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.

© 1968 Optical Society of America

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References

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  1. O. H. Schade, J. Opt. Soc. Am. 46, 721 (1956).
    [Crossref] [PubMed]
  2. J. D. Bliss and W. B. Macurdy, J. Opt. Soc. Am. 51, 1373 (1961).
    [Crossref]
  3. O. Bryngdahl, J. Opt. Soc. Am. 54, 1152 (1964).
    [Crossref]
  4. G. von Békésy, J. Opt. Soc. Am. 50, 1060 (1960).
    [Crossref]
  5. E. M. Lowry and J. J. DePalma, J. Opt. Soc. Am. 51, 740 (1961).
    [Crossref] [PubMed]
  6. G. A. Fry, J. Opt. Soc. Am. 53, 94 (1963).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  13. E. G. Heinemann, J. Exptl. Psychol. 50, 89 (1955).
    [Crossref]
  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).
    [Crossref] [PubMed]
  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).
    [PubMed]
  20. F. W. Weymouth, Am. J. Ophthalmol. 46, 102 (1958).
    [PubMed]
  21. C. H. Graham and H. H. Bartlett, J. Exptl. Psychol. 24, 574 (1939).
    [Crossref]
  22. C. H. Graham, R. H. Brown, and F. A. Mote, J. Exptl. Psychol. 24, 555 (1939).
    [Crossref]
  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).
    [Crossref] [PubMed]
  26. K. N. Ogle, J. Opt. Soc. Am. 51, 1265 (1961).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  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 frequencies less 0.06 cycle/min is 0.0861; I have taken the forced-choice 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).
    [Crossref]
  41. G. Westheimer and F. W. Campbell, J. Opt. Soc. Am. 52, 1040 (1962).
    [Crossref] [PubMed]
  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-μ sec 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 due to his use of light periods before and after the flash, although his use of square-wave gratings complicates the comparison in other ways.
    [Crossref]

1967 (2)

1966 (1)

G. Westheimer, Vision Res. 6, 669 (1966), Eq. (3).
[Crossref] [PubMed]

1965 (1)

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

1964 (1)

1963 (1)

1962 (3)

1961 (3)

1960 (1)

1958 (2)

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

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

1957 (1)

1956 (1)

1955 (1)

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

1952 (1)

1947 (2)

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

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

1939 (2)

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

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

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).
[Crossref]

Bliss, J. D.

Bouman, M. A.

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, J. Exptl. Psychol. 24, 555 (1939).
[Crossref]

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.

Campbell, F. W.

Cornsweet, T. N.

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

T. N. Cornsweet, Am. J. Psychol. 75, 485 (1962).
[Crossref] [PubMed]

DePalma, J. J.

Fry, G. A.

Graham, C. H.

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

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

Heinemann, E. G.

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

Higgins, G. C.

Jones, L. A.

Krauskopf, J.

Linfoot, E. H.

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

Lowry, E. M.

Macurdy, W. B.

Mandelbaum, J.

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

Mote, F. A.

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

Nachmias, J.

Ogle, K. N.

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,

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

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.

Schade, O. H.

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).
[PubMed]

Sneddon, I. N.

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

Taylor, A.

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

Toraldo di Francia, G.

van den Brink, G.

von Békésy, G.

Westheimer, G.

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

G. Westheimer, Vision Res. 6, 669 (1966), Eq. (3).
[Crossref] [PubMed]

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

Weymouth, F. W.

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

Woodworth, R. S.

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

Am. J. Ophthalmol. (2)

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

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

Am. J. Psychol. (1)

T. N. Cornsweet, Am. J. Psychol. 75, 485 (1962).
[Crossref] [PubMed]

J. Exptl. Psychol. (3)

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

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

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

J. Opt. Soc. Am. (13)

O. H. Schade, J. Opt. Soc. Am. 46, 721 (1956).
[Crossref] [PubMed]

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

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

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

E. M. Lowry and J. J. DePalma, J. Opt. Soc. Am. 51, 740 (1961).
[Crossref] [PubMed]

G. A. Fry, J. Opt. Soc. Am. 53, 94 (1963).
[Crossref] [PubMed]

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

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

K. N. Ogle, J. Opt. Soc. Am. 51, 1265 (1961).
[Crossref] [PubMed]

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

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

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

Although flashes of less than 500-μ sec 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 due to his use of light periods before and after the flash, although his use of square-wave gratings complicates the comparison in other ways.
[Crossref]

J. Physiol. (London) (3)

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

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

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

Vision Res. (1)

G. Westheimer, Vision Res. 6, 669 (1966), Eq. (3).
[Crossref] [PubMed]

Other (21)

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

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

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 frequencies less 0.06 cycle/min is 0.0861; I have taken the forced-choice threshold as the unit of subjective contrast (cf. Table I). The recognition threshold for BM was 1.64 units on this scale.

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

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.

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.

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.

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Figures (10)

Fig. 1
Fig. 1

Output from hypothetical spatial-summation system; input sinusoids of various frequencies. Horizontal axis, distance from foveal center in degrees of arc. Each sinusoid oscillates within envelope shown. Top, frequency 0.4 cycle/min of arc. Middle, 0.15 cycle/min. Bottom, 0.04 cycle/min. Horizontal bars, ten wavelengths.

Fig. 2
Fig. 2

Optical schematic of appratus. M, rotating mirror. Upper half, field objects and images. W1 and W2, sine-wave patterns. Images move across circular aperture C and retina as mirror rotates. F, fixation pattern; S, shutters. Lower half, light sources and images. L1 and L2 are for field patterns, LA for adapting field, LF for fixation pattern. P, artificial pupil.

Fig. 3
Fig. 3

Photographs through apparatus exit pupil of typical fields used in the experiment. Top, higher spatial frequency; bottom, lower spatial frequency. Left, higher contrast; right, lower contrast. In each case, photograph of slide image (mirror stationary) is shown beside field produced with mirror rotating.

Fig. 4
Fig. 4

Measured optical properties of apparatus. Main figure, apparatus transfer function (vertical axis). Nominal (input) amplitudes: triangles, 0.600; circles, 0.300; squares, 0.178. Inset: Mean log intensity of nominally uniform field (vertical axis) as a function of vertical distance from field center (horizontal axis).

Fig. 5
Fig. 5

Complete-system describing and transfer functions, subject LF. Describing functions. Vertical axis, reciprocal of relative variable-field amplitude required to match contrast of standard field. Triangles, vertices up, data at 1.86 units subjective contrast; triangles, vertices down, 5.84 units; diamonds, 7.73 units; squares, 9.61 units. Transfer function. Open circles, extrapolation from unsmoothed data (data at the particular wavelength only); solid circles, extrapolation from smoothed-data curves.

Fig. 6
Fig. 6

Complete-system describing and transfer functions, subject BM. Describing functions. Vertical axis, reciprocal of relative variable-field amplitude required to match contrast of standard field. Triangles, vertices up, data at 1.83 units subjective contrast; triangles, vertices down, 3.11 units; squares, 7.81 units. Transfer function. Open circles, extrapolation from unsmoothed data (data at the particular wavelength only); solid circles, extrapolation from smoothed-data curves.

Fig. 7
Fig. 7

Effect of field diameter on describing function (vertical axis). Subject BM, 1.83 units subjective contrast. Circles, 6°40′ field diameter; squares, 5°20′; triangles, 4°0′.

Fig. 8
Fig. 8

Comparison of threshold-for-contrast measurements (points) to extrapolated transfer functions (curves from Figs. 5 and 6). Vertical axis, reciprocal threshold (average of two values). Triangles, recognition threshold (subject BM); circles, forced-choice threshold (subject LF).

Fig. 9
Fig. 9

Spatial interaction mechanisms in the experimental situation. LN, nominal luminance distribution; L0, actual objective luminance distribution; LR, retinal illuminance distribution. F, input to spatial interaction system (logarithm of LR); Φ, brightness distribution.

Fig. 10
Fig. 10

Visual-system transfer functions. Upper group, subject BM; lower group, subject LF. Within each group, lower curve (circles) is for complete visual system, upper curve for post-receptor system only. Latter obtained by using transfer functions for visual optics, only, from Westheimer and Campbell (triangles) and Krauskopf (squares). Extrapolations at right are explained in text.

Tables (1)

Tables Icon

Table I Standard fields and subjective contrast, matching-experiment describing functions.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

Φ ( x , y ) = - - D { [ ( x - x ) 2 + ( y - y ) 2 ] 1 2 } × F ( x , y ) d x d y ,
F ( x , y ) = A cos ( ω y )
Φ ( x , y ) = A D ¯ ( ω ) cos ( ω y ) ,
D { r } = 1 2 π 0 D ¯ ( ω ) J 0 ( ω r ) ω d ω .
ϕ ( x , y ) = - - K ( x , y , x , y ) f ( x , y ) d x d y .
ϕ ( x , y ) = - - k ( x - x , y - y ) f ( x , y ) d x d y .
D R ( r ) = 1 2 π 0 [ D ¯ ( ω ) ] - 1 J 0 ( ω r ) ω d ω .