Abstract

A theory and model of the visual system are presented to explain the detection of static sinusoidal gratings near the threshold. The model incorporates a set of independent decision centers and associated photoreceptive fields (PRFs). The decision criterion value at each decision center is proportional to the standard deviation of the excitation current transmitted from a PRF to its associated decision center caused by quantum fluctuations in the absorption of light. It is well known that the spatial-frequency-response (SFR) function and the spatial-impulse-response (SIR) function of a photodetector are a Fourier transform pair. A systematic examination of the SIR and SFR functions of PRF configurations consisting of rectangular regions of alternately excitatory and inhibitory response reveals that modulation sensitivity of the visual system is explained at scotopic and photopic illuminance by a set of PRFs composed of a single excitatory region and a central excitatory region bordered by inhibitory regions, respectively. The complete model is shown to yield a high degree of conformity between theoretical and experimental threshold modulation curves.

© 1976 Optical Society of America

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References

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  1. P. M. Duffieux and G. Lansraux, Rev. d’Opt. 24, 65 (1945); Rev. d’Opt. 24, 151 (1945); Rev. d’Opt. 24, 215 (1945).
  2. For a general review of spatial-frequency-response characteristics, see L. Levi, Progress in Optics VIII, edited by E. Wolf (North-Holland, Amsterdam, 1970), p. 343.
    [Crossref]
  3. Recognition depends on detection of structural detail. For a useful rule of thumb expressed in terms of resolution frequency by J. Johnson see Perception of Displayed Information, edited by L. M. Biberman (Plenum, New York, 1973), p. 4.
  4. A. Van Meeteren, Visual Aspects of Image Intensification (Bronder-offset, Rotterdam, 1973), p. 77.
  5. F. W. Campbell and R. W. Gubisch, J. Physiol. (Lond.) 186, 558 (1966).
  6. D. H. Kelly, J. Opt. Soc. Am. 64, 983 (1974).
    [Crossref]
  7. F. L. Van Nes and M. Bouman, J. Opt. Soc. Am. 57, 401 (1967).
    [Crossref]
  8. O. H. Schade, J. Opt. Soc. Am. 46, 721 (1956).
    [Crossref] [PubMed]
  9. J. J. De Palma and E. M. Lowry, J. Opt. Soc. Am. 52, 328 (1962).
    [Crossref]
  10. D. H. Kelly, J. Opt. Soc. Am. 56, 1628 (1966).
    [Crossref]
  11. J. G. Robson, J. Opt. Soc. Am. 56, 1141 (1966).
    [Crossref]
  12. The author is indebted to one of the reviewers for calling attention to the sensitivity of the slope to temporal rate, shown by D. H. Kelly, Vision Res. 12, 89 (1972).
    [Crossref] [PubMed]
  13. F. W. Campbell and J. G. Robson, J. Physiol. (Lond.) 197, 551 (1968).
  14. To explain Mach band phenomena, G. V. Békésy [J. Opt. Soc. Am. 50, 1060 (1960)] proposed a spatial function, qualitatively similar to the SIR function deduced from threshold modulation data by Ref. 8, to describe the interaction of suprathreshold responses from proximate regions of the visual field at some higher level of the visual pathway.
    [Crossref] [PubMed]
  15. C. Blakemore and F. W. Campbell, J. Physiol. (Lond.) 203, 237 (1969).
  16. M. B. Sachs, J. Nachmias, and J. G. Robson, J. Opt. Soc. Am. 61, 1176 (1971).
    [Crossref] [PubMed]
  17. D. H. Hubel and T. N. Wiesel, J. Physiol. (Lond.) 154, 572 (1960).
  18. A. D. Schnitzler, J. Opt. Soc. Am. 63, 1357 (1973).
    [Crossref] [PubMed]
  19. For early references to the concept of a signal-to-noise limitation in visual detection, see Ref. 18.
  20. A. Papoulis, System and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  21. Y. W. Lee, Statistical Theory of Communications (Wiley, New York, 1960).
  22. R. A. Moses, Adler’s Physiology of the Eye (Mosby, St. Louis, 1970).
  23. D. H. Kelly and R. E. Savoie, Percept. Psychophys. 14, 313 (1973).
    [Crossref]
  24. H. R. Blackwell, J. Opt. Soc. Am. 53, 129 (1963).
    [Crossref] [PubMed]

1974 (1)

1973 (2)

A. D. Schnitzler, J. Opt. Soc. Am. 63, 1357 (1973).
[Crossref] [PubMed]

D. H. Kelly and R. E. Savoie, Percept. Psychophys. 14, 313 (1973).
[Crossref]

1972 (1)

The author is indebted to one of the reviewers for calling attention to the sensitivity of the slope to temporal rate, shown by D. H. Kelly, Vision Res. 12, 89 (1972).
[Crossref] [PubMed]

1971 (1)

1969 (1)

C. Blakemore and F. W. Campbell, J. Physiol. (Lond.) 203, 237 (1969).

1968 (1)

F. W. Campbell and J. G. Robson, J. Physiol. (Lond.) 197, 551 (1968).

1967 (1)

1966 (3)

1963 (1)

1962 (1)

1960 (2)

1956 (1)

1945 (1)

P. M. Duffieux and G. Lansraux, Rev. d’Opt. 24, 65 (1945); Rev. d’Opt. 24, 151 (1945); Rev. d’Opt. 24, 215 (1945).

Békésy, G. V.

Blackwell, H. R.

Blakemore, C.

C. Blakemore and F. W. Campbell, J. Physiol. (Lond.) 203, 237 (1969).

Bouman, M.

Campbell, F. W.

C. Blakemore and F. W. Campbell, J. Physiol. (Lond.) 203, 237 (1969).

F. W. Campbell and J. G. Robson, J. Physiol. (Lond.) 197, 551 (1968).

F. W. Campbell and R. W. Gubisch, J. Physiol. (Lond.) 186, 558 (1966).

De Palma, J. J.

Duffieux, P. M.

P. M. Duffieux and G. Lansraux, Rev. d’Opt. 24, 65 (1945); Rev. d’Opt. 24, 151 (1945); Rev. d’Opt. 24, 215 (1945).

Gubisch, R. W.

F. W. Campbell and R. W. Gubisch, J. Physiol. (Lond.) 186, 558 (1966).

Hubel, D. H.

D. H. Hubel and T. N. Wiesel, J. Physiol. (Lond.) 154, 572 (1960).

Johnson, J.

Recognition depends on detection of structural detail. For a useful rule of thumb expressed in terms of resolution frequency by J. Johnson see Perception of Displayed Information, edited by L. M. Biberman (Plenum, New York, 1973), p. 4.

Kelly, D. H.

D. H. Kelly, J. Opt. Soc. Am. 64, 983 (1974).
[Crossref]

D. H. Kelly and R. E. Savoie, Percept. Psychophys. 14, 313 (1973).
[Crossref]

The author is indebted to one of the reviewers for calling attention to the sensitivity of the slope to temporal rate, shown by D. H. Kelly, Vision Res. 12, 89 (1972).
[Crossref] [PubMed]

D. H. Kelly, J. Opt. Soc. Am. 56, 1628 (1966).
[Crossref]

Lansraux, G.

P. M. Duffieux and G. Lansraux, Rev. d’Opt. 24, 65 (1945); Rev. d’Opt. 24, 151 (1945); Rev. d’Opt. 24, 215 (1945).

Lee, Y. W.

Y. W. Lee, Statistical Theory of Communications (Wiley, New York, 1960).

Levi, L.

For a general review of spatial-frequency-response characteristics, see L. Levi, Progress in Optics VIII, edited by E. Wolf (North-Holland, Amsterdam, 1970), p. 343.
[Crossref]

Lowry, E. M.

Moses, R. A.

R. A. Moses, Adler’s Physiology of the Eye (Mosby, St. Louis, 1970).

Nachmias, J.

Papoulis, A.

A. Papoulis, System and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Robson, J. G.

Sachs, M. B.

Savoie, R. E.

D. H. Kelly and R. E. Savoie, Percept. Psychophys. 14, 313 (1973).
[Crossref]

Schade, O. H.

Schnitzler, A. D.

Van Meeteren, A.

A. Van Meeteren, Visual Aspects of Image Intensification (Bronder-offset, Rotterdam, 1973), p. 77.

Van Nes, F. L.

Wiesel, T. N.

D. H. Hubel and T. N. Wiesel, J. Physiol. (Lond.) 154, 572 (1960).

J. Opt. Soc. Am. (10)

J. Physiol. (Lond.) (4)

D. H. Hubel and T. N. Wiesel, J. Physiol. (Lond.) 154, 572 (1960).

C. Blakemore and F. W. Campbell, J. Physiol. (Lond.) 203, 237 (1969).

F. W. Campbell and R. W. Gubisch, J. Physiol. (Lond.) 186, 558 (1966).

F. W. Campbell and J. G. Robson, J. Physiol. (Lond.) 197, 551 (1968).

Percept. Psychophys. (1)

D. H. Kelly and R. E. Savoie, Percept. Psychophys. 14, 313 (1973).
[Crossref]

Rev. d’Opt. (1)

P. M. Duffieux and G. Lansraux, Rev. d’Opt. 24, 65 (1945); Rev. d’Opt. 24, 151 (1945); Rev. d’Opt. 24, 215 (1945).

Vision Res. (1)

The author is indebted to one of the reviewers for calling attention to the sensitivity of the slope to temporal rate, shown by D. H. Kelly, Vision Res. 12, 89 (1972).
[Crossref] [PubMed]

Other (7)

For a general review of spatial-frequency-response characteristics, see L. Levi, Progress in Optics VIII, edited by E. Wolf (North-Holland, Amsterdam, 1970), p. 343.
[Crossref]

Recognition depends on detection of structural detail. For a useful rule of thumb expressed in terms of resolution frequency by J. Johnson see Perception of Displayed Information, edited by L. M. Biberman (Plenum, New York, 1973), p. 4.

A. Van Meeteren, Visual Aspects of Image Intensification (Bronder-offset, Rotterdam, 1973), p. 77.

For early references to the concept of a signal-to-noise limitation in visual detection, see Ref. 18.

A. Papoulis, System and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Y. W. Lee, Statistical Theory of Communications (Wiley, New York, 1960).

R. A. Moses, Adler’s Physiology of the Eye (Mosby, St. Louis, 1970).

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Figures (4)

FIG. 1
FIG. 1

Typical experimental curves of modulation sensitivity. Photopic curve (Ref. 6) obtained with 2.3 mm diameter artificial pupils at 1300 Td. Scotopic curve (Ref. 7) obtained with 2 mm diameter artificial pupils at 0.0009 Td.

FIG. 2
FIG. 2

Theoretical curves of relative modulation sensitivity of a single decision center and associated photoreceptive field (PRF). Long-dash, short-dash, and solid curves result from first, second, and third PRFs, respectively, described in text. Note corresponding slopes in low-frequency regime are 0, 1, and 2, respectively.

FIG. 3
FIG. 3

Theoretical curves of relative modulation sensitivity of over-all visual system. Curves labeled n = 1, 2, and 3 correspond to the set of first, second, and third PRFs, respectively, described in text.

FIG. 4
FIG. 4

Comparison of theoretical curves of noise-required modulation (NRM) and experimental threshold modulation data (Ref. 23). Squares, triangles, and circles are the data points of three different subjects, respectively. Solid curve is a graph of our NRM function fitted to data via two adjustable parameters. Dash curve is the resulting modulation on retina. Dash-dot curve is the graph of an NRM function based on assuming decision criterion value is independent of frequency.

Equations (32)

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E ( x , y , t ) = Ē + E exp [ i 2 π ( f x x + f y y + f t t ) ] ,
s i = S i exp [ i 2 π ( f x x + f y y + f t t ) ] ,
s 0 ( x , y , t ) = - g ( x , y , t ) × s i ( x - x , y - y , t - t ) d x d y d t ,
S 0 = G ( f x , f y , f t ) S i ( f x , f y , f t ) ,
σ 0 2 = - ϕ 0 ( f x , f y , f t ) d f x d f y d f t ,
ϕ 0 = G ( f x , f y , f t ) 2 ϕ i ,
ϕ i ( f x , f y , f t ) = R Ē .
σ 0 = ( - G ( f x , f y , f t ) 2 ϕ i ( f x , f y , f t ) d f x d f y d f t ) 1 / 2 .
k 0 = G ( f x , f y , f t ) S i ( f x , f y , f t ) [ - G ( f x , f y , f t 2 ϕ i ( f x , f y , f t ) d f x d f y d f t ] 1 / 2 .
k 0 = M I G ( f x , 0 , 0 ) × ( R Ē - G ( f x , f y , f t ) 2 d f x d f y d f t ) 1 / 2 .
M T N = 2 k T F D ( π R L ¯ ) 1 / 2 [ - G ( f x , f y , f t ) 2 d f x d f y d f t ] 1 / 2 G ( f x , 0 , 0 ) T 0 ( f x ) .
g ( x , y ) = 1 ,             - w / 2 x w / 2 ,             - h / 2 y h / 2 ,
G ( f x , f y ) = [ sin ( π f x w ) ] [ sin ( π f y h ) ] / π 2 f x f y ,
g ( x , y ) = - 1 , - w x 0 , - h / 2 y h / 2 , g ( x , y ) = 1 , 0 x w , - h / 2 y h / 2 ,
G ( f x , f y ) = i 2 [ sin ( π f x w ) ] 2 [ sin ( π f y h ) ] / π 2 f x f y ,
g ( x , y ) = - 0.5 ,             - 3 w / 2 x - w / 2 ,             w / 2 x 3 w / 2 ,             - h / 2 y h / 2 , g ( x , y ) = 1 ,             - w / 2 x w / 2 ,             - h / 2 y h / 2 ,
G ( f x , f y ) = 2 [ sin ( π f x w ) ] 3 [ sin ( π f y h ) ] / π 2 f x f y .
g ( t ) = ( 1 / τ ) exp ( - t / τ ) ,
G ( f t ) = ( 1 - i 2 π f t τ ) / ( 1 + 4 π 2 f t 2 τ 2 ) .
M T N ( ν , α ) = 2 m ν / sin ( π ν α ) T 0 ( ν ) ,
m = π k T / D ( π R L ¯ τ ) 1 / 2 .
M T N ( ν ) = 2 m ν / T 0 ( ν ) ,             1 / 2 α F ν ,
M T N ( ν ) = 2 m ν / sin ( π ν α F ) T 0 ( ν ) ,             ν 1 / 2 α F .
M T N ( ν , α ) = m ν / sin 2 ( π ν α ) T 0 ( ν ) ,
M T N ( ν ) = m ν / T 0 ( ν ) ,             1 / 2 α F ν ,
M T N ( ν ) = m ν / sin 2 ( π ν α F ) T 0 ( ν ) ,             ν 1 / 2 α F .
M T N ( ν , α ) = 3 m ν / 2 sin 3 ( π ν α ) T 0 ( ν ) ,
M T N ( ν ) = 3 m ν / 2 T 0 ( ν ) ,             1 / 2 α F ν ,
M T N ( ν ) = 3 m ν / 2 sin 3 ( π ν α F ) T 0 ( ν ) ,             ν 1 / 2 α F .
M T S ( ν ) = m ν 2 / T 0 ( ν ) ,             1 / 2 α F ν ,
M T S ( ν ) = m / 2 α F T 0 ( ν ) sin 3 ( π ν α F ) ,             ν 1 / 2 α F ,
m = 4 S T / R L ¯ D 2 .