Abstract

The tools of Fourier analysis can be used to explain visual phenomena of spatial brightness interaction provided that attention is confined to small perturbations of spatially uniform fields. A perturbation approach is outlined here, and a transfer function is presented which is appropriate for small perturbations. The transfer function was obtained from human subjects with psychophysical methods, for the case of briefly flashed, achromatic fields at photopic levels of illumination. For the frequency range of 0.005 to 0.15 cycles per minute of arc, the transfer function is roughly proportional to spatial frequency, thus reflecting, in large part, nonoptical properties of the system. A simple mechanism of lateral inhibition could underlie this transfer function.

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  1. O. H. Schade, J. Opt. Soc. Am. 46, 721 (1956).
  2. J. D. Bliss and W. B. Macurdy, J. Opt. Soc. Am. 51, 1373 (1961).
  3. O. Bryngdahl, J. Opt. Soc. Am. 54, 1152 (1964).
  4. G. von Békésy, J. Opt. Soc. Am. 50, 1060 (1960).
  5. E. M. Lowery and J. J. DePalma, J. Opt. Soc. Am. 51, 740 (1961).
  6. G. A. Fry, J. Opt. Soc. Am. 53, 94 (1963).
  7. F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, 1965), p. 151.
  8. L. A. Riggs, in Vision and Visual Perception, C. H. Graham, Ed., (John Wiley & Sons, Inc., New York, 1965), p. 334.
  9. I assume that ∫0 |D{r}|rdr converges.
  10. I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Co., New York, 1951).
  11. I have arbitrarily chosen sinusoids varying along y and uniform over x. There is no loss of generality; since D depends only on distance, the system is invariant under rotations, and the same D¯ would be obtained from any other orientation. But see the section on isotropy.
  12. L. Ronchi and G. Toraldo di Francia, J. Opt. Soc. Am. 47, 639 (1957).
  13. E. G. Heinemann, J. Exptl. Psychol. 50, 89 (1955).
  14. See e.g. A. Taylor, Advanced Calculus (Ginn and Co., Boston, Mass., 1955), p. 228.
  15. See W. M. Brown, Analysis of Linear Time-Invariant Systems (McGraw-Hill Book Co., New York, 1963), for a proof in the case of temporal interaction systems. Time-invariance there parallels space-invariance (homogeneity) here.
  16. It is usually said that nonlinearity prevents Fourier analysis of this problem. But nonlinearity is easily handled with the perturbation approach. The more serious obstacle is lack of homogeneity, which is both often overlooked and logically independent of the question of linearity.
  17. L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 37, 217 (1947).
  18. R. S. Woodworth and H. Schlosberg, Experimental Psychology (Henry Holt & Co., Inc., New York, 1954), p. 386.
  19. J. Mandelbaum and L. L. Sloan, Am. J. Ophthalmol. 30, 581 (1947).
  20. F. W. Weymouth, Am. J. Ophthalmol. 46, 102 (1958).
  21. C. H. Graham and H. H. Bartlett, J. Exptl. Psychol. 24, 574 (1939).
  22. C. H. Graham, R. H. Brown, and F. A. Mote, Jr., J. Exptl. Psychol. 24, 555 (1939).
  23. H. B. Barlow, J. Physiol. (London) 141, 337 (1958).
  24. G. S. Brindley, Physiology of the Retina and the Visual Pathway (Edward Arnold and Co., London, 1960), pp. 173, 236.
  25. M. A. Bouman and G. van den Brink, J. Opt. Soc. Am. 42, 617 (1952).
  26. K. N. Ogle, J. Opt. Soc. Am. 51, 1265 (1961).
  27. G. Westheimer, J. Physiol. (London) 190, 139 (1967).
  28. S. Polyak, The Vertebrate Visual System (Univ. of Chicago Press, 1957).
  29. J. L. Brown, in Ref. 8, p. 50.
  30. S. S. Stevens, Ed., Handbook of Experimental Psychology (John Wiley & Sons, Inc., New York, 1951), p. 929.
  31. E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964), p. 15ff.
  32. See the work of Hartline and Ratliff, as summarized by Ratliff (Ref. 7), pp. 105–117.
  33. G. Westheimer, Vision Res. 6, 669 (1966), Eq. (3).
  34. This may not be serious, since the results indicate that the visual system being studied strongly attenuates low frequencies.
  35. T. N. Cornsweet, Am. J. Psychol. 75, 485 (1962).
  36. T. N. Cornsweet and H. Pinsker, J. Physiol. (London) 176, 294 (1965).
  37. One unit of subjective contrast is the perturbation amplitude at the absolute threshold for the perception of contrast. Cf. Ref. 39.
  38. This effect is not evident in Fig. 4, but only moderately low amplitudes were used in these measurements.
  39. The average value of Aω at LF's forced-choice threshold for frequenc'es less 0.06 cycle/min is 0.0861; I have taken the forcedcho.ce threshold as the unit of subjective contrast (cf. Table I). The recognition threshold for BM was 1.64 units on this scale.
  40. J. Krauskopf, J. Opt. Soc. Am. 52, 1046 (1962).
  41. G. Westheimer and F. W. Campbell, J. Opt. Soc. Am. 52, 1040 (1962).
  42. See Ref. 2; Eq. (7) is the two-dimensional equivalent of the result given in this reference.
  43. This lateral travel time is not the same as the latency for inhibition. It is the difference of the onset time for inhibition as a function of difference of distance between inhibiting and inhibited units.
  44. Although flashes of less than 500-;usec duration were used here, the results do not imply necessarily that inhibition is effective in such a time. It is possible that the system's excitation persists for some time following the flash, and, since there was darkness both before and after the flash, I can say only that inhibition is fast enough to catch up with the excitation. The negative results of J. Nachmias, J. Opt. Soc. Am. 57, 421 (1967), may be clue to his use of light periods before and after the flash, although his use of square-wave gratings complicates the comparison in other ways.

di Francia, G. Toraldo

L. Ronchi and G. Toraldo di Francia, J. Opt. Soc. Am. 47, 639 (1957).

van den Brink, G.

M. A. Bouman and G. van den Brink, J. Opt. Soc. Am. 42, 617 (1952).

vonBékésy, G.

G. von Békésy, J. Opt. Soc. Am. 50, 1060 (1960).

Barlow, H. B.

H. B. Barlow, J. Physiol. (London) 141, 337 (1958).

Bartlett, H. H.

C. H. Graham and H. H. Bartlett, J. Exptl. Psychol. 24, 574 (1939).

Bliss, J. D.

J. D. Bliss and W. B. Macurdy, J. Opt. Soc. Am. 51, 1373 (1961).

Bouman, M. A.

M. A. Bouman and G. van den Brink, J. Opt. Soc. Am. 42, 617 (1952).

Brindley, G. S.

G. S. Brindley, Physiology of the Retina and the Visual Pathway (Edward Arnold and Co., London, 1960), pp. 173, 236.

Brown, J. L.

J. L. Brown, in Ref. 8, p. 50.

Brown, R. H.

C. H. Graham, R. H. Brown, and F. A. Mote, Jr., J. Exptl. Psychol. 24, 555 (1939).

Brown, W. M.

See W. M. Brown, Analysis of Linear Time-Invariant Systems (McGraw-Hill Book Co., New York, 1963), for a proof in the case of temporal interaction systems. Time-invariance there parallels space-invariance (homogeneity) here.

Bryngdahl, O.

O. Bryngdahl, J. Opt. Soc. Am. 54, 1152 (1964).

Campbell, F. W.

G. Westheimer and F. W. Campbell, J. Opt. Soc. Am. 52, 1040 (1962).

Cornsweet, T. N.

T. N. Cornsweet, Am. J. Psychol. 75, 485 (1962).

T. N. Cornsweet and H. Pinsker, J. Physiol. (London) 176, 294 (1965).

DePalma, J. J.

E. M. Lowery and J. J. DePalma, J. Opt. Soc. Am. 51, 740 (1961).

Fry, G. A.

G. A. Fry, J. Opt. Soc. Am. 53, 94 (1963).

Graham, C. H.

C. H. Graham and H. H. Bartlett, J. Exptl. Psychol. 24, 574 (1939).

C. H. Graham, R. H. Brown, and F. A. Mote, Jr., J. Exptl. Psychol. 24, 555 (1939).

Heinemann, E. G.

E. G. Heinemann, J. Exptl. Psychol. 50, 89 (1955).

Higgins, G. C.

L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 37, 217 (1947).

Jones, L. A.

L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 37, 217 (1947).

Krauskopf, J.

J. Krauskopf, J. Opt. Soc. Am. 52, 1046 (1962).

Linfoot, E. H.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964), p. 15ff.

Lowery, E. M.

E. M. Lowery and J. J. DePalma, J. Opt. Soc. Am. 51, 740 (1961).

Macurdy, W. B.

J. D. Bliss and W. B. Macurdy, J. Opt. Soc. Am. 51, 1373 (1961).

Mandelbaum, J.

J. Mandelbaum and L. L. Sloan, Am. J. Ophthalmol. 30, 581 (1947).

Mote, Jr., F. A.

C. H. Graham, R. H. Brown, and F. A. Mote, Jr., J. Exptl. Psychol. 24, 555 (1939).

Ogle, K. N.

K. N. Ogle, J. Opt. Soc. Am. 51, 1265 (1961).

Pinsker, H.

T. N. Cornsweet and H. Pinsker, J. Physiol. (London) 176, 294 (1965).

Polyak, S.

S. Polyak, The Vertebrate Visual System (Univ. of Chicago Press, 1957).

Ratliff, F.

F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, 1965), p. 151.

Riggs, L. A.

L. A. Riggs, in Vision and Visual Perception, C. H. Graham, Ed., (John Wiley & Sons, Inc., New York, 1965), p. 334.

Ronchi, L.

L. Ronchi and G. Toraldo di Francia, J. Opt. Soc. Am. 47, 639 (1957).

Schade, O. H.

O. H. Schade, J. Opt. Soc. Am. 46, 721 (1956).

Schlosberg, H.

R. S. Woodworth and H. Schlosberg, Experimental Psychology (Henry Holt & Co., Inc., New York, 1954), p. 386.

Sloan, L. L.

J. Mandelbaum and L. L. Sloan, Am. J. Ophthalmol. 30, 581 (1947).

Sneddon, I. N.

I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Co., New York, 1951).

Stevens, S. S.

S. S. Stevens, Ed., Handbook of Experimental Psychology (John Wiley & Sons, Inc., New York, 1951), p. 929.

Taylor, A.

See e.g. A. Taylor, Advanced Calculus (Ginn and Co., Boston, Mass., 1955), p. 228.

Westheimer, G.

G. Westheimer, J. Physiol. (London) 190, 139 (1967).

G. Westheimer, Vision Res. 6, 669 (1966), Eq. (3).

G. Westheimer and F. W. Campbell, J. Opt. Soc. Am. 52, 1040 (1962).

Weymouth, F. W.

F. W. Weymouth, Am. J. Ophthalmol. 46, 102 (1958).

Woodworth, R. S.

R. S. Woodworth and H. Schlosberg, Experimental Psychology (Henry Holt & Co., Inc., New York, 1954), p. 386.

Other

O. H. Schade, J. Opt. Soc. Am. 46, 721 (1956).

J. D. Bliss and W. B. Macurdy, J. Opt. Soc. Am. 51, 1373 (1961).

O. Bryngdahl, J. Opt. Soc. Am. 54, 1152 (1964).

G. von Békésy, J. Opt. Soc. Am. 50, 1060 (1960).

E. M. Lowery and J. J. DePalma, J. Opt. Soc. Am. 51, 740 (1961).

G. A. Fry, J. Opt. Soc. Am. 53, 94 (1963).

F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina (Holden-Day, San Francisco, 1965), p. 151.

L. A. Riggs, in Vision and Visual Perception, C. H. Graham, Ed., (John Wiley & Sons, Inc., New York, 1965), p. 334.

I assume that ∫0 |D{r}|rdr converges.

I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Co., New York, 1951).

I have arbitrarily chosen sinusoids varying along y and uniform over x. There is no loss of generality; since D depends only on distance, the system is invariant under rotations, and the same D¯ would be obtained from any other orientation. But see the section on isotropy.

L. Ronchi and G. Toraldo di Francia, J. Opt. Soc. Am. 47, 639 (1957).

E. G. Heinemann, J. Exptl. Psychol. 50, 89 (1955).

See e.g. A. Taylor, Advanced Calculus (Ginn and Co., Boston, Mass., 1955), p. 228.

See W. M. Brown, Analysis of Linear Time-Invariant Systems (McGraw-Hill Book Co., New York, 1963), for a proof in the case of temporal interaction systems. Time-invariance there parallels space-invariance (homogeneity) here.

It is usually said that nonlinearity prevents Fourier analysis of this problem. But nonlinearity is easily handled with the perturbation approach. The more serious obstacle is lack of homogeneity, which is both often overlooked and logically independent of the question of linearity.

L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 37, 217 (1947).

R. S. Woodworth and H. Schlosberg, Experimental Psychology (Henry Holt & Co., Inc., New York, 1954), p. 386.

J. Mandelbaum and L. L. Sloan, Am. J. Ophthalmol. 30, 581 (1947).

F. W. Weymouth, Am. J. Ophthalmol. 46, 102 (1958).

C. H. Graham and H. H. Bartlett, J. Exptl. Psychol. 24, 574 (1939).

C. H. Graham, R. H. Brown, and F. A. Mote, Jr., J. Exptl. Psychol. 24, 555 (1939).

H. B. Barlow, J. Physiol. (London) 141, 337 (1958).

G. S. Brindley, Physiology of the Retina and the Visual Pathway (Edward Arnold and Co., London, 1960), pp. 173, 236.

M. A. Bouman and G. van den Brink, J. Opt. Soc. Am. 42, 617 (1952).

K. N. Ogle, J. Opt. Soc. Am. 51, 1265 (1961).

G. Westheimer, J. Physiol. (London) 190, 139 (1967).

S. Polyak, The Vertebrate Visual System (Univ. of Chicago Press, 1957).

J. L. Brown, in Ref. 8, p. 50.

S. S. Stevens, Ed., Handbook of Experimental Psychology (John Wiley & Sons, Inc., New York, 1951), p. 929.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964), p. 15ff.

See the work of Hartline and Ratliff, as summarized by Ratliff (Ref. 7), pp. 105–117.

G. Westheimer, Vision Res. 6, 669 (1966), Eq. (3).

This may not be serious, since the results indicate that the visual system being studied strongly attenuates low frequencies.

T. N. Cornsweet, Am. J. Psychol. 75, 485 (1962).

T. N. Cornsweet and H. Pinsker, J. Physiol. (London) 176, 294 (1965).

One unit of subjective contrast is the perturbation amplitude at the absolute threshold for the perception of contrast. Cf. Ref. 39.

This effect is not evident in Fig. 4, but only moderately low amplitudes were used in these measurements.

The average value of Aω at LF's forced-choice threshold for frequenc'es less 0.06 cycle/min is 0.0861; I have taken the forcedcho.ce threshold as the unit of subjective contrast (cf. Table I). The recognition threshold for BM was 1.64 units on this scale.

J. Krauskopf, J. Opt. Soc. Am. 52, 1046 (1962).

G. Westheimer and F. W. Campbell, J. Opt. Soc. Am. 52, 1040 (1962).

See Ref. 2; Eq. (7) is the two-dimensional equivalent of the result given in this reference.

This lateral travel time is not the same as the latency for inhibition. It is the difference of the onset time for inhibition as a function of difference of distance between inhibiting and inhibited units.

Although flashes of less than 500-;usec duration were used here, the results do not imply necessarily that inhibition is effective in such a time. It is possible that the system's excitation persists for some time following the flash, and, since there was darkness both before and after the flash, I can say only that inhibition is fast enough to catch up with the excitation. The negative results of J. Nachmias, J. Opt. Soc. Am. 57, 421 (1967), may be clue to his use of light periods before and after the flash, although his use of square-wave gratings complicates the comparison in other ways.

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