Abstract

To some degree, all current models of visual motion-perception mechanisms depend on the power of the visual signal in various spatiotemporal-frequency bands. Here we show how to construct counterexamples: visual stimuli that are consistently perceived as obviously moving in a fixed direction yet for which Fourier-domain power analysis yields no systematic motion components in any given direction. We provide a general theoretical framework for investigating non-Fourier motion-perception mechanisms; central are the concepts of drift-balanced and microbalanced random stimuli. A random stimulus S is drift balanced if its expected power in the frequency domain is symmetric with respect to temporal frequency, that is, if the expected power in S of every drifting sinusoidal component is equal to the expected power of the sinusoid of the same spatial frequency, drifting at the same rate in the opposite direction. Additionally, S is microbalanced if the result WS of windowing S by any space–time-separable function W is drift balanced. We prove that (i) any space–time-separable random (or nonrandom) stimulus is microbalanced; (ii) any linear combination of pairwise independent microbalanced (respectively, drift-balanced) random stimuli is microbalanced and drift balanced if the expectation of each component is uniformly zero; (iii) the convolution of independent microbalanced and drift-balanced random stimuli is microbalanced and drift balanced; (iv) the product of independent microbalanced random stimuli is microbalanced; and (v) the expected response of any Reichardt detector to any microbalanced random stimulus is zero at every instant in time. Examples are provided of classes of microbalanced random stimuli that display consistent and compelling motion in one direction. All the results and examples from the domain of motion perception are transposable to the space-domain problem of detecting orientation in a texture pattern.

© 1988 Optical Society of America

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  1. J. P. H. van Santen, G. Sperling, “Temporal covariance model of motion perception,” J. Opt. Soc. Am. A 1, 451–473 (1984).
    [CrossRef] [PubMed]
  2. J. P. H. van Santen, G. Sperling, “Elaborated Reichardt detectors,” J. Opt. Soc. Am. A 2, 300–321 (1985).
    [CrossRef] [PubMed]
  3. E. H. Adelson, J. R. Bergen, “Spatiotemporal energy models for the perception of motion,” J. Opt. Soc. Am. A 2, 284–299 (1985).
    [CrossRef] [PubMed]
  4. A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,” NASA Tech. Memo. 84352 (National Aeronautics and Space Administration, Washington, D.C., 1983).
  5. D. J. Fleet, A. D. Jepson, “On the hierarchical construction of orientation and velocity selective filters,” Tech. Rep. RBCV-TR-85-8 (Department of Computer Science, University of Toronto, Toronto, 1985).
  6. D. J. Heeger, “Model for the extraction of image flow,” J. Opt. Soc. Am. A 4, 1455–1471 (1987).
    [CrossRef] [PubMed]
  7. A. Pantle, L. Picciano, “A multistable movement display: evidence for two separate motion systems in human vision,” Science 193, 500–502 (1976).
    [CrossRef] [PubMed]
  8. M. Green, “What determines correspondence strength in apparent motion,” Vision Res. 26, 599–607, 1986.
    [CrossRef]
  9. A. M. Derrington, G. B. Henning, “Errors in direction-of-motion discrimination with complex stimuli,” Vision Res. 27, 61–75 (1987).
    [CrossRef] [PubMed]
  10. A. M. Derrington, D. R. Badcock, “Separate detectors for simple and complex grating patterns?” Vision Res. 25, 1869–1878 (1985).
    [CrossRef] [PubMed]
  11. A. Pantle, K. Turano, “Direct comparisons of apparent motions produced with luminance, contrast-modulated (CM), and texture gratings,” Invest. Ophthalmol. Vis. Sci. 27, 141 (1986).
  12. K. Turano, A. Pantle, “On the mechanism that encodes the movement of contrast variations. I. velocity discrimination,” submitted to Vision Res.
  13. G. Sperling, “Movement perception in computer-driven visual displays,” Behav. Res. Methods Instrum. 8, 144–151 (1976).
    [CrossRef]
  14. J. T. Petersik, K. I. Hicks, A. J. Pantle, “Apparent movement of successively generated subjective figures,” Perception 7, 371–383 (1978).
    [CrossRef] [PubMed]
  15. C. Chubb, G. Sperling, “Drift-balanced random stimuli: a general basis for studying non-Fourier motion perception,” Invest. Ophthalmol. Vis. Sci. 28, 233 (1987).
  16. The main demonstrations and results described herein were first reported at the Symposium on Computational Models in Vision, Center for Visual Science, University of Rochester, June 20, 1986, and at the meeting of the Association for Research in Vision and Opthalmology, Sarasota, Florida, May 7, 1987.
  17. A. B. Watson, A. J. Ahumada, “Model of human visual-motion sensing,” J. Opt. Soc. Am. A 2, 322–342 (1985).
    [CrossRef] [PubMed]
  18. J. Victor, “Nonlinear processes in spatial vision: analysis with stochastic visual textures,” Invest. Ophthalmol. Vis. Sci. 29, 118 (1988).
  19. C. Chubb, G. Sperling, “Texture quilts: basic tools for studying motion from texture,” Publ. 88-1 of Mathematical Studies in Perception and Cognition (Department of Psychology, New York University, New York, 1988).
  20. W. Reichardt, “Autokorrelationsauswertung als Funktionsprinzip des Zentralnervensystems,” Z. Naturforschung Teil B 12, 447–457 (1957).
  21. J. P. H. van Santen, G. Sperling, “Applications of a Reichardt-type model of two-frame motion,” Invest. Ophthalmol. Vis. Sci. 25, 14 (1984).
  22. A. B. Watson, A. J. Ahumada, J. E. Farrell, “The window of visibility: a psychophysical theory of fidelity in time-sampled motion displays,” NASA Tech. Paper 2211 (National Aeronautics and Space Administration, Washington, D.C., 1983).
  23. A. B. Watson, A. J. Ahumada, J. E. Farrell, “Window of visibility: a psychophysical theory of fidelity in time-sampled visual motion displays,” J. Opt. Soc. Am. A 3, 300–307 (1986).
    [CrossRef]
  24. S. Grossberg, E. Mingolla, “Neural dynamics of form perception boundary completion, illusory figures and neon color spreading,” Psychol. Rev. 92, 173–211 (1985).
    [CrossRef] [PubMed]
  25. S. Grossberg, E. Mingolla, “Neural dynamics of form perception: textures, boundaries, and emergent segmentations,” Percept. Psychophys 38, 141–171 (1985).
    [CrossRef] [PubMed]
  26. R. J. Watt, M. J. Morgan, “The recognition and representation of edge blur: evidence of spatial primitives in human vision,” Vision Res. 23, 1465–1477 (1983).
    [CrossRef]
  27. R. J. Watt, M. J. Morgan, “Spatial filters and the localization of luminance changes in human vision,” Vision Res. 24, 1387–1397 (1984).
    [CrossRef] [PubMed]
  28. R. J. Watt, M. J. Morgan, “A theory of the primitive spatial code in human vision,” Vision Res. 25, 1661–1674 (1985).
    [CrossRef] [PubMed]
  29. G. J. Burton, “Evidence of nonlinear response processes in the human visual system from measurements on the thresholds of spatial beat frequencies,” Vision Res. 13, 1211–1225 (1973).
    [CrossRef] [PubMed]
  30. A. Y. Maudarbocus, K. H. Ruddock, “Non-linearity of visual signals in relation to shape-sensitive adaptation responses,” Vision Res. 13, 1713–1737 (1973).
    [CrossRef] [PubMed]
  31. E. Peli, “Perception of high-pass filtered images,” in Visual Communications and Image Processing II, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.845, 140–146 (1987).
    [CrossRef]
  32. T. Caelli, “Three processing characteristics of visual texture segmentation,” Spatial Vision 1, 19–30 (1985).
    [CrossRef] [PubMed]
  33. C. Chubb, G. Sperling, D. H. Parish, “Designing psychophysical discriminations tasks for which ideal performance is computationally tractable,” Publ. 88-2 of Mathematical Studies in Perception and Cognition (Department of Psychology, New York University, New York, 1988).
  34. D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966), pp. 209–231.
  35. C. Chubb, G. Sperling, “Processing stages in non-Fourier motion perception,” Invest. Ophthalmol. Vis. Sci. 29, 266 (1988).
  36. G. A. Miller, W. G. Taylor, “The perception of repeated bursts of noise,” J. Acoust. Soc. Am. 20, 171–182 (1948).
    [CrossRef]
  37. W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1966), Vol. 2, p. 151.

1988

J. Victor, “Nonlinear processes in spatial vision: analysis with stochastic visual textures,” Invest. Ophthalmol. Vis. Sci. 29, 118 (1988).

C. Chubb, G. Sperling, “Processing stages in non-Fourier motion perception,” Invest. Ophthalmol. Vis. Sci. 29, 266 (1988).

1987

D. J. Heeger, “Model for the extraction of image flow,” J. Opt. Soc. Am. A 4, 1455–1471 (1987).
[CrossRef] [PubMed]

A. M. Derrington, G. B. Henning, “Errors in direction-of-motion discrimination with complex stimuli,” Vision Res. 27, 61–75 (1987).
[CrossRef] [PubMed]

C. Chubb, G. Sperling, “Drift-balanced random stimuli: a general basis for studying non-Fourier motion perception,” Invest. Ophthalmol. Vis. Sci. 28, 233 (1987).

1986

A. Pantle, K. Turano, “Direct comparisons of apparent motions produced with luminance, contrast-modulated (CM), and texture gratings,” Invest. Ophthalmol. Vis. Sci. 27, 141 (1986).

M. Green, “What determines correspondence strength in apparent motion,” Vision Res. 26, 599–607, 1986.
[CrossRef]

A. B. Watson, A. J. Ahumada, J. E. Farrell, “Window of visibility: a psychophysical theory of fidelity in time-sampled visual motion displays,” J. Opt. Soc. Am. A 3, 300–307 (1986).
[CrossRef]

1985

E. H. Adelson, J. R. Bergen, “Spatiotemporal energy models for the perception of motion,” J. Opt. Soc. Am. A 2, 284–299 (1985).
[CrossRef] [PubMed]

J. P. H. van Santen, G. Sperling, “Elaborated Reichardt detectors,” J. Opt. Soc. Am. A 2, 300–321 (1985).
[CrossRef] [PubMed]

A. B. Watson, A. J. Ahumada, “Model of human visual-motion sensing,” J. Opt. Soc. Am. A 2, 322–342 (1985).
[CrossRef] [PubMed]

T. Caelli, “Three processing characteristics of visual texture segmentation,” Spatial Vision 1, 19–30 (1985).
[CrossRef] [PubMed]

A. M. Derrington, D. R. Badcock, “Separate detectors for simple and complex grating patterns?” Vision Res. 25, 1869–1878 (1985).
[CrossRef] [PubMed]

S. Grossberg, E. Mingolla, “Neural dynamics of form perception boundary completion, illusory figures and neon color spreading,” Psychol. Rev. 92, 173–211 (1985).
[CrossRef] [PubMed]

S. Grossberg, E. Mingolla, “Neural dynamics of form perception: textures, boundaries, and emergent segmentations,” Percept. Psychophys 38, 141–171 (1985).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, “A theory of the primitive spatial code in human vision,” Vision Res. 25, 1661–1674 (1985).
[CrossRef] [PubMed]

1984

J. P. H. van Santen, G. Sperling, “Applications of a Reichardt-type model of two-frame motion,” Invest. Ophthalmol. Vis. Sci. 25, 14 (1984).

R. J. Watt, M. J. Morgan, “Spatial filters and the localization of luminance changes in human vision,” Vision Res. 24, 1387–1397 (1984).
[CrossRef] [PubMed]

J. P. H. van Santen, G. Sperling, “Temporal covariance model of motion perception,” J. Opt. Soc. Am. A 1, 451–473 (1984).
[CrossRef] [PubMed]

1983

R. J. Watt, M. J. Morgan, “The recognition and representation of edge blur: evidence of spatial primitives in human vision,” Vision Res. 23, 1465–1477 (1983).
[CrossRef]

1978

J. T. Petersik, K. I. Hicks, A. J. Pantle, “Apparent movement of successively generated subjective figures,” Perception 7, 371–383 (1978).
[CrossRef] [PubMed]

1976

G. Sperling, “Movement perception in computer-driven visual displays,” Behav. Res. Methods Instrum. 8, 144–151 (1976).
[CrossRef]

A. Pantle, L. Picciano, “A multistable movement display: evidence for two separate motion systems in human vision,” Science 193, 500–502 (1976).
[CrossRef] [PubMed]

1973

G. J. Burton, “Evidence of nonlinear response processes in the human visual system from measurements on the thresholds of spatial beat frequencies,” Vision Res. 13, 1211–1225 (1973).
[CrossRef] [PubMed]

A. Y. Maudarbocus, K. H. Ruddock, “Non-linearity of visual signals in relation to shape-sensitive adaptation responses,” Vision Res. 13, 1713–1737 (1973).
[CrossRef] [PubMed]

1957

W. Reichardt, “Autokorrelationsauswertung als Funktionsprinzip des Zentralnervensystems,” Z. Naturforschung Teil B 12, 447–457 (1957).

1948

G. A. Miller, W. G. Taylor, “The perception of repeated bursts of noise,” J. Acoust. Soc. Am. 20, 171–182 (1948).
[CrossRef]

Adelson, E. H.

Ahumada, A. J.

A. B. Watson, A. J. Ahumada, J. E. Farrell, “Window of visibility: a psychophysical theory of fidelity in time-sampled visual motion displays,” J. Opt. Soc. Am. A 3, 300–307 (1986).
[CrossRef]

A. B. Watson, A. J. Ahumada, “Model of human visual-motion sensing,” J. Opt. Soc. Am. A 2, 322–342 (1985).
[CrossRef] [PubMed]

A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,” NASA Tech. Memo. 84352 (National Aeronautics and Space Administration, Washington, D.C., 1983).

A. B. Watson, A. J. Ahumada, J. E. Farrell, “The window of visibility: a psychophysical theory of fidelity in time-sampled motion displays,” NASA Tech. Paper 2211 (National Aeronautics and Space Administration, Washington, D.C., 1983).

Badcock, D. R.

A. M. Derrington, D. R. Badcock, “Separate detectors for simple and complex grating patterns?” Vision Res. 25, 1869–1878 (1985).
[CrossRef] [PubMed]

Bergen, J. R.

Burton, G. J.

G. J. Burton, “Evidence of nonlinear response processes in the human visual system from measurements on the thresholds of spatial beat frequencies,” Vision Res. 13, 1211–1225 (1973).
[CrossRef] [PubMed]

Caelli, T.

T. Caelli, “Three processing characteristics of visual texture segmentation,” Spatial Vision 1, 19–30 (1985).
[CrossRef] [PubMed]

Chubb, C.

C. Chubb, G. Sperling, “Processing stages in non-Fourier motion perception,” Invest. Ophthalmol. Vis. Sci. 29, 266 (1988).

C. Chubb, G. Sperling, “Drift-balanced random stimuli: a general basis for studying non-Fourier motion perception,” Invest. Ophthalmol. Vis. Sci. 28, 233 (1987).

C. Chubb, G. Sperling, D. H. Parish, “Designing psychophysical discriminations tasks for which ideal performance is computationally tractable,” Publ. 88-2 of Mathematical Studies in Perception and Cognition (Department of Psychology, New York University, New York, 1988).

C. Chubb, G. Sperling, “Texture quilts: basic tools for studying motion from texture,” Publ. 88-1 of Mathematical Studies in Perception and Cognition (Department of Psychology, New York University, New York, 1988).

Derrington, A. M.

A. M. Derrington, G. B. Henning, “Errors in direction-of-motion discrimination with complex stimuli,” Vision Res. 27, 61–75 (1987).
[CrossRef] [PubMed]

A. M. Derrington, D. R. Badcock, “Separate detectors for simple and complex grating patterns?” Vision Res. 25, 1869–1878 (1985).
[CrossRef] [PubMed]

Farrell, J. E.

A. B. Watson, A. J. Ahumada, J. E. Farrell, “Window of visibility: a psychophysical theory of fidelity in time-sampled visual motion displays,” J. Opt. Soc. Am. A 3, 300–307 (1986).
[CrossRef]

A. B. Watson, A. J. Ahumada, J. E. Farrell, “The window of visibility: a psychophysical theory of fidelity in time-sampled motion displays,” NASA Tech. Paper 2211 (National Aeronautics and Space Administration, Washington, D.C., 1983).

Feller, W.

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1966), Vol. 2, p. 151.

Fleet, D. J.

D. J. Fleet, A. D. Jepson, “On the hierarchical construction of orientation and velocity selective filters,” Tech. Rep. RBCV-TR-85-8 (Department of Computer Science, University of Toronto, Toronto, 1985).

Green, D. M.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966), pp. 209–231.

Green, M.

M. Green, “What determines correspondence strength in apparent motion,” Vision Res. 26, 599–607, 1986.
[CrossRef]

Grossberg, S.

S. Grossberg, E. Mingolla, “Neural dynamics of form perception boundary completion, illusory figures and neon color spreading,” Psychol. Rev. 92, 173–211 (1985).
[CrossRef] [PubMed]

S. Grossberg, E. Mingolla, “Neural dynamics of form perception: textures, boundaries, and emergent segmentations,” Percept. Psychophys 38, 141–171 (1985).
[CrossRef] [PubMed]

Heeger, D. J.

Henning, G. B.

A. M. Derrington, G. B. Henning, “Errors in direction-of-motion discrimination with complex stimuli,” Vision Res. 27, 61–75 (1987).
[CrossRef] [PubMed]

Hicks, K. I.

J. T. Petersik, K. I. Hicks, A. J. Pantle, “Apparent movement of successively generated subjective figures,” Perception 7, 371–383 (1978).
[CrossRef] [PubMed]

Jepson, A. D.

D. J. Fleet, A. D. Jepson, “On the hierarchical construction of orientation and velocity selective filters,” Tech. Rep. RBCV-TR-85-8 (Department of Computer Science, University of Toronto, Toronto, 1985).

Maudarbocus, A. Y.

A. Y. Maudarbocus, K. H. Ruddock, “Non-linearity of visual signals in relation to shape-sensitive adaptation responses,” Vision Res. 13, 1713–1737 (1973).
[CrossRef] [PubMed]

Miller, G. A.

G. A. Miller, W. G. Taylor, “The perception of repeated bursts of noise,” J. Acoust. Soc. Am. 20, 171–182 (1948).
[CrossRef]

Mingolla, E.

S. Grossberg, E. Mingolla, “Neural dynamics of form perception boundary completion, illusory figures and neon color spreading,” Psychol. Rev. 92, 173–211 (1985).
[CrossRef] [PubMed]

S. Grossberg, E. Mingolla, “Neural dynamics of form perception: textures, boundaries, and emergent segmentations,” Percept. Psychophys 38, 141–171 (1985).
[CrossRef] [PubMed]

Morgan, M. J.

R. J. Watt, M. J. Morgan, “A theory of the primitive spatial code in human vision,” Vision Res. 25, 1661–1674 (1985).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, “Spatial filters and the localization of luminance changes in human vision,” Vision Res. 24, 1387–1397 (1984).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, “The recognition and representation of edge blur: evidence of spatial primitives in human vision,” Vision Res. 23, 1465–1477 (1983).
[CrossRef]

Pantle, A.

A. Pantle, K. Turano, “Direct comparisons of apparent motions produced with luminance, contrast-modulated (CM), and texture gratings,” Invest. Ophthalmol. Vis. Sci. 27, 141 (1986).

A. Pantle, L. Picciano, “A multistable movement display: evidence for two separate motion systems in human vision,” Science 193, 500–502 (1976).
[CrossRef] [PubMed]

K. Turano, A. Pantle, “On the mechanism that encodes the movement of contrast variations. I. velocity discrimination,” submitted to Vision Res.

Pantle, A. J.

J. T. Petersik, K. I. Hicks, A. J. Pantle, “Apparent movement of successively generated subjective figures,” Perception 7, 371–383 (1978).
[CrossRef] [PubMed]

Parish, D. H.

C. Chubb, G. Sperling, D. H. Parish, “Designing psychophysical discriminations tasks for which ideal performance is computationally tractable,” Publ. 88-2 of Mathematical Studies in Perception and Cognition (Department of Psychology, New York University, New York, 1988).

Peli, E.

E. Peli, “Perception of high-pass filtered images,” in Visual Communications and Image Processing II, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.845, 140–146 (1987).
[CrossRef]

Petersik, J. T.

J. T. Petersik, K. I. Hicks, A. J. Pantle, “Apparent movement of successively generated subjective figures,” Perception 7, 371–383 (1978).
[CrossRef] [PubMed]

Picciano, L.

A. Pantle, L. Picciano, “A multistable movement display: evidence for two separate motion systems in human vision,” Science 193, 500–502 (1976).
[CrossRef] [PubMed]

Reichardt, W.

W. Reichardt, “Autokorrelationsauswertung als Funktionsprinzip des Zentralnervensystems,” Z. Naturforschung Teil B 12, 447–457 (1957).

Ruddock, K. H.

A. Y. Maudarbocus, K. H. Ruddock, “Non-linearity of visual signals in relation to shape-sensitive adaptation responses,” Vision Res. 13, 1713–1737 (1973).
[CrossRef] [PubMed]

Sperling, G.

C. Chubb, G. Sperling, “Processing stages in non-Fourier motion perception,” Invest. Ophthalmol. Vis. Sci. 29, 266 (1988).

C. Chubb, G. Sperling, “Drift-balanced random stimuli: a general basis for studying non-Fourier motion perception,” Invest. Ophthalmol. Vis. Sci. 28, 233 (1987).

J. P. H. van Santen, G. Sperling, “Elaborated Reichardt detectors,” J. Opt. Soc. Am. A 2, 300–321 (1985).
[CrossRef] [PubMed]

J. P. H. van Santen, G. Sperling, “Applications of a Reichardt-type model of two-frame motion,” Invest. Ophthalmol. Vis. Sci. 25, 14 (1984).

J. P. H. van Santen, G. Sperling, “Temporal covariance model of motion perception,” J. Opt. Soc. Am. A 1, 451–473 (1984).
[CrossRef] [PubMed]

G. Sperling, “Movement perception in computer-driven visual displays,” Behav. Res. Methods Instrum. 8, 144–151 (1976).
[CrossRef]

C. Chubb, G. Sperling, D. H. Parish, “Designing psychophysical discriminations tasks for which ideal performance is computationally tractable,” Publ. 88-2 of Mathematical Studies in Perception and Cognition (Department of Psychology, New York University, New York, 1988).

C. Chubb, G. Sperling, “Texture quilts: basic tools for studying motion from texture,” Publ. 88-1 of Mathematical Studies in Perception and Cognition (Department of Psychology, New York University, New York, 1988).

Swets, J. A.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966), pp. 209–231.

Taylor, W. G.

G. A. Miller, W. G. Taylor, “The perception of repeated bursts of noise,” J. Acoust. Soc. Am. 20, 171–182 (1948).
[CrossRef]

Turano, K.

A. Pantle, K. Turano, “Direct comparisons of apparent motions produced with luminance, contrast-modulated (CM), and texture gratings,” Invest. Ophthalmol. Vis. Sci. 27, 141 (1986).

K. Turano, A. Pantle, “On the mechanism that encodes the movement of contrast variations. I. velocity discrimination,” submitted to Vision Res.

van Santen, J. P. H.

Victor, J.

J. Victor, “Nonlinear processes in spatial vision: analysis with stochastic visual textures,” Invest. Ophthalmol. Vis. Sci. 29, 118 (1988).

Watson, A. B.

A. B. Watson, A. J. Ahumada, J. E. Farrell, “Window of visibility: a psychophysical theory of fidelity in time-sampled visual motion displays,” J. Opt. Soc. Am. A 3, 300–307 (1986).
[CrossRef]

A. B. Watson, A. J. Ahumada, “Model of human visual-motion sensing,” J. Opt. Soc. Am. A 2, 322–342 (1985).
[CrossRef] [PubMed]

A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,” NASA Tech. Memo. 84352 (National Aeronautics and Space Administration, Washington, D.C., 1983).

A. B. Watson, A. J. Ahumada, J. E. Farrell, “The window of visibility: a psychophysical theory of fidelity in time-sampled motion displays,” NASA Tech. Paper 2211 (National Aeronautics and Space Administration, Washington, D.C., 1983).

Watt, R. J.

R. J. Watt, M. J. Morgan, “A theory of the primitive spatial code in human vision,” Vision Res. 25, 1661–1674 (1985).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, “Spatial filters and the localization of luminance changes in human vision,” Vision Res. 24, 1387–1397 (1984).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, “The recognition and representation of edge blur: evidence of spatial primitives in human vision,” Vision Res. 23, 1465–1477 (1983).
[CrossRef]

Behav. Res. Methods Instrum.

G. Sperling, “Movement perception in computer-driven visual displays,” Behav. Res. Methods Instrum. 8, 144–151 (1976).
[CrossRef]

Invest. Ophthalmol. Vis. Sci.

A. Pantle, K. Turano, “Direct comparisons of apparent motions produced with luminance, contrast-modulated (CM), and texture gratings,” Invest. Ophthalmol. Vis. Sci. 27, 141 (1986).

C. Chubb, G. Sperling, “Drift-balanced random stimuli: a general basis for studying non-Fourier motion perception,” Invest. Ophthalmol. Vis. Sci. 28, 233 (1987).

J. Victor, “Nonlinear processes in spatial vision: analysis with stochastic visual textures,” Invest. Ophthalmol. Vis. Sci. 29, 118 (1988).

J. P. H. van Santen, G. Sperling, “Applications of a Reichardt-type model of two-frame motion,” Invest. Ophthalmol. Vis. Sci. 25, 14 (1984).

C. Chubb, G. Sperling, “Processing stages in non-Fourier motion perception,” Invest. Ophthalmol. Vis. Sci. 29, 266 (1988).

J. Acoust. Soc. Am.

G. A. Miller, W. G. Taylor, “The perception of repeated bursts of noise,” J. Acoust. Soc. Am. 20, 171–182 (1948).
[CrossRef]

J. Opt. Soc. Am. A

Percept. Psychophys

S. Grossberg, E. Mingolla, “Neural dynamics of form perception: textures, boundaries, and emergent segmentations,” Percept. Psychophys 38, 141–171 (1985).
[CrossRef] [PubMed]

Perception

J. T. Petersik, K. I. Hicks, A. J. Pantle, “Apparent movement of successively generated subjective figures,” Perception 7, 371–383 (1978).
[CrossRef] [PubMed]

Psychol. Rev.

S. Grossberg, E. Mingolla, “Neural dynamics of form perception boundary completion, illusory figures and neon color spreading,” Psychol. Rev. 92, 173–211 (1985).
[CrossRef] [PubMed]

Science

A. Pantle, L. Picciano, “A multistable movement display: evidence for two separate motion systems in human vision,” Science 193, 500–502 (1976).
[CrossRef] [PubMed]

Spatial Vision

T. Caelli, “Three processing characteristics of visual texture segmentation,” Spatial Vision 1, 19–30 (1985).
[CrossRef] [PubMed]

Vision Res.

R. J. Watt, M. J. Morgan, “The recognition and representation of edge blur: evidence of spatial primitives in human vision,” Vision Res. 23, 1465–1477 (1983).
[CrossRef]

R. J. Watt, M. J. Morgan, “Spatial filters and the localization of luminance changes in human vision,” Vision Res. 24, 1387–1397 (1984).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, “A theory of the primitive spatial code in human vision,” Vision Res. 25, 1661–1674 (1985).
[CrossRef] [PubMed]

G. J. Burton, “Evidence of nonlinear response processes in the human visual system from measurements on the thresholds of spatial beat frequencies,” Vision Res. 13, 1211–1225 (1973).
[CrossRef] [PubMed]

A. Y. Maudarbocus, K. H. Ruddock, “Non-linearity of visual signals in relation to shape-sensitive adaptation responses,” Vision Res. 13, 1713–1737 (1973).
[CrossRef] [PubMed]

M. Green, “What determines correspondence strength in apparent motion,” Vision Res. 26, 599–607, 1986.
[CrossRef]

A. M. Derrington, G. B. Henning, “Errors in direction-of-motion discrimination with complex stimuli,” Vision Res. 27, 61–75 (1987).
[CrossRef] [PubMed]

A. M. Derrington, D. R. Badcock, “Separate detectors for simple and complex grating patterns?” Vision Res. 25, 1869–1878 (1985).
[CrossRef] [PubMed]

Z. Naturforschung Teil B

W. Reichardt, “Autokorrelationsauswertung als Funktionsprinzip des Zentralnervensystems,” Z. Naturforschung Teil B 12, 447–457 (1957).

Other

C. Chubb, G. Sperling, D. H. Parish, “Designing psychophysical discriminations tasks for which ideal performance is computationally tractable,” Publ. 88-2 of Mathematical Studies in Perception and Cognition (Department of Psychology, New York University, New York, 1988).

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966), pp. 209–231.

E. Peli, “Perception of high-pass filtered images,” in Visual Communications and Image Processing II, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.845, 140–146 (1987).
[CrossRef]

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1966), Vol. 2, p. 151.

K. Turano, A. Pantle, “On the mechanism that encodes the movement of contrast variations. I. velocity discrimination,” submitted to Vision Res.

A. B. Watson, A. J. Ahumada, J. E. Farrell, “The window of visibility: a psychophysical theory of fidelity in time-sampled motion displays,” NASA Tech. Paper 2211 (National Aeronautics and Space Administration, Washington, D.C., 1983).

C. Chubb, G. Sperling, “Texture quilts: basic tools for studying motion from texture,” Publ. 88-1 of Mathematical Studies in Perception and Cognition (Department of Psychology, New York University, New York, 1988).

The main demonstrations and results described herein were first reported at the Symposium on Computational Models in Vision, Center for Visual Science, University of Rochester, June 20, 1986, and at the meeting of the Association for Research in Vision and Opthalmology, Sarasota, Florida, May 7, 1987.

A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,” NASA Tech. Memo. 84352 (National Aeronautics and Space Administration, Washington, D.C., 1983).

D. J. Fleet, A. D. Jepson, “On the hierarchical construction of orientation and velocity selective filters,” Tech. Rep. RBCV-TR-85-8 (Department of Computer Science, University of Toronto, Toronto, 1985).

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Figures (7)

Fig. 1
Fig. 1

Spatiotemporal Fourier analysis of a rightward-stepping bar. The abscissa represents horizontal space; the ordinate represents time. a, One frame of a movie of a rightward-stepping vertical bar. b, Horizontal–temporal cross section of a rightward-stepping vertical bar. c, Approximation to the rightward-stepping bar obtained by taking an equally weighted sum of {cos(2πn(x/Xt/T))∥n = 1, 2}. d, Approximation to the rightward-stepping bar obtained by taking an equally weighted sum of {cos(2πn(x/Xt/T))∥n = 1, 2, …, 12}.

Fig. 2
Fig. 2

Spatiotemporal Fourier analysis of stimulus h, a rightward-stepping, contrast-reversing vertical bar. a, Horizontal–temporal cross section of h. b, Horizontal–temporal cross section of a vertical, leftward-drifting sinusoid, which correlates well with h: cos(2π(2x/X + 2t/T) − π/2). c, Horizontal–temporal cross section of a more slowly leftward-drifting sinusoid, which also correlates well with h: cos(2π(3x/Xz + t/T) − π/2).

Fig. 3
Fig. 3

Rightward-stepping, randomly contrast-reversing vertical bar: a horizontal–temporal diagram of the random stimulus I, a vertical bar that appears with contrast C or −C randomly assigned and steps its width rightward three times over a zero-contrast visual field, assuming contrast C or −C with equal probability with each step. The expected power in I of any given drifting sinusoid is equal to the expected power of the sinusoid of the same spatial frequency drifting at the same rate but in the opposite direction.

Fig. 4
Fig. 4

a, Rightward-stepping, randomly contrast-reversing vertical bar: a horizontal–temporal cross section of a realization of the random stimulus I (see demonstration 1). I is the sum of pairwise independent space–time-separable random stimuli, each of which has an expectation of 0; consequently I is drift balanced (by corollary 1). b, Modulation of the contrast of a static noise field by a drifting sinusoidal grating: a horizontal–temporal cross section of a realization of the random stimulus K (demonstration 2). That K is drift balanced follows from corollary 1. c, Traveling contrast reversal of a noise field: a horizontal–temporal cross section of a realization of the random stimulus j (demonstration 3). J is the sum of pairwise independent space–time-separable random stimuli, each of which has an expectation of 0 and is thus drift balanced (by corollary 1). Note that, in contrast to |I| (for I of Fig. 4a), |J| is devoid of motion information. d, Modulation of the flicker frequency of a flickering noise field by a drifting grating: a horizontal–temporal cross section of a realization of the random stimulus H (demonstration 4). That H is drift balanced is a consequence of corollary C1 (in Appendix C). The motion of H is derived from spatiotemporal modulation of the frequency of sinusoidal flicker, where the phase of the flicker is random over space. e, Modulation of the contrast of a flickering noise field by a drifting sinusoidal grating: a horizontal–temporal cross section of a realization of the random stimulus G (demonstration 5). G is drift balanced (by corollary C1). The motion of G is derived from spatiotemporal modulation of the amplitude of sinusoidal flicker, where the flicker phase is random over space.

Fig. 5
Fig. 5

Point-delay Reichardt detector and its component half-detectors. a, The right half-detector computes the covariance of the contrast fluctuations of the input stimulus at point (p, q) with the fluctuations δt frames earlier at point (x, y): (x, y) and (p, q) register signal contrast frame by frame. The contrast of the current frame at pixel (p, q) is multiplied by the contrast at pixel (x, y) δt frames in the past. (The box labeled δt outputs the input it received δt frames ago.) The output from the multiplier is accumulated over all the frames of the display. b, In a similar fashion, the left half-detector computes the covariance of the contrast fluctuations of the input stimulus at point (x, y) with the fluctuations δt frames earlier at point (p, q). c, The full point-delay Reichardt detector outputs the difference between the left and right half-detectors. A positive response thus signals leftward motion; a negative response signals rightward motion.

Fig. 6
Fig. 6

Diagram of the ERD. Let I be a random stimulus; then, in response to I, for i = 1, 2, the box containing the spatial function fi:Z2 → ℝ outputs the temporal function ( x , y ) Z 2 f i [ x , y ] I [ x , y , t ]; each of the boxes marked gi* outputs the convolution of its input with the temporal function gi:Z → ℝ; each of the boxes marked with a × outputs the product of its inputs; the box marked with a − outputs its left input minus its right; and the box containing h* outputs the convolution of its input with the temporal function h:Z → ℝ.

Fig. 7
Fig. 7

Consequences of full-wave and half-wave rectification. a, Space–time representation of a traveling, contrast-reversing bar; full-wave (fw) rectified representation; and positive (hw+) and negative (hw) half-wave rectified representations, showing that either of these rectifications suffices to expose the motion to Fourier motion-energy analysis. b, Space–time representation of a traveling contrast reversal of a random bar pattern; full-wave (fw) rectified representation; positive (hw+) and negative (hw) half-wave rectified representations, showing that none of these rectifications exposes motion. The analysis system for second-order motion stimuli is shown in the bottom row: c, the signal is linearly filtered (the impulse response of an appropriate space–time-separable linear filter is shown); d, the filtered signal is full-wave rectified; and e, it is subjected to motion-energy analysis (e.g., by an ERD). This is a sufficient sequence of operations to expose the directional motion in all the demonstrations of this paper.

Equations (89)

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Ω = { cos ( α x α t ) | α } .
C ( x , t ) = cos ( α x + β t + ρ )
C ( x , t ) = cos ( ω x + τ t + ρ )
β f ( ν ) d ν .
k f ( ν ) d ν = 1.
L = μ ( 1 + I )
I ¯ ( ω , θ , τ ) = ( x , y , t ) Z 3 I [ x , y , t ] exp ( j ( ω x + θ y + τ t ) ) ( analysis ) .
I [ x , y , t ] = 1 ( 2 π ) 3 π π π π π π I ¯ ( ω , θ , τ ) × exp ( j ( ω x + θ y + τ t ) ) d ω d θ d τ ( synthesis ) .
i [ x , y , t ] = { i ( x , y , t ) if ( x , y , t ) α 0 otherwise
E I [ x , y , t ] = k i [ x , y , t ] f ( i ) d i
E ¯ I ( ω , θ , τ ) = ( x , y , t ) Z 3 k i [ x , y , t ] f ( i ) d i × exp ( j ( ω x + θ y + τ f ) ) = k ( x , y , t ) Z 3 i [ x , y , t ] exp ( j ( ω x + θ y + τ t ) ) f ( i ) d i = k i ¯ ( ω , θ , τ ) f ( i ) d i = E [ I ¯ ( ω , θ , τ ) ] .
E [ I [ x , y , t ] ] = 0 ;
E [ | I ¯ ( ω , θ , τ ) | 2 ] = E [ | I ¯ ( ω , θ , τ ) | 2 ] .
E [ | I ¯ ( ω , θ , τ ) | 2 ] = E [ | I ¯ ( ω , θ , τ ) | 2 ] ;
I [ x , y , t ] = g [ x , y ] h [ t ] ,
I [ x , y , t ] = g [ x , y ] h [ t ]
| I ¯ ( ω , θ , τ ) | 2 = | g ¯ ( ω , θ ) | 2 | h ¯ ( τ ) | 2 .
| I ¯ ( ω , θ , τ ) | 2 = | g ¯ ( ω , θ ) | 2 | h ¯ ( τ ) | 2 = | I ¯ ( ω , θ , τ ) | 2 .
| I * J ¯ | 2 = | I ¯ | 2 | J ¯ | 2 .
E [ | I ¯ | 2 | J ¯ | 2 ] = E [ | I ¯ | 2 ] E [ | J ¯ | 2 ] .
E [ | I * J ¯ ( ω , θ , τ ) | 2 ] = E [ | I ¯ ( ω , θ , τ ) | 2 ] E [ | J ¯ ( ω , θ , τ ) | 2 ] = E [ | I ¯ ( ω , θ , τ ) | 2 ] E [ | J ¯ ( ω , θ , τ ) | 2 ] = E [ | I * J ¯ ( ω , θ , τ ) | 2 ] .
E [ | S ¯ | 2 ] = | E ¯ s | 2 + I Ω E [ | N ¯ I | 2 ] ,
E [ | S ¯ ( ω , θ , τ ) | 2 ] = | E ¯ S ( ω , θ , τ ) | 2 + J Ω E [ | N ¯ J ( ω , θ , τ ) | 2 ] = | I ¯ ( ω , θ , τ ) | 2 + J Ө E [ | J ¯ ( ω , θ , τ ) | 2 ] = | I ¯ ( ω , θ , τ ) | 2 + J Ө E [ | J ¯ ( ω , θ , τ ) | 2 ] = E [ | S ¯ ( ω , θ , τ ) | 2 ] .
I [ x , y , t ] = m = 0 M 1 ϕ m f m [ x , y ] g m [ t ] ,
g m [ t ] = cos [ 2 π ( α m / M β n / N ) ] + 1 2
H [ x , y , t ] = C cos ( 4 π ( 1 + cos ( 2 π ( m M n N ) ) ) + ρ m )
G [ x , y , t ] = C 2 ( cos ( 2 π ( α m M β n N ) ) + 1 ) × cos ( 2 π γ n N + ρ m ) ,
cos ( 2 π ( 2 m 128 2 n 32 ) ) + 1 ,
r ( I ) = r left ( I ) r right ( I ) = t Z I [ x , y , t ] I [ p , q , t δ t ] t Z I [ x , y , t δ t ] I [ p , q , t ] .
H I [ δ x , δ y , δ t ] = I [ x , y , t ] I [ p , q , r ] ,
R I [ δ x , δ y , δ t ] = H I [ δ x , δ y , δ t ] H I [ δ x , δ y , δ t ] .
H I [ δ ] = H I [ δ ] .
| I ¯ ( ω , θ , τ ) | 2 = I [ x , y , t ] I [ p , q , r ] × exp ( j ( ω ( x p ) + θ ( y q ) + τ ( t r ) ) ) ,
| I ¯ ( ω , θ , τ ) | 2 = H I [ δ x , δ y , δ t ] exp ( j ( ω δ x + θ δ y + τ δ t ) ) ,
Δ I ( ω , θ , τ ) = | I ¯ ( ω , θ , τ ) | 2 | I ¯ ( ω , θ , τ ) | 2 .
Δ I ( ω , θ , τ ) = ( H I [ δ x , δ y , δ t ] H I [ δ x , δ y , δ t ] ) × exp ( j ( ω δ x + θ δ y + τ δ t ) ) = ( H I [ δ x , δ y , δ t ] H I [ δ x , δ y , δ t ] ) × exp ( j ( ω δ x + θ δ y + τ δ t ) ) = R I [ δ x , δ y , δ t ] exp ( j ( ω δ x + θ δ y + τ δ t ) ) ,
| I ¯ ( ω , θ , τ ) | 2 = H I [ δ x , δ y , δ t ] exp ( j ( ω δ x + θ δ y τ δ t ) ) = H I [ δ x , δ y , δ t ] exp ( j ( ω δ x + θ δ y + τ δ t ) ) .
( H I [ δ x , δ y , δ t ] H I [ δ x , δ y , δ t ] ) exp ( j ( ω δ x + θ δ y + τ δ t ) )
( H I [ δ x , δ y , δ t ] H I [ δ x , δ y , δ t ] ) exp ( j ( ω δ x + θ δ y + τ δ t ) ) ,
Δ I ( ω , θ , τ ) = R I [ δ x , δ y , δ t ] exp ( j ( ω δ x + θ δ y + τ δ t ) ) ,
E [ R I [ δ x , δ y , δ t ] ] = 0
E [ R I [ δ x , δ y , δ t ] ] = 0
E [ Δ I ( ω , θ , τ ) ] = 0.
E 2 [ Δ I ( ω , θ , τ ) ] = E [ R I [ δ x , δ y , δ t ] ] E [ R I [ δ p , δ q , δ r ] ] × exp ( j ( ω ( δ x δ p ) + θ ( δ y δ q ) + τ ( δ t δ r ) ) ) ,
0 2 π 0 2 π 0 2 π exp ( j ( ω ( δ x δ p ) + θ ( δ y δ q ) + τ ( δ t δ r ) ) ) d ω d θ d τ = 0 2 π exp ( j ω ( δ x δ p ) ) d ω 0 2 π exp ( j θ ( δ y δ q ) ) d θ × 0 2 π exp ( j τ ( δ t δ r ) ) d τ = { ( 2 π ) 3 if δ x = δ p , δ y = δ q , δ t = δ r , 0 otherwise ,
E 2 [ R I [ δ x , δ y , δ t ] ] = 1 8 π 3 0 2 π 0 2 π 0 2 π E 2 [ Δ I ( ω , θ , τ ) ] d ω d θ d τ .
E [ I [ x , y , t ] I [ x , y , t ] ] = E [ I [ x , y , t ] I [ x , y , t ] ] .
E [ f [ x , y ] g [ t ] f [ x , y ] g [ t ] ] = E [ f [ x , y ] g [ t ] f [ x , y ] g [ t ] ] ,
f ( x , y ) = I [ x , y , t ]
g ( t ) = I [ x , y , t ] I [ x , y , t ] .
I [ x , y , t ] = f ( x , y ) g ( t ) .
I [ x , y , t ] = I [ x , y , t ] I [ x , y , t ] I [ x , y , t ] = f ( x , y ) g ( t ) .
E [ I J [ x , y , t ] I J [ x , y , t ] ] = E [ I [ x , y , t ] I [ x , y , t ] ] E [ J [ x , y , t ] J [ x , y , t ] ] = E [ I [ x , y , t ] I [ x , y , t ] ] E [ J [ x , y , t ] J [ x , y , t ] ] = E [ I J [ x , y , t ] I J [ x , y , t ] ] .
E [ H I [ δ x , δ y , δ t ] ] = E [ I [ x , y , t ] I [ x δ x , y δ y , t δ t ] ] = E [ I [ x , y , t ] I [ x δ x , y δ y , t δ t ] ] = E [ I [ x , y , t δ t ] I [ x δ x , y δ y , t ] ] = E [ I [ x , y , t δ t ] I [ x δ x , y δ y , t ] ] = E [ I [ x , y , t ] I [ x δ x , y δ y , t + δ t ] ] = E [ H I [ δ x , δ y , δ t ] ] ,
E [ I [ x , y , t ] I [ x , y , t ] ] E [ I [ x , y , t ] I [ x , y , t ] ] .
E [ H f g I [ x x , y y , t t ] ] = E [ I [ x , y , t ] I [ x , y , t ] ] E [ I [ x , y , t ] I [ x , y , t ] ] = E [ H f g I [ x x , y y , ( t t ) ] ] .
E [ I Γ I [ x , y , t ] J Γ I [ x , y , t ] ] = I Γ J Γ E [ I [ x , y , t ] J [ x , y , t ] ] .
E [ I [ x , y , t ] J [ x , y , t ] ] = E [ I [ x , y , t ] ] E [ J [ x , y , t ] ] = 0.
I Γ E [ I [ x , y , t ] I [ x , y , t ] ] = I Γ E [ I [ x , y , t ] I [ x , y , t ] ] = E [ I Γ I [ x , y , t ] J Γ J [ x , y , t ] ] .
a 1 , a 2 , , a n
E [ I * J [ x , y , t ] I * J [ x , y , t ] ] = E [ p , q , r I [ x p , y q , t r ] J [ p , q , r ] × p , q , r I [ x p , y q , t r ] J [ p , q , r ] ] = p , q , r , p , q , r E [ I [ x p , y q , t r ] I [ x p , y q , t r ] ] × E [ J [ p , q , r ] J [ p , q , r ] ] .
p , q , r , p , q , r E [ I [ x p , y q , t r ] I [ x p , y q , t r ] ] × E [ J [ p , q , r ] J [ p , q , r ] ] = E [ p , q , r I [ x p , y q , t r ] J [ p , q , r ] × p , q , r I [ x p , y q , t r ] J [ p , q , r ] ] = E [ I * J [ x , y , t ] I * J [ x , y , t ] ] .
y i [ t ] = ( x , y ) Z 2 f i [ x , y ] I [ x , y , t ] .
[ g 1 * y 1 [ t ] ] [ g 2 * y 2 [ t ] ] , [ g 1 * y 2 [ t ] ] [ g 2 * y 1 [ t ] ] ,
D [ t ] = [ g 1 * y 1 [ t ] ] [ g 2 * y 2 [ t ] ] [ g 1 * y 2 [ t ] ] [ g 2 * y 1 [ t ] ] .
a 1 , a 2 , , a n
D [ B ] = [ u g 1 [ u ] x , y f 1 [ x , y ] I [ x , y , B u ] ] × [ t g 2 [ t ] p , q f 2 [ p , q ] I [ p , q , B t ] ] [ u g 1 [ u ] p , q f 2 [ p , q ] I [ p , q , B u ] ] × [ t g 2 [ t ] x , y f 1 [ x , y ] I [ x , y , B t ] ] ,
D [ B ] = t , u , p , q , x , y g 1 [ u ] g 2 [ t ] f 1 [ x , y ] f 2 [ p , q ] × [ I [ x , y , B u ] I [ p , q , B t ] I [ x , y , B t ] I [ p , q , B u ] ] .
E [ | I ¯ ( ω , θ , τ ) | 2 ] = | D | i [ x , y , t ] i [ p , q , r ] × exp { j ( ω ( x p ) + θ ( y q ) + τ ( t r ) ) } f ( i ) d i = | D | i [ x , y , t ] i [ p , q , r ] f ( i ) d i × exp { j ( ω ( x p ) + θ ( y q ) + τ ( t r ) ) } ,
| D | i [ x , y , t ] i [ p , q , r ] f ( i ) d i = E [ I [ x , y , t ] I [ p , q , r ] ] .
E [ I [ x , y , t ] I [ p , q , r ] ] ( E [ I [ x , y , t ] 2 ] E [ I [ p , q , r ] 2 ] ) 1 / 2 .
E [ | S ¯ | 2 ] = | E ¯ S | 2 + I Ω E [ | N ¯ I | 2 ] ,
S = I Ω ( E I + N I ) .
S ¯ = I Ω ( E ¯ I + N ¯ I ) .
| S ¯ | 2 = [ E ¯ I ( E ¯ J ) * + N ¯ I ( N ¯ J ) * + E ¯ I ( N ¯ J ) * + N ¯ I ( E ¯ J ) * ] ,
E [ E ¯ I ( E ¯ J ) * + N ¯ I ( N ¯ J ) * + E ¯ I ( N ¯ J ) * + N ¯ I ( E ¯ J ) * ] = E ¯ I ( E ¯ J ) * ,
E [ N ¯ I ] = E [ ( N ¯ J ) * ] = 0 ¯ .
E [ E ¯ I ( E ¯ I ) * + N ¯ I ( N ¯ I ) * + E ¯ I ( N ¯ I ) * + N ¯ I ( E ¯ I ) * ] = E ¯ I ( E ¯ I ) * + E [ N ¯ I ( N ¯ I ) * ] .
E [ | S ¯ | 2 ] = J Ω K Ω E ¯ J ( E ¯ K ) * + I Ω E [ | N ¯ I | 2 ] = | J Ω E ¯ J | 2 + I Ω E [ | N ¯ I | 2 ] = | E ¯ S | 2 + I Ω E [ | N ¯ I | 2 ] .
I [ x , y , t ] = m = 0 M 1 d m [ x , y ] ( cos ( ρ m ) h m [ t ] sin ( ρ m ) k m [ t ] ) ,
I [ x , y , t ] = m = 0 M 1 n = 0 N 1 d m [ x , y ] p m , n [ t ] cos ( q m , n [ t ] + ρ m ) ,
I [ x , y , t ] = m = 0 M 1 d m [ x , y ] n = 0 N 1 p m , n [ t ] × ( cos ( q m , n [ t ] ) cos ( ρ m ) sin ( q m , n [ t ] ) sin ( ρ m ) ) = m = 0 M 1 d m [ x , y ] ( h m [ t ] cos ( ρ m ) k m [ t ] sin ( ρ m ) )
h m [ t ] = n = 0 N 1 p m , n [ t ] cos ( q m , n [ t ] ) , k m [ t ] = n = 0 N 1 p m , n [ t ] sin ( q m , n [ t ] ) .
H [ x , y , t ] = C m = 0 M 1 n = 0 N 1 d m [ x , y ] g n [ t ] × cos ( 4 π ( 1 + cos ( 2 π ( m M n N ) ) ) + ρ m ) .
p m , n [ t ] = C g n [ t ]
q m , n [ t ] = 4 π ( 1 + cos ( 2 π ( m M n N ) ) ) .
G [ x , y , t ] = C 2 m = 0 M 1 n = 0 N 1 d m [ x , y ] g n [ t ] × ( cos ( 2 π ( α m M β n N ) ) + 1 ) cos ( 2 π γ n N + ρ m ) .
p m , n [ t ] = C 2 g n [ t ] ( cos ( 2 π ( α m M β n N ) ) + 1 )
q m , n [ t ] = 2 π γ n N ,

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