Abstract

The elaborated Reichardt detector (ERD) proposed by van Santen and Sperling [ J. Opt. Soc. Am. A 1, 451 ( 1984)], based on Reichardt’s motion detector [ Z. Naturforsch. Teil B 12, 447 ( 1957)], is an opponent system of two mirror-image subunits. Each subunit receives inputs from two spatiotemporal filters (receptive fields), multiplies the filter outputs, and temporally integrates the product. Subunit outputs are algebraically subtracted to yield ERD output. ERD’s can correctly indicate direction of motion of drifting sine waves of any spatial and temporal frequency. Here we prove that with a careful choice of either temporal or spatial filters, the subunits can themselves become quite similar or equivalent to the whole ERD; with suitably chosen filters, the ERD is equivalent to an elaborated version of a motion detector proposed by Watson and Ahumada [NASA Tech. Memo. 84352 (1983)]; and for every choice of filters, the ERD is fully equivalent to the detector proposed by Adelson and Bergen [ J. Opt. Soc. Am. A2, 284– 299 ( 1985)]. Some equivalences between the motion detection (in x, t) by ERD’s and spatial pattern detection (in x, y) are demonstrated. The responses of the ERD and its variants to drifting sinusoidal gratings, to other sinusoidally modulated stimuli (on–off gratings, counterphase flicker), and to combinations of sinusoids are derived and compared with data. ERD responses to two-frame motion displays are derived, and several new experimental predictions are tested experimentally. It is demonstrated that a system containing ERD’s of various sizes can solve the correspondence problem in two-frame motion of random-bar stimuli and shows the predicted phase dependencies when confronted with displays composed of triple sinusoids combined either in amplitude modulation phase or in quasi-frequency modulation phase. Finally, it is shown that, while the ERD may in some instances give larger responses to nonrigid than to rigid displacements, the subunits (and hence the ERD) are especially well behaved with continuous movement of rigid or smoothly deforming objects.

© 1985 Optical Society of America

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  1. J. P. H. van Santen, G. Sperling, “A new class of models of motion-sensitive units in human vision,” presented at the 16th annual Meeting of the Society for Mathematical Psychology, Boulder, Colorado, 1983.
  2. J. P. H. van Santen, G. Sperling, “A temporal covariance model of motion perception,” Invest. Ophthalmol. Vis. Sci. Suppl. 24, 277 (1983).
  3. J. P. H. van Santen, G. Sperling, “Applications of a Reichardt-type model of two-frame motion,” Invest. Ophthalmol. Vis. Sci. Suppl. 25, 14 (1984).
  4. J. P. H. van Santen, G. Sperling, “A temporal covariance model of motion perception,” J. Opt. Soc. Am. A 1, 451–473 (1984).
    [CrossRef] [PubMed]
  5. A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,”NASA Tech. Memo. 84352 (1983).
  6. E. H. Adelson, J. Bergen, J. Opt. Soc. Am. A 2, 284–299 (1985).
    [CrossRef] [PubMed]
  7. W. Reichardt, “Autokorrelationsauswertung als Funktionsprinzip des Zentralnervensystems,”Z. Naturforsch. 12b, 447–457 (1957).
  8. J. J. Kulikowski, “Pattern and movement detection in man and rabbit; separation and comparison of occipital potentials,” Vision Res. 18, 183–189 (1978).
    [CrossRef]
  9. J. J. Kulikowski, “Spatial resolution for the detection of pattern and movement (real and apparent),” Vision Res. 18, 237–238 (1978).
    [CrossRef] [PubMed]
  10. I. Murray, F. McCana, J. J. Kulikowski, “Contribution to two movement detecting mechanisms to central and peripheral vision,” Vision Res. 23, 151–159 (1983).
    [CrossRef]
  11. G. G. Furman, “Comparison of models for subtractive and shunting lateral-inhibition in receptor-neuron fields,” Kybernetik 2, 257–274 (1965).
    [CrossRef] [PubMed]
  12. G. Sperling, M. M. Sondhi, “Model for visual luminance discrimination and flicker detection,”J. Opt. Soc. Am. 58, 1133–1145 (1968).
    [CrossRef] [PubMed]
  13. J. Thorson, “Small-signal analysis of a visual reflex in the locust,” II. Frequency dependence,” Kybernetik 3, 53–66 (1966).
    [CrossRef] [PubMed]
  14. V. Torre, T. Poggio, “A synaptic mechanism possibly underlying direction selectivity to motion,” Proc. Royal Soc. London Ser. B 202, 409–416 (1978).
    [CrossRef]
  15. G. Fermi, W. Reichardt, “Optomotorische Reaktionen der Fliege,” Musca Domestica Kybernetik 2, 15–28 (1963).
    [CrossRef]
  16. See Ref. 4, p. 456, Note 17.
  17. D. Gabor, “Theory of communication,”J. Inst. Electr. Eng. 93, 429–457 (1946).
  18. S. Marcelja, “Mathematical description of the responses of simple cortical cells,”J. Opt. Soc. Am. 70, 1297–1300 (1980).
    [CrossRef] [PubMed]
  19. D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
    [CrossRef] [PubMed]
  20. H. R. Wilson, “Spatiotemporal characterization of a transient mechanism in the human visual system,” Vision Res. 20, 443–452 (1980).
    [CrossRef] [PubMed]
  21. P. Burt, G. Sperling, “Time, distance, and feature trade-offs in visual apparent motion,” Psychol. Rev. 88, 171–195 (1981).
    [CrossRef] [PubMed]
  22. A. B. Watson, A. J. Ahumada, “A model of how humans sense image motion,” Invest. Ophthalmol. Vis. Sci. 25, 14 (1984).
  23. S. A. Rajalla, A. N. Riddle, W. E. Snyder, “Application of the one-dimensional Fourier transform for tracking moving objects in noise environments,” Comput. Vision Graphics Image Process. 21, 280–293 (1983).
    [CrossRef]
  24. E. H. Adelson, J. Bergen, “Motion channels based on spatiotemporal energy,” Invest. Ophthalmol. Vis. Sci. Suppl. 25, 14 (A) (1984).
  25. H. S. F. von Helmholtz, Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie derMusik (Dover, New York, 1863); (reprint, 1954).
  26. H. E. Ives, “A visual acuity test object,” Electron. World 55, 939–940 (1910).
  27. S. Anstis, C. W. Tyler, “Induced tilt from checkerboards: edged vs. Fourier components,” Invest. Ophthalmol. Vis. Sci. Suppl. 21, 165 (1980).
  28. G. Sperling, J. P. H. van Santen, “Models of motion perception,” presented at the U.S. Air Force Office of Scientific Research Review of Sponsored Basic Research in Human Information Processing, Sarasota, Florida, 1982.
  29. G. Sperling, “Theories of motion perception,” presented at the 17th Annual Meeting of the Society for Mathematical Psychology, Chicago, Illinois, 1984.
  30. N. Graham, “Spatial-frequency channels in the human visual system: detecting edges without edge detectors,” in Visual Coding and Adaptability, C. S. Harris, ed. (Erlbaum/Halstead, Potomac, Md., 1980).
  31. A. B. Watson, “Detection and recognition of simple spatial forms,” in Physical and Biological Processing of Images, O. J. Braddick, A. C. Sleigh, eds. (Springer-Verlag, Berlin, 1982), pp. 100–114.
  32. H. R. Wilson, “Psychophysical evidence for spatial channels,” in Physical and Biological Processingof Images, O. J. Braddick, A. C. Sleigh, eds. (Springer-Verlag, Berlin, 1982), pp. 88–99.
  33. G. Sperling, “Linear theory and the psychophysics of flicker,” Doc. Ophthalmol. 18, 3–15 (1964).
    [CrossRef] [PubMed]
  34. O. Braddick, “A short-range process in apparent motion,” Vision Res. 14, 519–529 (1974).
    [CrossRef] [PubMed]
  35. J. J. Chang, B. Julesz, “Displacement limits for spatial frequency filtered random-dot cinematograms in apparent motion,” Vision Res. 23, 1379–1385 (1983).
    [CrossRef] [PubMed]
  36. K. Nakayama, G. H. Silverman, “Temporal and spatial properties of the upper displacement limit in random dots,” Vision Res. 24, 293–300 (1984).
    [CrossRef]
  37. K. Nakayama, G. H. Silverman, “Detection and discrimination of sinusoidal grating displacements,” J. Opt. Soc. Am. A 2, 267–274 (1985).
    [CrossRef] [PubMed]
  38. E. Levinson, R. Sekuler, “The independence of channels in human vision selective for direction of movement,”J. Physiol. 250, 347–366 (1975).
    [PubMed]
  39. A. B. Watson, P. G. Thompson, B. J. Murphy, J. Nachmias, “Summation and discrimination of gratings moving in opposite directions,” Vision Res. 20, 341–348 (1980).
    [CrossRef] [PubMed]

1985

1984

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

K. Nakayama, G. H. Silverman, “Temporal and spatial properties of the upper displacement limit in random dots,” Vision Res. 24, 293–300 (1984).
[CrossRef]

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

A. B. Watson, A. J. Ahumada, “A model of how humans sense image motion,” Invest. Ophthalmol. Vis. Sci. 25, 14 (1984).

E. H. Adelson, J. Bergen, “Motion channels based on spatiotemporal energy,” Invest. Ophthalmol. Vis. Sci. Suppl. 25, 14 (A) (1984).

1983

S. A. Rajalla, A. N. Riddle, W. E. Snyder, “Application of the one-dimensional Fourier transform for tracking moving objects in noise environments,” Comput. Vision Graphics Image Process. 21, 280–293 (1983).
[CrossRef]

J. P. H. van Santen, G. Sperling, “A temporal covariance model of motion perception,” Invest. Ophthalmol. Vis. Sci. Suppl. 24, 277 (1983).

I. Murray, F. McCana, J. J. Kulikowski, “Contribution to two movement detecting mechanisms to central and peripheral vision,” Vision Res. 23, 151–159 (1983).
[CrossRef]

J. J. Chang, B. Julesz, “Displacement limits for spatial frequency filtered random-dot cinematograms in apparent motion,” Vision Res. 23, 1379–1385 (1983).
[CrossRef] [PubMed]

1981

D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
[CrossRef] [PubMed]

P. Burt, G. Sperling, “Time, distance, and feature trade-offs in visual apparent motion,” Psychol. Rev. 88, 171–195 (1981).
[CrossRef] [PubMed]

1980

S. Anstis, C. W. Tyler, “Induced tilt from checkerboards: edged vs. Fourier components,” Invest. Ophthalmol. Vis. Sci. Suppl. 21, 165 (1980).

H. R. Wilson, “Spatiotemporal characterization of a transient mechanism in the human visual system,” Vision Res. 20, 443–452 (1980).
[CrossRef] [PubMed]

A. B. Watson, P. G. Thompson, B. J. Murphy, J. Nachmias, “Summation and discrimination of gratings moving in opposite directions,” Vision Res. 20, 341–348 (1980).
[CrossRef] [PubMed]

S. Marcelja, “Mathematical description of the responses of simple cortical cells,”J. Opt. Soc. Am. 70, 1297–1300 (1980).
[CrossRef] [PubMed]

1978

V. Torre, T. Poggio, “A synaptic mechanism possibly underlying direction selectivity to motion,” Proc. Royal Soc. London Ser. B 202, 409–416 (1978).
[CrossRef]

J. J. Kulikowski, “Pattern and movement detection in man and rabbit; separation and comparison of occipital potentials,” Vision Res. 18, 183–189 (1978).
[CrossRef]

J. J. Kulikowski, “Spatial resolution for the detection of pattern and movement (real and apparent),” Vision Res. 18, 237–238 (1978).
[CrossRef] [PubMed]

1975

E. Levinson, R. Sekuler, “The independence of channels in human vision selective for direction of movement,”J. Physiol. 250, 347–366 (1975).
[PubMed]

1974

O. Braddick, “A short-range process in apparent motion,” Vision Res. 14, 519–529 (1974).
[CrossRef] [PubMed]

1968

1966

J. Thorson, “Small-signal analysis of a visual reflex in the locust,” II. Frequency dependence,” Kybernetik 3, 53–66 (1966).
[CrossRef] [PubMed]

1965

G. G. Furman, “Comparison of models for subtractive and shunting lateral-inhibition in receptor-neuron fields,” Kybernetik 2, 257–274 (1965).
[CrossRef] [PubMed]

1964

G. Sperling, “Linear theory and the psychophysics of flicker,” Doc. Ophthalmol. 18, 3–15 (1964).
[CrossRef] [PubMed]

1963

G. Fermi, W. Reichardt, “Optomotorische Reaktionen der Fliege,” Musca Domestica Kybernetik 2, 15–28 (1963).
[CrossRef]

1957

W. Reichardt, “Autokorrelationsauswertung als Funktionsprinzip des Zentralnervensystems,”Z. Naturforsch. 12b, 447–457 (1957).

1946

D. Gabor, “Theory of communication,”J. Inst. Electr. Eng. 93, 429–457 (1946).

1910

H. E. Ives, “A visual acuity test object,” Electron. World 55, 939–940 (1910).

Adelson, E. H.

E. H. Adelson, J. Bergen, J. Opt. Soc. Am. A 2, 284–299 (1985).
[CrossRef] [PubMed]

E. H. Adelson, J. Bergen, “Motion channels based on spatiotemporal energy,” Invest. Ophthalmol. Vis. Sci. Suppl. 25, 14 (A) (1984).

Ahumada, A. J.

A. B. Watson, A. J. Ahumada, “A model of how humans sense image motion,” Invest. Ophthalmol. Vis. Sci. 25, 14 (1984).

A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,”NASA Tech. Memo. 84352 (1983).

Anstis, S.

S. Anstis, C. W. Tyler, “Induced tilt from checkerboards: edged vs. Fourier components,” Invest. Ophthalmol. Vis. Sci. Suppl. 21, 165 (1980).

Bergen, J.

E. H. Adelson, J. Bergen, J. Opt. Soc. Am. A 2, 284–299 (1985).
[CrossRef] [PubMed]

E. H. Adelson, J. Bergen, “Motion channels based on spatiotemporal energy,” Invest. Ophthalmol. Vis. Sci. Suppl. 25, 14 (A) (1984).

Braddick, O.

O. Braddick, “A short-range process in apparent motion,” Vision Res. 14, 519–529 (1974).
[CrossRef] [PubMed]

Burt, P.

P. Burt, G. Sperling, “Time, distance, and feature trade-offs in visual apparent motion,” Psychol. Rev. 88, 171–195 (1981).
[CrossRef] [PubMed]

Chang, J. J.

J. J. Chang, B. Julesz, “Displacement limits for spatial frequency filtered random-dot cinematograms in apparent motion,” Vision Res. 23, 1379–1385 (1983).
[CrossRef] [PubMed]

Fermi, G.

G. Fermi, W. Reichardt, “Optomotorische Reaktionen der Fliege,” Musca Domestica Kybernetik 2, 15–28 (1963).
[CrossRef]

Furman, G. G.

G. G. Furman, “Comparison of models for subtractive and shunting lateral-inhibition in receptor-neuron fields,” Kybernetik 2, 257–274 (1965).
[CrossRef] [PubMed]

Gabor, D.

D. Gabor, “Theory of communication,”J. Inst. Electr. Eng. 93, 429–457 (1946).

Graham, N.

N. Graham, “Spatial-frequency channels in the human visual system: detecting edges without edge detectors,” in Visual Coding and Adaptability, C. S. Harris, ed. (Erlbaum/Halstead, Potomac, Md., 1980).

Ives, H. E.

H. E. Ives, “A visual acuity test object,” Electron. World 55, 939–940 (1910).

Julesz, B.

J. J. Chang, B. Julesz, “Displacement limits for spatial frequency filtered random-dot cinematograms in apparent motion,” Vision Res. 23, 1379–1385 (1983).
[CrossRef] [PubMed]

Kulikowski, J. J.

I. Murray, F. McCana, J. J. Kulikowski, “Contribution to two movement detecting mechanisms to central and peripheral vision,” Vision Res. 23, 151–159 (1983).
[CrossRef]

J. J. Kulikowski, “Spatial resolution for the detection of pattern and movement (real and apparent),” Vision Res. 18, 237–238 (1978).
[CrossRef] [PubMed]

J. J. Kulikowski, “Pattern and movement detection in man and rabbit; separation and comparison of occipital potentials,” Vision Res. 18, 183–189 (1978).
[CrossRef]

Levinson, E.

E. Levinson, R. Sekuler, “The independence of channels in human vision selective for direction of movement,”J. Physiol. 250, 347–366 (1975).
[PubMed]

Marcelja, S.

McCana, F.

I. Murray, F. McCana, J. J. Kulikowski, “Contribution to two movement detecting mechanisms to central and peripheral vision,” Vision Res. 23, 151–159 (1983).
[CrossRef]

Murphy, B. J.

A. B. Watson, P. G. Thompson, B. J. Murphy, J. Nachmias, “Summation and discrimination of gratings moving in opposite directions,” Vision Res. 20, 341–348 (1980).
[CrossRef] [PubMed]

Murray, I.

I. Murray, F. McCana, J. J. Kulikowski, “Contribution to two movement detecting mechanisms to central and peripheral vision,” Vision Res. 23, 151–159 (1983).
[CrossRef]

Nachmias, J.

A. B. Watson, P. G. Thompson, B. J. Murphy, J. Nachmias, “Summation and discrimination of gratings moving in opposite directions,” Vision Res. 20, 341–348 (1980).
[CrossRef] [PubMed]

Nakayama, K.

K. Nakayama, G. H. Silverman, “Detection and discrimination of sinusoidal grating displacements,” J. Opt. Soc. Am. A 2, 267–274 (1985).
[CrossRef] [PubMed]

K. Nakayama, G. H. Silverman, “Temporal and spatial properties of the upper displacement limit in random dots,” Vision Res. 24, 293–300 (1984).
[CrossRef]

Poggio, T.

V. Torre, T. Poggio, “A synaptic mechanism possibly underlying direction selectivity to motion,” Proc. Royal Soc. London Ser. B 202, 409–416 (1978).
[CrossRef]

Pollen, D. A.

D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
[CrossRef] [PubMed]

Rajalla, S. A.

S. A. Rajalla, A. N. Riddle, W. E. Snyder, “Application of the one-dimensional Fourier transform for tracking moving objects in noise environments,” Comput. Vision Graphics Image Process. 21, 280–293 (1983).
[CrossRef]

Reichardt, W.

G. Fermi, W. Reichardt, “Optomotorische Reaktionen der Fliege,” Musca Domestica Kybernetik 2, 15–28 (1963).
[CrossRef]

W. Reichardt, “Autokorrelationsauswertung als Funktionsprinzip des Zentralnervensystems,”Z. Naturforsch. 12b, 447–457 (1957).

Riddle, A. N.

S. A. Rajalla, A. N. Riddle, W. E. Snyder, “Application of the one-dimensional Fourier transform for tracking moving objects in noise environments,” Comput. Vision Graphics Image Process. 21, 280–293 (1983).
[CrossRef]

Ronner, S. F.

D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
[CrossRef] [PubMed]

Sekuler, R.

E. Levinson, R. Sekuler, “The independence of channels in human vision selective for direction of movement,”J. Physiol. 250, 347–366 (1975).
[PubMed]

Silverman, G. H.

K. Nakayama, G. H. Silverman, “Detection and discrimination of sinusoidal grating displacements,” J. Opt. Soc. Am. A 2, 267–274 (1985).
[CrossRef] [PubMed]

K. Nakayama, G. H. Silverman, “Temporal and spatial properties of the upper displacement limit in random dots,” Vision Res. 24, 293–300 (1984).
[CrossRef]

Snyder, W. E.

S. A. Rajalla, A. N. Riddle, W. E. Snyder, “Application of the one-dimensional Fourier transform for tracking moving objects in noise environments,” Comput. Vision Graphics Image Process. 21, 280–293 (1983).
[CrossRef]

Sondhi, M. M.

Sperling, G.

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

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

J. P. H. van Santen, G. Sperling, “A temporal covariance model of motion perception,” Invest. Ophthalmol. Vis. Sci. Suppl. 24, 277 (1983).

P. Burt, G. Sperling, “Time, distance, and feature trade-offs in visual apparent motion,” Psychol. Rev. 88, 171–195 (1981).
[CrossRef] [PubMed]

G. Sperling, M. M. Sondhi, “Model for visual luminance discrimination and flicker detection,”J. Opt. Soc. Am. 58, 1133–1145 (1968).
[CrossRef] [PubMed]

G. Sperling, “Linear theory and the psychophysics of flicker,” Doc. Ophthalmol. 18, 3–15 (1964).
[CrossRef] [PubMed]

J. P. H. van Santen, G. Sperling, “A new class of models of motion-sensitive units in human vision,” presented at the 16th annual Meeting of the Society for Mathematical Psychology, Boulder, Colorado, 1983.

G. Sperling, “Theories of motion perception,” presented at the 17th Annual Meeting of the Society for Mathematical Psychology, Chicago, Illinois, 1984.

G. Sperling, J. P. H. van Santen, “Models of motion perception,” presented at the U.S. Air Force Office of Scientific Research Review of Sponsored Basic Research in Human Information Processing, Sarasota, Florida, 1982.

Thompson, P. G.

A. B. Watson, P. G. Thompson, B. J. Murphy, J. Nachmias, “Summation and discrimination of gratings moving in opposite directions,” Vision Res. 20, 341–348 (1980).
[CrossRef] [PubMed]

Thorson, J.

J. Thorson, “Small-signal analysis of a visual reflex in the locust,” II. Frequency dependence,” Kybernetik 3, 53–66 (1966).
[CrossRef] [PubMed]

Torre, V.

V. Torre, T. Poggio, “A synaptic mechanism possibly underlying direction selectivity to motion,” Proc. Royal Soc. London Ser. B 202, 409–416 (1978).
[CrossRef]

Tyler, C. W.

S. Anstis, C. W. Tyler, “Induced tilt from checkerboards: edged vs. Fourier components,” Invest. Ophthalmol. Vis. Sci. Suppl. 21, 165 (1980).

van Santen, J. P. H.

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

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

J. P. H. van Santen, G. Sperling, “A temporal covariance model of motion perception,” Invest. Ophthalmol. Vis. Sci. Suppl. 24, 277 (1983).

G. Sperling, J. P. H. van Santen, “Models of motion perception,” presented at the U.S. Air Force Office of Scientific Research Review of Sponsored Basic Research in Human Information Processing, Sarasota, Florida, 1982.

J. P. H. van Santen, G. Sperling, “A new class of models of motion-sensitive units in human vision,” presented at the 16th annual Meeting of the Society for Mathematical Psychology, Boulder, Colorado, 1983.

von Helmholtz, H. S. F.

H. S. F. von Helmholtz, Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie derMusik (Dover, New York, 1863); (reprint, 1954).

Watson, A. B.

A. B. Watson, A. J. Ahumada, “A model of how humans sense image motion,” Invest. Ophthalmol. Vis. Sci. 25, 14 (1984).

A. B. Watson, P. G. Thompson, B. J. Murphy, J. Nachmias, “Summation and discrimination of gratings moving in opposite directions,” Vision Res. 20, 341–348 (1980).
[CrossRef] [PubMed]

A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,”NASA Tech. Memo. 84352 (1983).

A. B. Watson, “Detection and recognition of simple spatial forms,” in Physical and Biological Processing of Images, O. J. Braddick, A. C. Sleigh, eds. (Springer-Verlag, Berlin, 1982), pp. 100–114.

Wilson, H. R.

H. R. Wilson, “Spatiotemporal characterization of a transient mechanism in the human visual system,” Vision Res. 20, 443–452 (1980).
[CrossRef] [PubMed]

H. R. Wilson, “Psychophysical evidence for spatial channels,” in Physical and Biological Processingof Images, O. J. Braddick, A. C. Sleigh, eds. (Springer-Verlag, Berlin, 1982), pp. 88–99.

Comput. Vision Graphics Image Process.

S. A. Rajalla, A. N. Riddle, W. E. Snyder, “Application of the one-dimensional Fourier transform for tracking moving objects in noise environments,” Comput. Vision Graphics Image Process. 21, 280–293 (1983).
[CrossRef]

Doc. Ophthalmol.

G. Sperling, “Linear theory and the psychophysics of flicker,” Doc. Ophthalmol. 18, 3–15 (1964).
[CrossRef] [PubMed]

Electron. World

H. E. Ives, “A visual acuity test object,” Electron. World 55, 939–940 (1910).

Invest. Ophthalmol. Vis. Sci.

A. B. Watson, A. J. Ahumada, “A model of how humans sense image motion,” Invest. Ophthalmol. Vis. Sci. 25, 14 (1984).

Invest. Ophthalmol. Vis. Sci. Suppl.

S. Anstis, C. W. Tyler, “Induced tilt from checkerboards: edged vs. Fourier components,” Invest. Ophthalmol. Vis. Sci. Suppl. 21, 165 (1980).

E. H. Adelson, J. Bergen, “Motion channels based on spatiotemporal energy,” Invest. Ophthalmol. Vis. Sci. Suppl. 25, 14 (A) (1984).

J. P. H. van Santen, G. Sperling, “A temporal covariance model of motion perception,” Invest. Ophthalmol. Vis. Sci. Suppl. 24, 277 (1983).

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

J. Inst. Electr. Eng.

D. Gabor, “Theory of communication,”J. Inst. Electr. Eng. 93, 429–457 (1946).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Physiol.

E. Levinson, R. Sekuler, “The independence of channels in human vision selective for direction of movement,”J. Physiol. 250, 347–366 (1975).
[PubMed]

Kybernetik

G. G. Furman, “Comparison of models for subtractive and shunting lateral-inhibition in receptor-neuron fields,” Kybernetik 2, 257–274 (1965).
[CrossRef] [PubMed]

J. Thorson, “Small-signal analysis of a visual reflex in the locust,” II. Frequency dependence,” Kybernetik 3, 53–66 (1966).
[CrossRef] [PubMed]

Musca Domestica Kybernetik

G. Fermi, W. Reichardt, “Optomotorische Reaktionen der Fliege,” Musca Domestica Kybernetik 2, 15–28 (1963).
[CrossRef]

Proc. Royal Soc. London Ser. B

V. Torre, T. Poggio, “A synaptic mechanism possibly underlying direction selectivity to motion,” Proc. Royal Soc. London Ser. B 202, 409–416 (1978).
[CrossRef]

Psychol. Rev.

P. Burt, G. Sperling, “Time, distance, and feature trade-offs in visual apparent motion,” Psychol. Rev. 88, 171–195 (1981).
[CrossRef] [PubMed]

Science

D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
[CrossRef] [PubMed]

Vision Res.

H. R. Wilson, “Spatiotemporal characterization of a transient mechanism in the human visual system,” Vision Res. 20, 443–452 (1980).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

J. J. Chang, B. Julesz, “Displacement limits for spatial frequency filtered random-dot cinematograms in apparent motion,” Vision Res. 23, 1379–1385 (1983).
[CrossRef] [PubMed]

K. Nakayama, G. H. Silverman, “Temporal and spatial properties of the upper displacement limit in random dots,” Vision Res. 24, 293–300 (1984).
[CrossRef]

J. J. Kulikowski, “Pattern and movement detection in man and rabbit; separation and comparison of occipital potentials,” Vision Res. 18, 183–189 (1978).
[CrossRef]

J. J. Kulikowski, “Spatial resolution for the detection of pattern and movement (real and apparent),” Vision Res. 18, 237–238 (1978).
[CrossRef] [PubMed]

I. Murray, F. McCana, J. J. Kulikowski, “Contribution to two movement detecting mechanisms to central and peripheral vision,” Vision Res. 23, 151–159 (1983).
[CrossRef]

A. B. Watson, P. G. Thompson, B. J. Murphy, J. Nachmias, “Summation and discrimination of gratings moving in opposite directions,” Vision Res. 20, 341–348 (1980).
[CrossRef] [PubMed]

Z. Naturforsch.

W. Reichardt, “Autokorrelationsauswertung als Funktionsprinzip des Zentralnervensystems,”Z. Naturforsch. 12b, 447–457 (1957).

Other

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J. P. H. van Santen, G. Sperling, “A new class of models of motion-sensitive units in human vision,” presented at the 16th annual Meeting of the Society for Mathematical Psychology, Boulder, Colorado, 1983.

G. Sperling, J. P. H. van Santen, “Models of motion perception,” presented at the U.S. Air Force Office of Scientific Research Review of Sponsored Basic Research in Human Information Processing, Sarasota, Florida, 1982.

G. Sperling, “Theories of motion perception,” presented at the 17th Annual Meeting of the Society for Mathematical Psychology, Chicago, Illinois, 1984.

N. Graham, “Spatial-frequency channels in the human visual system: detecting edges without edge detectors,” in Visual Coding and Adaptability, C. S. Harris, ed. (Erlbaum/Halstead, Potomac, Md., 1980).

A. B. Watson, “Detection and recognition of simple spatial forms,” in Physical and Biological Processing of Images, O. J. Braddick, A. C. Sleigh, eds. (Springer-Verlag, Berlin, 1982), pp. 100–114.

H. R. Wilson, “Psychophysical evidence for spatial channels,” in Physical and Biological Processingof Images, O. J. Braddick, A. C. Sleigh, eds. (Springer-Verlag, Berlin, 1982), pp. 88–99.

See Ref. 4, p. 456, Note 17.

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Figures (14)

Fig. 1
Fig. 1

(a) The ERD. The input is a luminance pattern with contrast c(x, t); it is sampled by linear spatial filters (receptive fields, SF’s) with spatial responses rleft and rright centered at locations xleft and xright; yi,H (H = left, right) represents the signal at the various stages i for the left and right subunits. TF indicates a linear, time-invariant filter with Fourier transform D(ω), × indicates a multiplication unit, TA indicates a temporal integration operation, and − indicates a unit that subtracts its left from its right input. (b) The right subunit of an ERD. (c) The ERD of (a) with three pairs of additional temporal filters (TF1, TF2, TF3) that leave its operation essentially unchanged.

Fig. 2
Fig. 2

Watson and Ahumada’s5 motion subunit and the elaborated W-A detector. (a) Their original linear subunit. SF, TF1 indicate separable spatial and temporal input filters, TF indicates linear, temporal delay filters, DELAY indicates an absolute delay inserted to simplify the description of the (noncausal) linear filters, +/− indicates that the operation is addition or subtraction according to whether the subunit signals motion to the right or left, respectively. (b) An elaborated W-A detector that includes squaring, temporal filtering TA and the subtraction of outputs of two mirror-image W-A subunits and is equivalent to an ERD.

Fig. 3
Fig. 3

Adelson and Bergen’s24 motion detector. Notation as in Figs. 1 and 2. We have provided the temporal filters TA. The A-B detector is equivalent to an ERD [Fig. 1(c)].

Fig. 4
Fig. 4

(a) Artist’s rendition of a single frame of a one-dimensional drifting sinusoidal grating. The various shadings represent stimulus intensities. The arrow indicates the direction of movement. (b) Two-dimensional (x, t) representation of (a). The horizontal dimension x represents the instantaneous spatial luminance pattern. The vertical dimension represents time. The slope inversely indicates the speed of movement. (b) Also can be interpreted as a static, diagonally oriented grating in two dimensions, that is, a grating pattern in (x, y) space. (c) A linear, u, v separable, receptive field (approximation to a simple cell in cortical area 17 that receives (b) as an input (spatial interpretation). The + and − symbols are a conventional notation used to indicate areas in which the response to a point of light is positive and negative, respectively. The receptive field is oriented along its primary axes u, v, and the local responsivity of the field (its impulse response) is indicated in the cross-sectional graphs, u, v. (d) Transformation of (b) via (c) onto a receptor field. The small square dxdy indicates the physical area occupied by a receptor located at x, y (spatial interpretation). In the motion interpretation, the vertical slit represents the physical domain of a motion receptor, that is, a motion receptor at location x responds to inputs from a times t.

Fig. 5
Fig. 5

Examples of two-dimensional functions considered in this paper and their Fourier transforms. For the functions c(x, t), the horizontal dimension is x, the vertical is t, and the shading indicates the value of c(x, t). For the Fourier transforms C(f, ω), the horizontal dimension is spatial frequency f; the vertical dimension is temporal frequency ω; the shading indicates the magnitude of the complex number C(f, ω). The one-dimensional motion interpretation (x, t) rather than the two-dimensional space (x, y) interpretation is used in the following descriptions. Functions (a) to (e) are periodic; one period of an infinite two-dimensional mosaic is shown. (a) Right-drifting sinusoid. (b) Right-drifting sinusoid with double the spatial frequency and double the temporal frequency of (a) and hence the same velocity. (c) Spatially uniform, stationary flicker. (d) Left-moving sinusoid. (e) A sampled (stroboscopic) motion display derived from (d). The motion path is ambiguous; three possible paths (1, 2, 3) and their transforms are shown. The contrast functions (f–j) are not periodic in x but are spatially confined to the x interval shown; functions (h–j) are also temporally confined. (f) Continuously present, stationary single line. (g) Continuously present, stationary random bar pattern. (h) A briefly flashed sinusoidal grating pattern. (i) A two-flash presentation of a random bar pattern moved to the right. (j) A two-flash presentation of a sinusoidal grating pattern moved to the right.

Fig. 6
Fig. 6

Frequency responses of an ERD. (a) Impulse response function of the temporal delay filter (TF, Fig. 1), a first-order low-pass filter with time constant of 1 Hz. (b) Spatial filters (left and right receptive fields) of the ERD. The abscissa indicates the spatial location of a line stimulus input; the horizontal line indicates zero response; the ordinate indicates output amplitude in response to this stimulus. The one-dimensional receptive fields are even Gabor functions, separated by one-quarter cycle of the optimal spatial frequency for the individual receptive fields [Eq. (24)]. (c) Response of this ERD to drifting sinusoidal gratings as a function of their spatial and temporal frequency. Two contour lines are shown, representing the 50 and 1% contours of the absolute value of the response relative to the maximum response. The locations where the actual maxima occur are indicated by the centers of the + and − signs. These signs also indicate the sign of ERD output within each quadrant.

Fig. 7
Fig. 7

The range of responses to combinations of drifting sinusoids of: the GFA (with arbitrary combination rules), the ERD, and two ERD subunits (V1 = π/2 temporal phase-delay subunit; V2 = π/2 spatial phase-shift subunit).

Fig. 8
Fig. 8

Fourier representations of counterphase and on–off gratings. (a) Freeze-frame x, y representation of a grating. (b) Counterphase grating; temporal modulation of luminance (ordinate) of adjacent bars around a mean luminance L0 as a function of time (abscissa). Ordinate: luminance; abscissa, time. (c) On–off modulated grating. (d) Fourier transform of counterphase and (e) on–off modulated grating. Note that these two stimuli differ only by the presence of a static sinusoidal grating shown in (e).

Fig. 9
Fig. 9

The response to two-frame random bar motion of ERD’s with the same spatial filters as in Fig. 6. A representative display (frames 1, 2) is shown at the top of the figure. Average distance between bars is 2.5 min of visual angle (mva). The between-frame displacement is 6 mva rightward. ERD responses are given as a function of their x location; up indicates rightward response (R), and down indicates leftward response (L). Zero-response reference lines are drawn for each detector size to highlight the sign—the indicated direction—of detector outputs. Incorrect direction responses (leftward) are indicated by dark shading. Different lines represent different sizes of proportionally scaled ERD’s; the label indicates their size in terms of the distance Δx between their receptive field centers.

Fig. 10
Fig. 10

Two-frame two-component sinusoidal stimuli in rigid motion. The two frames are shown at top; they consist of superimposed sine waves whose spatial frequencies have a ratio of 3:1. The displacement is π/4 of the lower frequency, resulting in a rigid displacement of the entire pattern. Responses as a function of spatial location x and ERD size Δx (left-center to right-center distance of scaled receptive fields) are shown for ERD’s with the same parameters as in Figs. 6 and 9. Despite the rigid displacement, small ERD’s (Δx ≤ 4 mva) indicate leftward motion, and large ERD’s (Δx ≥ 6 mva) indicate rightward motion, resulting in an ambiguous overall response.

Fig. 11
Fig. 11

Two-frame nonrigid motion. Same as Fig. 10, except that sine-wave components are each displaced by one quarter of their respective cycles. In this nonrigid displacement, the luminance pattern itself changes from a peaks-subtract to a peaks-add pattern. Nevertheless, ERD’s of all sizes correctly indicate rightward motion.

Fig. 12
Fig. 12

Two-frame, three-component sinusoids in an amplitude-modulated grating displaced rigidly in the right. Parameters as in Figs. 6 and 9; ERD’s of all sizes and at all locations have nonnegative responses, indicating unambiguous rightward motion.

Fig. 13
Fig. 13

Two-frame, three-component sinusoids in a quasi-frequency-modulated grating displaced rigidly to the right. Although this grating contains exactly the same spatial sine-wave components with the same respective amplitudes as in Fig. 12, ERD responses are quite different. Except for the largest size (Δx = 16 mva), all detectors have outputs of diferent signs depending on location, indicating an ambiguous overall response.

Fig. 14
Fig. 14

Data from a two-frame direction-discrimination experiment with sinusoidal grating stimuli. The coordinates represent contrast thresholds for 71% correct direction discrimination with frame-to-frame displacements of 48 and 132 deg. The straight line represents exactly equal thresholds. Data for four subjects are shown; each point represents 2 × 500 trials in an interleaved-staircases procedure.

Tables (2)

Tables Icon

Table 1 Restrictive Auxiliary Assumptions Used in Derivations of Principal Results.a

Tables Icon

Table 2 Response of the Elaborated Reichardt Detector to Combinations of Drifting Sinusoids as a Function of Their Differences in Spatial and/or Temporal Frequencya

Equations (46)

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C ( f , ω ) = c ( x , t ) exp [ - 2 π i ( f x + ω t ) ] d x d t .
c ( x , t ) = C ( f , ω ) exp [ 2 π i ( f x + ω t ) ] d ω d f .
y H , 0 ( t ) = r H ( x ) c ( x , t ) d x = C ( f , ω ) exp ( 2 π i ω t ) R * H ( f ) d ω d f ,
H H , 0 ( ω ) = C ( f , ω ) R * H ( f ) d f .
y H , 1 ( t ) = D ( ω ) Y H , 0 ( ω ) exp ( 2 π i ω t ) d ω .
y H , 2 ( y ) = [ Y H , 0 ( ω ) exp ( 2 π i ω t ) d ω ] × [ D ( ν ) Y H , 0 ( ν ) exp ( 2 π i ν t ) d ν ] .
T I t ( g 1 , g 2 ) = T 1 t g 1 ( τ ) g 2 ( τ ) d τ ( t T 1 ) = - g 1 ( τ ) g 2 ( τ ) d τ ( t T 2 ) .
T I t ( g 1 , g 2 ) = G 1 ( ω ) G * 2 ( ω ) d ω .
y H , 3 = Y * H , 0 ( ω ) Y H , 0 ( ω ) D ( ω ) d ω ,
y 4 = [ Y * r i g h t , 0 ( ω ) Y , 0 ( ω ) - Y * l e f t , 0 ( ω ) Y r , 0 ( ω ) ] D ( ω ) d ω .
y 4 ( ω 0 ) = [ Y r , 0 * ( ω 0 ) Y l , 0 ( ω 0 ) - Y l , 0 * ( ω 0 ) Y r , 0 ( ω 0 ) ] D ( ω 0 ) .
y H , 3 = - i Y H , 0 * ( ω ) Y H , 0 ( ω ) D ( ω ) d ω .
y 4 = 2 y right , 3 = - 2 y left , 3 .
y left , 3 = - i C * ( f , ω ) × sign ( f ) C ( g , ω ) R right ( f ) R right * ( g ) D ( ω ) d ω f d g
y right , 3 = + i C * ( g , ω ) × sign ( f ) C ( f , ω ) R right ( g ) R right * ( f ) D ( ω ) d ω d f d g .
y H , 2 ( t ) = y H , 0 ( t ) y H , 1 ( t ) ;
y H , 2 ( t ) = [ y H , 0 ( t ) + y H , 1 ( t ) ] 2 + [ y H , 0 ( t ) - y H , 1 ( t ) ] 2 .
y H , 2 ( t ) = 2 [ y H , 2 ( t ) - y H , 2 ( t ) ] + k ( t ) ,
k ( t ) = y right , 0 2 ( t ) + y right , 1 2 ( t ) + y left , 0 2 ( t ) + y left , 1 2 ( t ) .
y H , 3 ( t ) = 2 [ y H , 3 ( t ) - y H , 3 ( t ) ] + K ( t ) = 2 y 4 ( t ) + K ( t ) ,
y 4 ( t ) = y right , 3 ( t ) - y left , 3 ( t ) . = 2 y 4 ( t )
c ( x , t ) = L 0 + m cos ( 2 π f 0 x + 2 π ω 0 d t ) .
y H , 3 = m 2 D ( ω 0 ) R H ( f 0 ) R H ( f 0 ) × cos [ τ ω 0 - d ( ρ H , f 0 - ρ H , f 0 ) ] .
y 4 = m 2 D ( ω 0 ) R left ( f 0 ) R right ( f 0 ) d × sin ( τ ω 0 ) sin ( ρ right , f 0 - ρ left , f 0 ) .
D ( ω 0 ) d sin ( τ ω 0 ) 0
R left ( f 0 ) R rigth ( f 0 ) sin ( ρ right , f 0 - ρ left , f 0 ) 0.
r H ( x ) = W ( x - x H ) cos [ ( x - x H ) f 0 ]             H = left , right .
c ( x , t ) = L 0 + m cos ( 2 π ω 0 t ) cos ( 2 π f 0 x ) .
C ( f , ω ) = m , f = ± f 0         and ω = ± ω 0 = 0 , elsewhere .
y H , 3 = m 2 D ( ω 0 ) R left ( f 0 ) × cos ( τ ω 0 ) cos ( ρ left , f 0 ) cos ( ρ right , f 0 ) .
c ( x , t ) = L 0 + m [ cos ( 2 π ω 0 t ) + 1 ] cos ( 2 π d f 0 x ) .
c ( x , t ) = j = 1 N c j ( x ) m j ( t ) .
C ( f , ω ) = j = 1 N C j ( f ) M j ( ω ) .
r H , j , c ( x ) = c j ( x ) r H ( x - x ) d x
= C j ( f ) R H * ( f ) d f .
y 4 ( x ) = 1 j < k N [ R right , j ( x ) R left , k ( x ) - R right , k ( x ) R left , j ( x ) ] × { D ( ω ) [ M j * ( ω ) M k ( ω ) - M k * ( ω ) M j ( ω ) ] d ω } .
y 4 ( x ) = Δ 12 ( x ) D ( ω ) [ M 1 * ( ω ) M 2 ( ω ) - M 2 * ( ω ) M 1 ( ω ) ] d ω .
c ( x , t ) = [ L 0 + A 1 sin ( 2 π f x - φ 1 ] m 1 ( t ) + [ L 0 + A 2 sin ( 2 π f x - φ 2 ) ] m 2 ( t ) .
Δ 12 ( x ) = A 1 A 2 sin ( φ 1 - φ 2 ) = A 1 A 2 sin ( φ ) .
c ( x , t ) = γ ( x - v t ) .
C ( f , ω ) = Γ ( f ) δ ( f + ω / v ) ,
y left , 3 = - i Γ * ( f ) sin ( f ) Γ ( f ) R right ( f ) R * right ( f ) D ( - f v ) d f = - y right , 3 .
C ( f , ω ) = Γ ( f ) δ [ f + H ( ω ) ] ,
c ( x , t ) = [ L 0 + A 1 sin ( 2 π f x - φ 1 ) ] m 1 ( t ) + [ L 0 + A 2 sin ( 2 π f x - φ 2 ) ] m 2 ( t ) ,
Δ 12 ( x ) = α A 2 sin ( φ ) + β L 0 U A ( 1 - cos φ ) 1 / 2 cos ( x - γ ) .
Δ 12 = α A 2 sin φ + β L 0 U A ( 1 - cos φ ) 1 / 2 .

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