Abstract

The Julesz paradigm involving statistical constraints for the study of texture discrimination is extended to continuous-contrast repetitive patterns by using nth-order autocorrelation functions rather than Julesz’s nth-order statistics. Second-order autocorrelations specify the power spectrum of the pattern, and the higher-order autocorrelations specify different levels of phase relationships in the pattern. A method is presented for generating patterns with statistical constraints of any order, in one and two dimensions. We show that, without scrutiny by foveal attention, discrimination of continuous textures fails at about the level of fourth-order constraints. An explanation for this failure based on the bandwidth of spatial-frequency-tuned mechanisms is excluded. The autocorrelation approach therefore may provide a general metric for the description of phase discrimination of repetitive textures.

© 1986 Optical Society of America

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References

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  1. C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in vision: The detectability of frequency modulated gratings,” Vision Res. 15, 899–910 (1975).
    [Crossref]
  2. H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
    [Crossref]
  3. H. Mostafavi, D. J. Sakrison, “Structure and properties of a single channel in the human visual system,” Vision Res. 16, 957–968 (1976).
    [Crossref] [PubMed]
  4. S. A. Klein, D. M. Levi, “Hyperacuity thresholds of 1 sec: theoretical predictions and empirical validation,” J. Opt. Soc. Am. A 2, 1170–1190 (1985).
    [Crossref] [PubMed]
  5. B. Julesz, “Visual pattern discrimination,”IRE Trans. Inf. Theory IT-8, 84–92 (1962). This is Julesz’s first paper on texture discrimination. It shows clearly that the Julesz statistics provide stronger constraints than autocorrelation statistics.
    [Crossref]
  6. H. B. Barlow, “The efficiency of detecting changes of density in random dot patterns,” Vision Res. 18, 637–650 (1978).
    [Crossref] [PubMed]
  7. B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics: revisited,” Perception 2, 391–405 (1973).
    [Crossref]
  8. T. M. Caelli, B. Julesz, “On perceptual analyzers underlying visual texture discrimination: Part I,” Biol. Cybern. 28, 167–175 (1978).
    [Crossref] [PubMed]
  9. T. M. Caelli, B. Julesz, E. N. Gilbert, “On perceptual analyzers underlying visual texture discrimination: Part II,” Biol. Cybern. 29, 201–214 (1978).
    [Crossref] [PubMed]
  10. B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137–140 (1978).
    [Crossref] [PubMed]
  11. B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature 290, 91–97 (1981).
    [Crossref] [PubMed]
  12. J. R. Bergen, B. Julesz, “Parallel versus serial processing in rapid pattern discrimination,” Nature 303, 696–698 (1983).
    [Crossref] [PubMed]
  13. S. A. Klein, C. W. Tyler, “Phase discrimination of two-component gratings,” Invest. Ophthalmol. Suppl. 20, 124 (1981).
  14. M. C. Lawden, “An investigation of the ability of the human visual system to encode spatial phase relationships,” Vision Res. 23, 1451–1463 (1983).
    [Crossref] [PubMed]
  15. C. Stephenson, O. Braddick, “Discrimination of relative spatial phase in fovea and periphery,” Invest. Ophthalmol. Suppl. 24, 146 (1983).
  16. D. J. Field, J. Nachmias, “Phase reversal discrimination,” Vision Res. 24, 333–340 (1984).
    [Crossref] [PubMed]
  17. I. Rentschler, B. Treutwein, “Loss of spatial phase relationships in extrafoveal vision,” Nature 313, 308–310 (1985).
    [Crossref] [PubMed]
  18. J. Nachmias, B. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1629 (1983).
    [Crossref] [PubMed]
  19. C. F. Stromeyer, A. F. Lange, L. Ganz, “Spatial frequency phase effects in human vision,” Vision Res. 13, 2345–2360 (1973).
    [Crossref] [PubMed]
  20. C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
    [Crossref] [PubMed]
  21. S. Klein, C. F. Stromeyer, “On inhibition between spatial frequency channels: Adaptation to complex gratings,” Vision Res. 20, 459–466 (1980).
    [Crossref] [PubMed]
  22. G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency,” Vision Res. 16, 887–898 (1975).
    [Crossref]
  23. D. Swift, R. A. Smith, “Adaptation of contrast-modulated gratings: evidence for a compressive non-linearity,” Invest. Ophthalmol. Suppl. 26, 139 (1985).
  24. T. B. Lawton, “The effect of phase structures on spatial phase discrimination,” Vision Res. 24, 139–148 (1984).
    [Crossref] [PubMed]

1985 (3)

I. Rentschler, B. Treutwein, “Loss of spatial phase relationships in extrafoveal vision,” Nature 313, 308–310 (1985).
[Crossref] [PubMed]

D. Swift, R. A. Smith, “Adaptation of contrast-modulated gratings: evidence for a compressive non-linearity,” Invest. Ophthalmol. Suppl. 26, 139 (1985).

S. A. Klein, D. M. Levi, “Hyperacuity thresholds of 1 sec: theoretical predictions and empirical validation,” J. Opt. Soc. Am. A 2, 1170–1190 (1985).
[Crossref] [PubMed]

1984 (2)

T. B. Lawton, “The effect of phase structures on spatial phase discrimination,” Vision Res. 24, 139–148 (1984).
[Crossref] [PubMed]

D. J. Field, J. Nachmias, “Phase reversal discrimination,” Vision Res. 24, 333–340 (1984).
[Crossref] [PubMed]

1983 (4)

J. Nachmias, B. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1629 (1983).
[Crossref] [PubMed]

M. C. Lawden, “An investigation of the ability of the human visual system to encode spatial phase relationships,” Vision Res. 23, 1451–1463 (1983).
[Crossref] [PubMed]

C. Stephenson, O. Braddick, “Discrimination of relative spatial phase in fovea and periphery,” Invest. Ophthalmol. Suppl. 24, 146 (1983).

J. R. Bergen, B. Julesz, “Parallel versus serial processing in rapid pattern discrimination,” Nature 303, 696–698 (1983).
[Crossref] [PubMed]

1981 (2)

S. A. Klein, C. W. Tyler, “Phase discrimination of two-component gratings,” Invest. Ophthalmol. Suppl. 20, 124 (1981).

B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature 290, 91–97 (1981).
[Crossref] [PubMed]

1980 (1)

S. Klein, C. F. Stromeyer, “On inhibition between spatial frequency channels: Adaptation to complex gratings,” Vision Res. 20, 459–466 (1980).
[Crossref] [PubMed]

1979 (1)

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

1978 (4)

H. B. Barlow, “The efficiency of detecting changes of density in random dot patterns,” Vision Res. 18, 637–650 (1978).
[Crossref] [PubMed]

T. M. Caelli, B. Julesz, “On perceptual analyzers underlying visual texture discrimination: Part I,” Biol. Cybern. 28, 167–175 (1978).
[Crossref] [PubMed]

T. M. Caelli, B. Julesz, E. N. Gilbert, “On perceptual analyzers underlying visual texture discrimination: Part II,” Biol. Cybern. 29, 201–214 (1978).
[Crossref] [PubMed]

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137–140 (1978).
[Crossref] [PubMed]

1976 (1)

H. Mostafavi, D. J. Sakrison, “Structure and properties of a single channel in the human visual system,” Vision Res. 16, 957–968 (1976).
[Crossref] [PubMed]

1975 (2)

C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in vision: The detectability of frequency modulated gratings,” Vision Res. 15, 899–910 (1975).
[Crossref]

G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency,” Vision Res. 16, 887–898 (1975).
[Crossref]

1974 (1)

C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
[Crossref] [PubMed]

1973 (2)

C. F. Stromeyer, A. F. Lange, L. Ganz, “Spatial frequency phase effects in human vision,” Vision Res. 13, 2345–2360 (1973).
[Crossref] [PubMed]

B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics: revisited,” Perception 2, 391–405 (1973).
[Crossref]

1962 (1)

B. Julesz, “Visual pattern discrimination,”IRE Trans. Inf. Theory IT-8, 84–92 (1962). This is Julesz’s first paper on texture discrimination. It shows clearly that the Julesz statistics provide stronger constraints than autocorrelation statistics.
[Crossref]

Barlow, H. B.

H. B. Barlow, “The efficiency of detecting changes of density in random dot patterns,” Vision Res. 18, 637–650 (1978).
[Crossref] [PubMed]

Bergen, J. R.

J. R. Bergen, B. Julesz, “Parallel versus serial processing in rapid pattern discrimination,” Nature 303, 696–698 (1983).
[Crossref] [PubMed]

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

Braddick, O.

C. Stephenson, O. Braddick, “Discrimination of relative spatial phase in fovea and periphery,” Invest. Ophthalmol. Suppl. 24, 146 (1983).

Broadbent, D. E.

G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency,” Vision Res. 16, 887–898 (1975).
[Crossref]

Caelli, T. M.

T. M. Caelli, B. Julesz, “On perceptual analyzers underlying visual texture discrimination: Part I,” Biol. Cybern. 28, 167–175 (1978).
[Crossref] [PubMed]

T. M. Caelli, B. Julesz, E. N. Gilbert, “On perceptual analyzers underlying visual texture discrimination: Part II,” Biol. Cybern. 29, 201–214 (1978).
[Crossref] [PubMed]

Field, D. J.

D. J. Field, J. Nachmias, “Phase reversal discrimination,” Vision Res. 24, 333–340 (1984).
[Crossref] [PubMed]

Frisch, H. L.

B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics: revisited,” Perception 2, 391–405 (1973).
[Crossref]

Ganz, L.

C. F. Stromeyer, A. F. Lange, L. Ganz, “Spatial frequency phase effects in human vision,” Vision Res. 13, 2345–2360 (1973).
[Crossref] [PubMed]

Gilbert, E. N.

T. M. Caelli, B. Julesz, E. N. Gilbert, “On perceptual analyzers underlying visual texture discrimination: Part II,” Biol. Cybern. 29, 201–214 (1978).
[Crossref] [PubMed]

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137–140 (1978).
[Crossref] [PubMed]

B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics: revisited,” Perception 2, 391–405 (1973).
[Crossref]

Henning, G. B.

G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency,” Vision Res. 16, 887–898 (1975).
[Crossref]

Hertz, B. G.

G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency,” Vision Res. 16, 887–898 (1975).
[Crossref]

Julesz, B.

J. R. Bergen, B. Julesz, “Parallel versus serial processing in rapid pattern discrimination,” Nature 303, 696–698 (1983).
[Crossref] [PubMed]

B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature 290, 91–97 (1981).
[Crossref] [PubMed]

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137–140 (1978).
[Crossref] [PubMed]

T. M. Caelli, B. Julesz, E. N. Gilbert, “On perceptual analyzers underlying visual texture discrimination: Part II,” Biol. Cybern. 29, 201–214 (1978).
[Crossref] [PubMed]

T. M. Caelli, B. Julesz, “On perceptual analyzers underlying visual texture discrimination: Part I,” Biol. Cybern. 28, 167–175 (1978).
[Crossref] [PubMed]

B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics: revisited,” Perception 2, 391–405 (1973).
[Crossref]

B. Julesz, “Visual pattern discrimination,”IRE Trans. Inf. Theory IT-8, 84–92 (1962). This is Julesz’s first paper on texture discrimination. It shows clearly that the Julesz statistics provide stronger constraints than autocorrelation statistics.
[Crossref]

Klein, S.

S. Klein, C. F. Stromeyer, “On inhibition between spatial frequency channels: Adaptation to complex gratings,” Vision Res. 20, 459–466 (1980).
[Crossref] [PubMed]

C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in vision: The detectability of frequency modulated gratings,” Vision Res. 15, 899–910 (1975).
[Crossref]

C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
[Crossref] [PubMed]

Klein, S. A.

S. A. Klein, D. M. Levi, “Hyperacuity thresholds of 1 sec: theoretical predictions and empirical validation,” J. Opt. Soc. Am. A 2, 1170–1190 (1985).
[Crossref] [PubMed]

S. A. Klein, C. W. Tyler, “Phase discrimination of two-component gratings,” Invest. Ophthalmol. Suppl. 20, 124 (1981).

Lange, A. F.

C. F. Stromeyer, A. F. Lange, L. Ganz, “Spatial frequency phase effects in human vision,” Vision Res. 13, 2345–2360 (1973).
[Crossref] [PubMed]

Lawden, M. C.

M. C. Lawden, “An investigation of the ability of the human visual system to encode spatial phase relationships,” Vision Res. 23, 1451–1463 (1983).
[Crossref] [PubMed]

Lawton, T. B.

T. B. Lawton, “The effect of phase structures on spatial phase discrimination,” Vision Res. 24, 139–148 (1984).
[Crossref] [PubMed]

Levi, D. M.

Mostafavi, H.

H. Mostafavi, D. J. Sakrison, “Structure and properties of a single channel in the human visual system,” Vision Res. 16, 957–968 (1976).
[Crossref] [PubMed]

Nachmias, J.

D. J. Field, J. Nachmias, “Phase reversal discrimination,” Vision Res. 24, 333–340 (1984).
[Crossref] [PubMed]

J. Nachmias, B. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1629 (1983).
[Crossref] [PubMed]

Rentschler, I.

I. Rentschler, B. Treutwein, “Loss of spatial phase relationships in extrafoveal vision,” Nature 313, 308–310 (1985).
[Crossref] [PubMed]

Rogowitz, B. E.

J. Nachmias, B. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1629 (1983).
[Crossref] [PubMed]

Sakrison, D. J.

H. Mostafavi, D. J. Sakrison, “Structure and properties of a single channel in the human visual system,” Vision Res. 16, 957–968 (1976).
[Crossref] [PubMed]

Shepp, L. A.

B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics: revisited,” Perception 2, 391–405 (1973).
[Crossref]

Smith, R. A.

D. Swift, R. A. Smith, “Adaptation of contrast-modulated gratings: evidence for a compressive non-linearity,” Invest. Ophthalmol. Suppl. 26, 139 (1985).

Stephenson, C.

C. Stephenson, O. Braddick, “Discrimination of relative spatial phase in fovea and periphery,” Invest. Ophthalmol. Suppl. 24, 146 (1983).

Stromeyer, C. F.

S. Klein, C. F. Stromeyer, “On inhibition between spatial frequency channels: Adaptation to complex gratings,” Vision Res. 20, 459–466 (1980).
[Crossref] [PubMed]

C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in vision: The detectability of frequency modulated gratings,” Vision Res. 15, 899–910 (1975).
[Crossref]

C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
[Crossref] [PubMed]

C. F. Stromeyer, A. F. Lange, L. Ganz, “Spatial frequency phase effects in human vision,” Vision Res. 13, 2345–2360 (1973).
[Crossref] [PubMed]

Swift, D.

D. Swift, R. A. Smith, “Adaptation of contrast-modulated gratings: evidence for a compressive non-linearity,” Invest. Ophthalmol. Suppl. 26, 139 (1985).

Treutwein, B.

I. Rentschler, B. Treutwein, “Loss of spatial phase relationships in extrafoveal vision,” Nature 313, 308–310 (1985).
[Crossref] [PubMed]

Tyler, C. W.

S. A. Klein, C. W. Tyler, “Phase discrimination of two-component gratings,” Invest. Ophthalmol. Suppl. 20, 124 (1981).

Victor, J. D.

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137–140 (1978).
[Crossref] [PubMed]

Wilson, H. R.

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

Biol. Cybern. (3)

T. M. Caelli, B. Julesz, “On perceptual analyzers underlying visual texture discrimination: Part I,” Biol. Cybern. 28, 167–175 (1978).
[Crossref] [PubMed]

T. M. Caelli, B. Julesz, E. N. Gilbert, “On perceptual analyzers underlying visual texture discrimination: Part II,” Biol. Cybern. 29, 201–214 (1978).
[Crossref] [PubMed]

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137–140 (1978).
[Crossref] [PubMed]

Invest. Ophthalmol. Suppl. (3)

S. A. Klein, C. W. Tyler, “Phase discrimination of two-component gratings,” Invest. Ophthalmol. Suppl. 20, 124 (1981).

D. Swift, R. A. Smith, “Adaptation of contrast-modulated gratings: evidence for a compressive non-linearity,” Invest. Ophthalmol. Suppl. 26, 139 (1985).

C. Stephenson, O. Braddick, “Discrimination of relative spatial phase in fovea and periphery,” Invest. Ophthalmol. Suppl. 24, 146 (1983).

IRE Trans. Inf. Theory (1)

B. Julesz, “Visual pattern discrimination,”IRE Trans. Inf. Theory IT-8, 84–92 (1962). This is Julesz’s first paper on texture discrimination. It shows clearly that the Julesz statistics provide stronger constraints than autocorrelation statistics.
[Crossref]

J. Opt. Soc. Am. A (1)

Nature (3)

B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature 290, 91–97 (1981).
[Crossref] [PubMed]

J. R. Bergen, B. Julesz, “Parallel versus serial processing in rapid pattern discrimination,” Nature 303, 696–698 (1983).
[Crossref] [PubMed]

I. Rentschler, B. Treutwein, “Loss of spatial phase relationships in extrafoveal vision,” Nature 313, 308–310 (1985).
[Crossref] [PubMed]

Perception (1)

B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics: revisited,” Perception 2, 391–405 (1973).
[Crossref]

Vision Res. (12)

D. J. Field, J. Nachmias, “Phase reversal discrimination,” Vision Res. 24, 333–340 (1984).
[Crossref] [PubMed]

J. Nachmias, B. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1629 (1983).
[Crossref] [PubMed]

C. F. Stromeyer, A. F. Lange, L. Ganz, “Spatial frequency phase effects in human vision,” Vision Res. 13, 2345–2360 (1973).
[Crossref] [PubMed]

C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
[Crossref] [PubMed]

S. Klein, C. F. Stromeyer, “On inhibition between spatial frequency channels: Adaptation to complex gratings,” Vision Res. 20, 459–466 (1980).
[Crossref] [PubMed]

G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyses spatial patterns in independent bands of spatial frequency,” Vision Res. 16, 887–898 (1975).
[Crossref]

T. B. Lawton, “The effect of phase structures on spatial phase discrimination,” Vision Res. 24, 139–148 (1984).
[Crossref] [PubMed]

C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in vision: The detectability of frequency modulated gratings,” Vision Res. 15, 899–910 (1975).
[Crossref]

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

H. Mostafavi, D. J. Sakrison, “Structure and properties of a single channel in the human visual system,” Vision Res. 16, 957–968 (1976).
[Crossref] [PubMed]

H. B. Barlow, “The efficiency of detecting changes of density in random dot patterns,” Vision Res. 18, 637–650 (1978).
[Crossref] [PubMed]

M. C. Lawden, “An investigation of the ability of the human visual system to encode spatial phase relationships,” Vision Res. 23, 1451–1463 (1983).
[Crossref] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

These third-order luminance gratings are composed of the first, third, and fifth harmonics. The left-hand panel is in square-wave phase (edges-add) and the right-hand panel is in triangle phase (peaks-add).

Fig. 2
Fig. 2

These third-order gratings are composed of the third, fourth, and fifth harmonics. The upper left, upper right, and lower left panels are in AM phase and the lower right panel is in FM phase.

Fig. 3
Fig. 3

These fourth-order gratings are composed of a first, a fourth, and an eleventh harmonic. Each of these four panels presents a different phase relationship between the components.

Fig. 4
Fig. 4

The spectral components are given in the text. The upper left and upper right panels have peaks-add and peaks-subtract phase, the lower left panel has edges-add phase in order to maximize local contrast, and the lower right panel has scrambled phases in order to minimize the maximum local contrast.

Fig. 5
Fig. 5

Phase discrimination of a fundamental plus nth harmonic for two observers. The harmonic n is on the abscissa. The data points (e.g., peripheral viewing of a first plus a second harmonic), which are plotted at 99.5% correct, are actually 100% correct. The upper component was 12 c/deg.

Fig. 6
Fig. 6

Phase discrimination of two sinusoids. The symbols l, m, and h represent three contrast levels as indicated. The abscissa gives the frequency ratio of the two components. The higher component was always 12 c/deg.

Fig. 7
Fig. 7

A, Four phase relationships of a second-order compound grating consisting of a first plus a second harmonic given by cos x + cos(2x + φ). From top to bottom the phases are φ = 0°, 180°, −90°, +90°. B, Four phase relationships of the second-order pattern given by ½ cos(x + φ) + cos 5x(1 + cos x). From top to bottom the phases are φ = 0°, 180°, −90°, +90°. C, The waveform of the four third-order patterns of Fig. 2. The luminances given by Eq. (7) are cos(4x + φ1) + 2 cos(4x + φ1 + φ2)cos(x). From top to bottom the phases are (φ1, φ2) = (180°, 0°),(0°, 0°), (90°, 0°), and (−90°, 90°). The first three phases are AM gratings, and the bottom waveform is quasi-FM.

Equations (40)

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

k = 1 K p k ( r ) = 1 ,
S k 1 , k n ( d 1 , d n ) = i = 1 n p k i ( r + d i ) d r / d r .
S 13 ( 0 , d ) = p 1 ( r ) p 3 ( r + d ) d r / d r = p 1 ( r ) [ 1 - p 1 ( r + d ) - p 2 ( r + d ) ] d r / d r = S 1 ( 0 ) - S 11 ( 0 , d ) - S 12 ( 0 , d ) .
L ( r ) = k = 1 K L k p k ( r ) for discrete levels = L k p k ( r ) d k for the continuous case ,
A n ( d n , d n ) = i = 1 n L ( r + d i ) d r / d r .
A n ( d 1 , , d n ) = i = 1 n L k i S k 1 k n ( d 1 , d n ) d k 1 d k n for continuous levels = k i i = 1 n L k i S k 1 k n ( d 1 , d n ) for discrete levels .            
A ˜ n ( f 1 , , f n - 1 ) = L ˜ * ( f 1 + + f n - 1 ) L ˜ ( f 1 ) L ˜ ( f n - 1 ) .
A ˜ 3 ( f 1 , f 2 ) = i = 1 3 L ˜ ( f i ) ,             where f 1 + f 2 + f 3 = 0 , = L ˜ * ( f 1 + f 2 ) L ˜ ( f 1 ) L ˜ ( f 2 ) .
cos x - 1 / 3 cos 3 x + 1 / 8 cos 5 x ( Fig .1 , left panel ) , cos x + 1 / 3 cos 3 x + 1 / 8 cos 5 x ( Fig .1 , right panel ) .
- cos 3 x - cos 4 x - cos 5 x ( Fig .2 , upper left panel ) , cos 3 x + cos 4 x + cos 5 x ( Fig .2 , upper right panel ) , - sin 3 x - sin 4 x - sin 5 z ( Fig .2 , lower left panel ) , cos 3 x + sin 4 x + cos 5 x ( Fig .2 , lower right panel ) .
A ˜ 4 ( f 1 , f 2 , f 3 ) = L ˜ * ( f 1 + f 2 + f 3 ) L ˜ ( f 1 ) L ˜ ( f 2 ) L ˜ ( f 3 ) .
A ˜ 4 ( f 1 , - f 1 , f 3 ) = + [ L ˜ * ( f 3 ) L ˜ ( f 3 ) ] [ L ˜ * ( f 1 ) L ˜ ( f 1 ) ] .
f 1 ± f 2 ± f 3 ± f 4 = 0 ,
cos x + cos 4 x + cos 11 x ( Fig .3 , upper left panel ) , - cos x - cos 4 x - cos 11 x ( Fig .3 , upper right panel ) , sin x + sin 4 x + sin 11 x ( Fig .3 , lower left panel ) , cos x + sin 4 x + cos 11 x ( Fig .3 , lower right panel ) .
L ( x ) = L 0 + a cos f 1 x + b cos ( f 2 x - φ ) = L 0 + a cos f 1 x + ( b cos φ ) cos f 2 x - ( b sin φ ) sin f 2 x .
L ˜ ( 0 ) = L 0 , L ˜ ( 1 ) = L ˜ ( - 1 ) = L ˜ * ( 1 ) = a , L ˜ ( 3 ) = L ˜ * ( - 3 ) = b ( cos φ + i sin φ ) , L ˜ ( - 3 ) = L ˜ * ( 3 ) = b ( cos φ ) - i sin φ ) ,
A 4 ( 3 , 1 , - 1 ) = A 4 ( 3 , - 3 , 1 ) = a 2 b 2 ,
Z n = [ L ( r ) ] n d r / d r ,
Z n = [ L ( x ) ] n d x / d x .
L ( x ) = 1 + c 1 cos ( f x ) + c n cos ( n f x + φ ) .
cos n θ cos n θ d θ / d θ = 2 - n ,
Z n + 1 = 2 - n c 1 n c n cos φ + ( terms independent of φ ) .
1 + c 1 cos ( f x ) + c 2 cos ( 2 f x + φ ) .
Z 3 = ( 1 / 8 ) c 1 c 2 2 cos ( φ ) + ( terms independent of φ ) .
L ( x ) = 1 + c 1 cos ( f x ) + c 3 cos ( 3 f x + φ m ± φ ) ,
Z 4 = ( 1 / 16 ) c 1 3 c 3 cos ( φ m ± φ ) + ( terms independent of φ ) .
1 + c T cos ( f x + φ ) + c M cos ( g x ) [ 1 + cos ( f x ) ] ,
Z 3 = ( 1 / 4 ) c T c M 2 cos ( φ ) + ( terms independent of φ ) .
1 + c 1 cos ( f x + φ 1 ) + c 2 cos ( f x + φ 1 + φ 2 ) cos ( g x ) ,
Z 4 = c 2 2 c 1 2 cos ( 2 φ 2 ) + ( terms independent of φ ) .
L ( r ) = f L ˜ ( f ) exp ( - i f · r ) ,
L ˜ ( f ) = D 2 0 D 0 D exp ( i f · r ) L ( r ) d r ,
A n ( d 1 , d n ) = D - 2 0 D 0 D i = 1 n L ( r + d i ) d r .
F ˜ n ( f 1 , f n ) = D - 2 n d d 1 d d n × exp [ i ( f 1 · d 1 + f n · d n ) ] A n ( d 1 , d n ) .
F ˜ n ( f 1 , f n ) = D - 2 n - 2 d r d d 1 exp ( i f 1 d 1 ) × L ( r + d 1 ) d d n exp ( i f n · d n ) L ( r , d n ) = D - 2 n - 2 d r exp ( - i r · f T ) i = 1 n d d i × exp [ i f i · ( r + d i ) ] L ( r + d i ) ,
F ˜ n ( f 1 f n = D - 2 d r exp ( - i r · f T ) i = 1 n L ˜ ( f i ) = i = 1 n L ˜ ( f i ) δ ( f T ) = A ˜ n ( f 1 , f n - 1 ) δ ( f T , )
δ ( f ) = 0 if f x 0 or f y 0 , δ ( f ) = 1 if f x = f y = 0.
A ˜ 1 = L ˜ ( 0 ) ,
A 2 ( f ) = L ˜ ( f ) L ˜ ( - f ) . = L ˜ ( f ) L ˜ * ( f ) .
A ˜ n ( f 1 , , f n - 1 ) = L ˜ * ( f 1 + f n - 1 ) L ˜ ( f 1 ) L ˜ ( f n - 1 ) .

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