Baumgardt, E.

E. Baumgardt, "The quantic and statistical basis of visual excitation," J. Gen. Physiol. 31, 269–290 (1950).

E. Baumgardt, "Threshold quantal problems," in Handbook of Sensory Physiology: Visual Psychophysics, D. Jameson and L. M. Hurvich, eds. (Springer-Verlag, Berlin, 1972), Vol. VII/4, Chap. 2, pp. 29–55.

Baylor, D. A.

D. A. Baylor, G. Matthews, and K.-W. Yau, "Two components of electrical dark noise in toad retinal rod outer segments," J. Physiol. 309, 591–621 (1980).

K.-W. Yau, G. Matthews, and D. A. Baylor, "Thermal activation of the visual transduction mechanism in retinal rods," Nature 279, 785–786 (1979).

D. A. Baylor, T. D. Lamb, and K.-W. Yau, "Responses of retinal rods to single photons," J. Physiol. 288, 613–634 (1979).

K.-W. Yau, T. D. Lamb, and D. A. Baylor, "Light-induced fluctuations in membrane current of single toad rod outer segments," Nature 269, 78–80 (1977).

Bergen, J. R.

H. R. Wilson and J. R. Bergen, "A four mechanism model for threshold spatial vision," Vision Res. 19, 19–32 (1979).

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W. W. Peterson, T. G. Birdsall, and W. C. Fox, "The theory of signal detectability," IRE Trans. Prof. Group Inf. Theory PGIT-4, 171–212 (1954).

Bishop, P. O.

P. O. Bishop, W. R. Levick, and W. O. Williams, "Statistical analysis of the dark discharge of lateral geniculate neurones," J. Physiol. 170, 598–612 (1964).

Blackwell, H. R.

Bouman, M. A.

M. A. Bouman and H. A. van der Velden, "The two-quanta explanation of the dependences of the threshold values and visual acuity on the visual angle and the time of observation," J. Opt. Soc. Am. 37, 908–919 (1947).

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M. A. Bouman and J. J. Koenderink, "Psychophysical basis of coincidence mechanisms in the human visual system," in Reviews of Physiology: Biochemistry and Experimental Pharmacology (Springer-Verlag, Berlin, 1972), Vol. 65, pp. 125–172.

Breton, M. E.

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton and W. J. McGill, "Role of quantum fluctuations and the Neyman Type-A distribution in human vision," J. Opt. Soc. Am. 69, 1469(A) (1979).

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, and W. J. McGill, "Non-Poisson nature of the effective noise in the visual system near threshold," J. Opt. Soc. Am. 68, 1454(A) (1978).

M. E. Breton, M. C. Teich, and L. Matin, "Intensity fluctuations produced by multimode lasers in combination with dielectric beamsplitters," Behavioral Res. Methods Instrum. 9, 324–325 (1977).

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, and W. J. McGill, "Multiplication noise in the human visual system at threshold: 3. The role of non-Poisson quantum fluctuations," Biol. Cybern. (submitted for publication).

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G. S. Brindley, "The relation of frequency of detection to intensity of a stimulus for a system of many independent detectors each of which is stimulated by a m-quantum coincidence," J. Physiol. 169, 412–415 (1963).

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T. E. Cohn, "A new hypothesis to explain why the increment threshold exceeds the decrement threshold," Vision Res. 14, 1277–1279 (1974).

T. E. Cohn, L. N. Thibos, and R. N. Kleinstein, "Detection of a luminance increment," J. Opt. Soc. Am. 64, 1321–1327 (1974); T. E. Cohn and D. J. Lasley, "Detectability of a luminance increment: Effect of spatial uncertainty," J. Opt. Soc. Am. 64, 1715–1719 (1974).

H. B. Barlow, T. E. Cohn (School of Optometry; University of California, Berkeley; Calif. 94720), W. R. Levick, and L. N. Thibos, "Low light level performance of cat retinal ganglion cells" (personal communication).

Cox, D. R.

D. R. Cox and W. L. Smith, "On the superposition of renewal processes," Biometrika 41, 91–99 (1954).

D. R. Cox and W. L. Smith, "The superposition of several strictly periodic sequences of events," Biometrika 40, 1–11 (1953).

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W. J. Crozier, "On the sensory discrimination of intensities," Proc. Nat. Acad. Sci. U.S.A. 22, 412–416 (1936).

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HI. de Vries, "The quantum character of light and its bearing upon the threshold of vision, the differential sensitivity and acuity of the eye," Physica 10, 553–564 (1943).

Denton, E. J.

E. J. Denton and M. H. Pirenne, "The absolute sensitivity and functional stability of the human eye," J. Physiol. 123, 417–442 (1954).

Diament, P.

M. C. Teich and P. Diament, "Relative refractoriness in visual information processing," Biol. Cybern. 38, 187–191 (1980).

FitzHugh, R.

S. W. Kuffler, R. FitzHugh, and H. B. Barlow, "Maintained activity in the cat's retina in light and darkness," J. Gen. Physiol. 40, 683–702 (1957).

Fox, W. C.

W. W. Peterson, T. G. Birdsall, and W. C. Fox, "The theory of signal detectability," IRE Trans. Prof. Group Inf. Theory PGIT-4, 171–212 (1954).

Green, D. M.

D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966) (reprinted by Krieger, Huntington, N.Y., 1974).

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H. Grimm, "Tafeln der Neyman-Verteilung Typ-A," Biom. Z. 6, 10–23 (1964).

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S. Hecht, S. Shlaer, and M. H. Pirenne, "Energy, quanta, and vision," J. Gen. Physiol. 25, 819–840 (1942).

S. Hecht, "Energy and vision." in Science In Progress, A Baitsell, ed. (Yale U. Press, New Haven, Conn., 1945), pp. 75–95, 309–310.

Ishii, N.

H. Nakahama, N. Ishii, M. Yamamoto, and H. Saito, "Stochastic properties of spontaneous impulse activity in central single neurons," Tohoku J. Exp. Med. 104, 373–409 (1971).

Kleinstein, R. N.

Kocher, E. C.

Koenderink, J. J.

M. A. Bouman and J. J. Koenderink, "Psychophysical basis of coincidence mechanisms in the human visual system," in Reviews of Physiology: Biochemistry and Experimental Pharmacology (Springer-Verlag, Berlin, 1972), Vol. 65, pp. 125–172.

Kuffler, S. W.

S. W. Kuffler, R. FitzHugh, and H. B. Barlow, "Maintained activity in the cat's retina in light and darkness," J. Gen. Physiol. 40, 683–702 (1957).

Lamb, T. D.

D. A. Baylor, T. D. Lamb, and K.-W. Yau, "Responses of retinal rods to single photons," J. Physiol. 288, 613–634 (1979).

K.-W. Yau, T. D. Lamb, and D. A. Baylor, "Light-induced fluctuations in membrane current of single toad rod outer segments," Nature 269, 78–80 (1977).

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S. P. Langley, "Energy and vision," Philos. Mag. Ser. 5, 27, 1–23 (1889).

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P. Lennie, "Scotopic increment thresholds in retinal ganglion cells," Vision Res. 19, 425–430 (1979).

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H. B. Barlow and W. R. Levick, "Threshold setting by the surround of cat retinal ganglion cells," J. Physiol. 259, 737–757 (1976).

W. R. Levick, "Form and function of cat retinal ganglion cells," Nature 254, 659–662 (1975).

H. B. Barlow, W. R. Levick, and M. Yooh, "Responses to single quanta of light in retinal ganglion cells of the cat," Vision Res. 11, Suppl. 3, 87–101 (1971).

W. R. Levick and J. L. Zacks, "Responses of cat retinal ganglion cells to brief flashes of light," J. Physiol. 206, 677–700 (1970).

P. O. Bishop, W. R. Levick, and W. O. Williams, "Statistical analysis of the dark discharge of lateral geniculate neurones," J. Physiol. 170, 598–612 (1964).

We are grateful to H. B. Barlow and W. R. Levick for providing us with experimental data.

H. B. Barlow, T. E. Cohn (School of Optometry; University of California, Berkeley; Calif. 94720), W. R. Levick, and L. N. Thibos, "Low light level performance of cat retinal ganglion cells" (personal communication).

Lillywhite, P. G.

P. G. Lillywhite, "Multiplicative intrinsic noise and the limits to visual performance," Vision Res. 21, 291–296 (1981).

Mangoubi, S. S.

Matin, L.

M. C. Teich, L. Matin, and B. I. Cantor, "Refractoriness in the maintained discharge of the cat's retinal ganglion cell," J. Opt. Soc. Am. 68, 386–402 (1978).

M. E. Breton, M. C. Teich, and L. Matin, "Intensity fluctuations produced by multimode lasers in combination with dielectric beamsplitters," Behavioral Res. Methods Instrum. 9, 324–325 (1977).

Matsuo, K.

K. Matsuo, B. E. A. Saleh, and M. C. Teich, "Cascaded Poisson processes," J. Stat. Phys. (submitted for publication).

Matthews, G.

D. A. Baylor, G. Matthews, and K.-W. Yau, "Two components of electrical dark noise in toad retinal rod outer segments," J. Physiol. 309, 591–621 (1980).

K.-W. Yau, G. Matthews, and D. A. Baylor, "Thermal activation of the visual transduction mechanism in retinal rods," Nature 279, 785–786 (1979).

McGill, W. J.

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton and W. J. McGill, "Role of quantum fluctuations and the Neyman Type-A distribution in human vision," J. Opt. Soc. Am. 69, 1469(A) (1979).

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, and W. J. McGill, "Non-Poisson nature of the effective noise in the visual system near threshold," J. Opt. Soc. Am. 68, 1454(A) (1978).

W. J. McGill, "Neural counting mechanisms and energy detection in audition," J. Math. Psychol. 4, 351–376 (1967).

W. J. McGill, "Poisson counting and detection in sensory systems," in Concepts of Communications: Interpersonal, Intrapersonal, and Mathematical, E. F. Beckenbach and C. B. Tompkins, eds. (Wiley, New York, 1971), Chap. 9, pp. 257–281.

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, and W. J. McGill, "Multiplication noise in the human visual system at threshold: 3. The role of non-Poisson quantum fluctuations," Biol. Cybern. (submitted for publication).

Nachmias, J.

J. Nachmias and E. C. Kocher, "Detection and discrimination of luminance increments," J. Opt. Soc. Am. 60, 382–389 (1970).

J. Nachmias and R. M. Steinman, "Study of absolute visual detection by the rating-scale method," J. Opt. Soc. Am. 53, 1206–1213 (1963).

J. Nachmias, "Signal detection theory and its application to problems in vision," in Handbook of Sensory Physiology: Visual Psychophysics, D. Jameson and L. M. Hurvich, eds. (Springer-Verlag, Berlin, 1972), Vol. VII/4, Chap. 3, pp. 56–77.

Nakahama, H.

H. Nakahama, N. Ishii, M. Yamamoto, and H. Saito, "Stochastic properties of spontaneous impulse activity in central single neurons," Tohoku J. Exp. Med. 104, 373–409 (1971).

Neyman, J.

J. Neyman, "On a new class of 'contagious' distributions, applicable in entomology and bacteriology," Ann. Math. Stat. 10, 35–57 (1939).

Østerberg, G.

G. Østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 1–103 (1935).

Peterson, W. W.

W. W. Peterson, T. G. Birdsall, and W. C. Fox, "The theory of signal detectability," IRE Trans. Prof. Group Inf. Theory PGIT-4, 171–212 (1954).

Pirenne, M. H.

E. J. Denton and M. H. Pirenne, "The absolute sensitivity and functional stability of the human eye," J. Physiol. 123, 417–442 (1954).

S. Hecht, S. Shlaer, and M. H. Pirenne, "Energy, quanta, and vision," J. Gen. Physiol. 25, 819–840 (1942).

Prucnal, P. R.

P. R. Prucnal and M. C. Teich, "An increment threshold law for stimuli of arbitrary statistics," J. Math. Psychol. 21, 168–177 (1980).

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton and W. J. McGill, "Role of quantum fluctuations and the Neyman Type-A distribution in human vision," J. Opt. Soc. Am. 69, 1469(A) (1979).

P. R. Prucnal and M. C. Teich, "Single-threshold detection of a random signal in noise with multiple independent observations: 1. Discrete case with application to optical communications," Appl. Opt. 17, 3576–3583 (1978).

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, and W. J. McGill, "Non-Poisson nature of the effective noise in the visual system near threshold," J. Opt. Soc. Am. 68, 1454(A) (1978).

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, and W. J. McGill, "Multiplication noise in the human visual system at threshold: 3. The role of non-Poisson quantum fluctuations," Biol. Cybern. (submitted for publication).

Prucnal, P.R.

P.R. Prucnal and M. C. Teich, "Multiplication noise in the human visual system at threshold: 2. Probit estimation of parameters," Biol. Cybern. 43, 87–96 (1982).

Pruenal, P. R.

Rose, A.

A. Rose, "The relative sensitivities of television pick-up tubes, photographic film, and the human eye," Proc. IRE 30, 293–300 (1942).

Saito, H.

H. Nakahama, N. Ishii, M. Yamamoto, and H. Saito, "Stochastic properties of spontaneous impulse activity in central single neurons," Tohoku J. Exp. Med. 104, 373–409 (1971).

Sakitt, B.

B. Sakitt, "Canonical ratings," Percept. Psychophys. 6, 478–488 (1974).

B. Sakitt, "Counting every quantum," J. Physiol. 223, 131–150 (1972).

Saleh, B. E. A.

M. C. Teich and B. E. A. Saleh, "Fluctuation properties of multiplied-Poisson light: Measurement of the photon-counting distribution for radioluminescence radiation from glass," Phys. Rev. A 24, 1651–1654 (1981).

M. C. Teich and B. E. A. Saleh, "Interevent-time statistics for shot-noise-driven self-exciting point processes in photon detection," J. Opt. Soc. Am. 71, 771–776 (1981).

B. E. A. Saleh, J. Tavolacci, and M. C. Teich, "Discrimination of shot-noise-driven Poisson processes by external dead time: application to radioluminescence from glass," IEEE J. Quantum Electron. QE-17, 2341–2350 (1981).

B. E. A. Saleh and M. C. Teich, "Multiplied-Poisson noise in pulse, particle, and photon detection," Proc. IEEE, 70 (to be published, 1982).

K. Matsuo, B. E. A. Saleh, and M. C. Teich, "Cascaded Poisson processes," J. Stat. Phys. (submitted for publication).

B. E. A. Saleh and M. C. Teich, "Statistical properties of a nonstationary Neyman-Scott cluster process," IEEE Trans. Inf. Theory (submitted for publication).

Shlaer, S.

S. Hecht, S. Shlaer, and M. H. Pirenne, "Energy, quanta, and vision," J. Gen. Physiol. 25, 819–840 (1942).

Smith, W. L.

D. R. Cox and W. L. Smith, "On the superposition of renewal processes," Biometrika 41, 91–99 (1954).

D. R. Cox and W. L. Smith, "The superposition of several strictly periodic sequences of events," Biometrika 40, 1–11 (1953).

Spiegler, J. B.

S. Yeandle and J. B. Spiegler, "Light evoked and spontaneous discrete waves in the ventral nerve photoreceptor of Limulus," J. Gen. Physiol. 61, 552–571 (1973).

Steinman, R. M.

Swets, J. A.

W. P. Tanner, Jr., and J. A. Swets, "The human use of information—I. Signal detection for the case of the signal known exactly," IRE Trans. Prof. Group Inf. Theory PGIT-4, 213–221 (1954).

W. P. Tanner, Jr., and J. A. Swets, "A decision-making theory of visual detection," Psychol. Rev. 61, 401–409 (1954).

D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966) (reprinted by Krieger, Huntington, N.Y., 1974).

Tanner, Jr., W. P.

W. P. Tanner, Jr., and J. A. Swets, "A decision-making theory of visual detection," Psychol. Rev. 61, 401–409 (1954).

W. P. Tanner, Jr., and J. A. Swets, "The human use of information—I. Signal detection for the case of the signal known exactly," IRE Trans. Prof. Group Inf. Theory PGIT-4, 213–221 (1954).

Tavolacci, J.

B. E. A. Saleh, J. Tavolacci, and M. C. Teich, "Discrimination of shot-noise-driven Poisson processes by external dead time: application to radioluminescence from glass," IEEE J. Quantum Electron. QE-17, 2341–2350 (1981).

Teich, M. C.

P.R. Prucnal and M. C. Teich, "Multiplication noise in the human visual system at threshold: 2. Probit estimation of parameters," Biol. Cybern. 43, 87–96 (1982).

M. C. Teich, "Role of the doubly stochastic Neyman Type-A and Thomas counting distributions in photon detection," Appl. Opt. 20, 2457–2467 (1981).

M. C. Teich and B. E. A. Saleh, "Fluctuation properties of multiplied-Poisson light: Measurement of the photon-counting distribution for radioluminescence radiation from glass," Phys. Rev. A 24, 1651–1654 (1981).

B. E. A. Saleh, J. Tavolacci, and M. C. Teich, "Discrimination of shot-noise-driven Poisson processes by external dead time: application to radioluminescence from glass," IEEE J. Quantum Electron. QE-17, 2341–2350 (1981).

M. C. Teich and B. E. A. Saleh, "Interevent-time statistics for shot-noise-driven self-exciting point processes in photon detection," J. Opt. Soc. Am. 71, 771–776 (1981).

M. C. Teich and P. Diament, "Relative refractoriness in visual information processing," Biol. Cybern. 38, 187–191 (1980).

P. R. Prucnal and M. C. Teich, "An increment threshold law for stimuli of arbitrary statistics," J. Math. Psychol. 21, 168–177 (1980).

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton and W. J. McGill, "Role of quantum fluctuations and the Neyman Type-A distribution in human vision," J. Opt. Soc. Am. 69, 1469(A) (1979).

P. R. Prucnal and M. C. Teich, "Single-threshold detection of a random signal in noise with multiple independent observations: 1. Discrete case with application to optical communications," Appl. Opt. 17, 3576–3583 (1978).

M. C. Teich, L. Matin, and B. I. Cantor, "Refractoriness in the maintained discharge of the cat's retinal ganglion cell," J. Opt. Soc. Am. 68, 386–402 (1978).

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, and W. J. McGill, "Non-Poisson nature of the effective noise in the visual system near threshold," J. Opt. Soc. Am. 68, 1454(A) (1978).

M. E. Breton, M. C. Teich, and L. Matin, "Intensity fluctuations produced by multimode lasers in combination with dielectric beamsplitters," Behavioral Res. Methods Instrum. 9, 324–325 (1977).

M. C. Teich, P. R. Pruenal, and G. Vannucci, "Optimum photon detection with a simple counting processor," Opt. Lett. 1, 208–210 (1977).

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, and W. J. McGill, "Multiplication noise in the human visual system at threshold: 3. The role of non-Poisson quantum fluctuations," Biol. Cybern. (submitted for publication).

B. E. A. Saleh and M. C. Teich, "Multiplied-Poisson noise in pulse, particle, and photon detection," Proc. IEEE, 70 (to be published, 1982).

B. E. A. Saleh and M. C. Teich, "Statistical properties of a nonstationary Neyman-Scott cluster process," IEEE Trans. Inf. Theory (submitted for publication).

K. Matsuo, B. E. A. Saleh, and M. C. Teich, "Cascaded Poisson processes," J. Stat. Phys. (submitted for publication).

Thibos, L. N.

T. E. Cohn, L. N. Thibos, and R. N. Kleinstein, "Detection of a luminance increment," J. Opt. Soc. Am. 64, 1321–1327 (1974); T. E. Cohn and D. J. Lasley, "Detectability of a luminance increment: Effect of spatial uncertainty," J. Opt. Soc. Am. 64, 1715–1719 (1974).

H. B. Barlow, T. E. Cohn (School of Optometry; University of California, Berkeley; Calif. 94720), W. R. Levick, and L. N. Thibos, "Low light level performance of cat retinal ganglion cells" (personal communication).

Treisman, M.

M. Treisman, "A statistical decision model for sensory discrimination which predicts Weber's law and other sensory laws: some results of a computer simulation," Percept. Psychophys. 1, 203–230 (1966).

M. Treisman, "Noise and Weber's law: the discrimination of brightness and other dimensions," Psychol. Rev. 71, 314–330 (1964).

van der Velden, H. A.

H. A. van der Velden, "Quanteuse verschijnselen bij het zien," Ned. Tijdschr. Natuurkd. 15, 147–151 (1949).

M. A. Bouman and H. A. van der Velden, "The two-quanta explanation of the dependences of the threshold values and visual acuity on the visual angle and the time of observation," J. Opt. Soc. Am. 37, 908–919 (1947).

H. A. van der Velden, "Over het aantal lichtquanta, dat nodig is voor een lichtprikkel bij het menselijk oog," Physica 11, 179–189 (1944); "The number of quanta necessary for the perception of light of the human eye," Ophthalmologica 111, 321–331 (1946) [translation].

van Trees, H. L.

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Vannucci, G.

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton and W. J. McGill, "Role of quantum fluctuations and the Neyman Type-A distribution in human vision," J. Opt. Soc. Am. 69, 1469(A) (1979).

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, and W. J. McGill, "Non-Poisson nature of the effective noise in the visual system near threshold," J. Opt. Soc. Am. 68, 1454(A) (1978).

M. C. Teich, P. R. Pruenal, and G. Vannucci, "Optimum photon detection with a simple counting processor," Opt. Lett. 1, 208–210 (1977).

M. C. Teich, P. R. Prucnal, G. Vannucci, M. E. Breton, and W. J. McGill, "Multiplication noise in the human visual system at threshold: 3. The role of non-Poisson quantum fluctuations," Biol. Cybern. (submitted for publication).

Williams, W. O.

P. O. Bishop, W. R. Levick, and W. O. Williams, "Statistical analysis of the dark discharge of lateral geniculate neurones," J. Physiol. 170, 598–612 (1964).

Wilson, H. R.

H. R. Wilson and J. R. Bergen, "A four mechanism model for threshold spatial vision," Vision Res. 19, 19–32 (1979).

Yamamoto, M.

H. Nakahama, N. Ishii, M. Yamamoto, and H. Saito, "Stochastic properties of spontaneous impulse activity in central single neurons," Tohoku J. Exp. Med. 104, 373–409 (1971).

Yau, K.-W.

D. A. Baylor, G. Matthews, and K.-W. Yau, "Two components of electrical dark noise in toad retinal rod outer segments," J. Physiol. 309, 591–621 (1980).

D. A. Baylor, T. D. Lamb, and K.-W. Yau, "Responses of retinal rods to single photons," J. Physiol. 288, 613–634 (1979).

K.-W. Yau, G. Matthews, and D. A. Baylor, "Thermal activation of the visual transduction mechanism in retinal rods," Nature 279, 785–786 (1979).

K.-W. Yau, T. D. Lamb, and D. A. Baylor, "Light-induced fluctuations in membrane current of single toad rod outer segments," Nature 269, 78–80 (1977).

Yeandle, S.

S. Yeandle and J. B. Spiegler, "Light evoked and spontaneous discrete waves in the ventral nerve photoreceptor of Limulus," J. Gen. Physiol. 61, 552–571 (1973).

Yooh, M.

H. B. Barlow, W. R. Levick, and M. Yooh, "Responses to single quanta of light in retinal ganglion cells of the cat," Vision Res. 11, Suppl. 3, 87–101 (1971).

Zacks, J. L.

W. R. Levick and J. L. Zacks, "Responses of cat retinal ganglion cells to brief flashes of light," J. Physiol. 206, 677–700 (1970).

Zeevi, Y. Y.

Acta Ophthalmol. Suppl. (1)

G. Østerberg, "Topography of the layer of rods and cones in the human retina," Acta Ophthalmol. Suppl. 6, 1–103 (1935).

Ann. Math. Stat. (1)

J. Neyman, "On a new class of 'contagious' distributions, applicable in entomology and bacteriology," Ann. Math. Stat. 10, 35–57 (1939).

Appl. Opt. (2)

Behavioral Res. Methods Instrum. (1)

M. E. Breton, M. C. Teich, and L. Matin, "Intensity fluctuations produced by multimode lasers in combination with dielectric beamsplitters," Behavioral Res. Methods Instrum. 9, 324–325 (1977).

Biol. Cybern. (2)

P.R. Prucnal and M. C. Teich, "Multiplication noise in the human visual system at threshold: 2. Probit estimation of parameters," Biol. Cybern. 43, 87–96 (1982).

M. C. Teich and P. Diament, "Relative refractoriness in visual information processing," Biol. Cybern. 38, 187–191 (1980).

Biom. Z. (1)

H. Grimm, "Tafeln der Neyman-Verteilung Typ-A," Biom. Z. 6, 10–23 (1964).

Biometrika (2)

D. R. Cox and W. L. Smith, "The superposition of several strictly periodic sequences of events," Biometrika 40, 1–11 (1953).

D. R. Cox and W. L. Smith, "On the superposition of renewal processes," Biometrika 41, 91–99 (1954).

IEEE J. Quantum Electron. (1)

B. E. A. Saleh, J. Tavolacci, and M. C. Teich, "Discrimination of shot-noise-driven Poisson processes by external dead time: application to radioluminescence from glass," IEEE J. Quantum Electron. QE-17, 2341–2350 (1981).

IRE Trans. Prof. Group Inf. Theory (2)

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W. P. Tanner, Jr., and J. A. Swets, "The human use of information—I. Signal detection for the case of the signal known exactly," IRE Trans. Prof. Group Inf. Theory PGIT-4, 213–221 (1954).

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The Poisson nature of the detected photons follows directly from the small value of the degeneracy parameter of the light, δ = 〈m〉τc/T, where τc, is the coherence time of the light [see L. Mandel, Proc. Phys. Soc. 74, 233–242 (1959); B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978)]. For the source used by HSP, 〈m〉 ≃ 10, T ≃ 10-3 sec, and τc ≃ 10-13 sec, whereas for our multimode Ar+ laser source τc 10-9 sec [see G. Vannucci and M. C. Teich, Appl. Opt. 19, 548–553 (1980)]. In both cases δ « 1. We shall see in Part 3 of this series32 that certain rather common sources of visible radiation, such as television and oscilloscope images produced by cathodoluminescence, as well as image-intensified light [see A. van Meeteren, Vision Res. 18, 257–267 (1978)], in general, generate non-Poisson photon statistics [see Refs. 43 and 47]. Their use in vision experiments therefore requires exceptional care.

Several of our stipulations on the format of detection are, no doubt, unnecessarily restrictive. For example, the assumptions of a single counting focus and a fixed integration time could likely be relaxed without seriously affecting the outcome.

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If we assume that the counting time is very large in comparison with the decay times τp, for all k stages, and that the multiplication parameter of each stage is given by αj, the variance-to-mean ratio is49 [equation]. This is the origin of the quantity ρ used in Part 2 of this series of papers.31 If the multiplication parameters of all stages are identical, and equal to α, then and R = 1 + α[(1 - αk-1)/(1 - α)] and ρ = α[(1 - αk-1)/(1 - α)]. For k = 1 and k = 2 we recover the usual expressions for the Poisson and Neyman Type-A distributions, respectively. Random deletion from an SNDP also maintains the cluster property of this process, although with a reduction in the multiplication parameter α.47

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