Abstract

We propose a model of how humans sense the velocity of moving images. The model exploits constraints provided by human psychophysics, notably that motion-sensing elements appear tuned for two-dimensional spatial frequency, and by the frequency spectrum of a moving image, namely, that its support lies in the plane in which the temporal frequency equals the dot product of the spatial frequency and the image velocity. The first stage of the model is a set of spatial-frequency-tuned, direction-selective linear sensors. The temporal frequency of the response of each sensor is shown to encode the component of the image velocity in the sensor direction. At the second stage, these components are resolved in order to measure the velocity of image motion at each of a number of spatial locations and spatial frequencies. The model has been applied to several illustrative examples, including apparent motion, coherent gratings, and natural image sequences. The model agrees qualitatively with human perception.

© 1985 Optical Society of America

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  32. E. Levinson, R. Sekuler, “Inhibition and disinhibition of direction-specific mechanisms in human vision,” Nature 254, 692–694 (1975).
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  35. F. W. Campbell, J. Nachmias, J. Jukes, “Spatial-frequency discrimination in human vision,”J. Opt. Soc. Am. 60, 555–559 (1970).
    [CrossRef] [PubMed]
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  38. The orientation of a 2D frequency component is the angle of a normal to the wavefront.
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  43. A. B. Watson, “Derivation of the impulse response: comments on the method of Roufs and Blommaert,” Vision Res. 22, 1335–1337 (1982).
    [CrossRef] [PubMed]
  44. S. Marcelja, “Mathematical description of the responses of simple cortical cells,”J. Opt. Soc. Am. 70, 1297–1300 (1980).
    [CrossRef] [PubMed]
  45. B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybern. 43, 97–108 (1982).
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    [CrossRef]
  48. P. J. Burt, E. H. Adelson, “The Laplacian pyramid as a compact image code,”IEEE Trans. Commun. COM-31, 532–540 (1983).
    [CrossRef]
  49. J. L. Crowley, R. M. Stern, “Fast computation of the difference of low-pass transform,”IEEE Trans. Pattern Anal. Mach. Intelligence PAMI-6, 212–222 (1984).
    [CrossRef]
  50. G. Sperling, J. P. H. van Santen, “Temporal covariance model of human motion perception,” J. Opt. Soc. Am. A 1, 451–473 (1984).
    [CrossRef] [PubMed]
  51. E. H. Adelson, J. R. Bergen, “Motion channels based on spatiotemporal energy,” Invest. Ophthalmol. Vis. Sci. Suppl. 25, 14 (A) (1984).

1984 (5)

P. Thompson, “The coding of velocity of movement in the human visual system,” Vision Res. 24, 41–45 (1984).
[CrossRef] [PubMed]

C. F. Stromeyer, R. E. Kronauer, J. C. Madsen, S. A. Klein, “Opponent-movement mechanisms in human vision,” J. Opt. Soc. Am. A 1, 876–884 (1984).
[CrossRef] [PubMed]

J. L. Crowley, R. M. Stern, “Fast computation of the difference of low-pass transform,”IEEE Trans. Pattern Anal. Mach. Intelligence PAMI-6, 212–222 (1984).
[CrossRef]

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

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

1983 (3)

P. J. Burt, E. H. Adelson, “The Laplacian pyramid as a compact image code,”IEEE Trans. Commun. COM-31, 532–540 (1983).
[CrossRef]

P. Thompson, “Discrimination of moving gratings at and above detection threshold,” Vision Res. 23, 1533–1538 (1983).
[CrossRef] [PubMed]

A. B. Watson, A. J. Ahumada, “A linear motion sensor,” Perception 12, A17 (1983).

1982 (6)

E. H. Adelson, J. A. Movshon, “Phenomenal coherence of moving visual patterns,” Nature 300, 523–525 (1982).
[CrossRef] [PubMed]

A. B. Watson, “Summation of grating patches indicates many types of detector at one retinal location,” Vision Res. 22, 17–25 (1982).
[CrossRef] [PubMed]

A. B. Watson, “Derivation of the impulse response: comments on the method of Roufs and Blommaert,” Vision Res. 22, 1335–1337 (1982).
[CrossRef] [PubMed]

D. C. Burr, J. Ross, “Contrast sensitivity at high velocities,” Vision Res. 22, 479–484 (1982).
[CrossRef] [PubMed]

A. B. Watson, A. J. Ahumada, “Sampling, filtering, and apparent motion,” Perception 11, A15 (1982).

B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybern. 43, 97–108 (1982).
[CrossRef] [PubMed]

1981 (5)

M. Fable, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. London Ser. B 213, 451–477 (1981).
[CrossRef]

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vision Res. 21, 1115–1122 (1981).
[CrossRef]

D. Marr, S. Ullman, “Directional selectivity and its use in early visual processing,” Proc. R. Soc. London Ser. B 211, 151–180 (1981).
[CrossRef]

S. P. McKee, “A local mechanism for differential velocity detection,” Vision Res. 21, 491–500 (1981).
[CrossRef] [PubMed]

R. J. W. Mansfield, J. Nachmias, “Perceived direction of motion under retinal image stabilization,” Vision Res. 21, 1423–1425 (1981).
[CrossRef] [PubMed]

1980 (4)

A. B. Watson, P. G. Thompson, B. J. Murphy, J. Nachmias, “Summation and discrimination of gratings moving in opposite directions,” Vision Res. 20, 341–347 (1980).
[CrossRef] [PubMed]

H. C. Longuet-Higgins, K. Prazdny, “The interpretation of moving retinal images,” Proc. R. Soc. London Ser. B 208, 385–387 (1980).
[CrossRef]

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

J. G. Moik, “Digital processing of remotely sensed images,”NASA Doc. SP-431 (1980).

1979 (2)

1978 (1)

1976 (2)

E. Levinson, R. Sekuler, “Adaptation alters perceived direction of motion,” Vision Res. 16, 779–781 (1976).
[CrossRef] [PubMed]

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

1975 (3)

E. Levinson, R. Sekuler, “Inhibition and disinhibition of direction-specific mechanisms in human vision,” Nature 254, 692–694 (1975).
[CrossRef] [PubMed]

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

S. Tanimoto, T. Pavlidis, “A hierarchical data structure for picture processing,” Comput. Graphics Image Process. 4, 104–119 (1975).
[CrossRef]

1974 (1)

A. Pantle, “Motion aftereffect magnitude as a measure of the spatiotemporal response properties of direction-sensitive analyzers,” Vision Res. 14, 1229–1236 (1974).
[CrossRef] [PubMed]

1973 (1)

D. J. Tolhurst, “Separate channels for the analysis of the shape and the movement of a moving visual stimulus,”J. Physiol. London 231, 385–402 (1973).

1972 (2)

N. Graham, “Spatial-frequency channels in the human visual system: effects of luminance and pattern drift rate,” Vision Res. 12, 53–68 (1972).
[CrossRef] [PubMed]

D. H. Kelly, “Adaptation effects on spatio-temporal sine-wave thresholds,” Vision Res. 12, 89–101 (1972).
[CrossRef] [PubMed]

1971 (1)

N. Graham, J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[CrossRef] [PubMed]

1970 (1)

1968 (1)

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,”J. Phys. London 197, 551–566 (1968).

1967 (1)

1966 (1)

1964 (1)

M. G. F. Fourtes, A. L. Hodgkin, “Changes in the time scale and sensitivity in the omatidia of limulus,”J. Physiol. London 172, 239–263 (1964).

Adelson, E. H.

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

P. J. Burt, E. H. Adelson, “The Laplacian pyramid as a compact image code,”IEEE Trans. Commun. COM-31, 532–540 (1983).
[CrossRef]

E. H. Adelson, J. A. Movshon, “Phenomenal coherence of moving visual patterns,” Nature 300, 523–525 (1982).
[CrossRef] [PubMed]

Ahumada, A. J.

A. B. Watson, A. J. Ahumada, “A linear motion sensor,” Perception 12, A17 (1983).

A. B. Watson, A. J. Ahumada, “Sampling, filtering, and apparent motion,” Perception 11, A15 (1982).

A. B. Watson, A. J. Ahumada, J. Farrell, “The window of visibility: a psychophysical theory of fidelity in time-sampled visual motion displays,”NASA Tech. Pap. 2211, (1983).

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

Barlow, H. B.

B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybern. 43, 97–108 (1982).
[CrossRef] [PubMed]

Bergen, J. R.

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

Bouman, M. A.

Burr, D. C.

D. C. Burr, J. Ross, “Contrast sensitivity at high velocities,” Vision Res. 22, 479–484 (1982).
[CrossRef] [PubMed]

Burt, P. J.

P. J. Burt, E. H. Adelson, “The Laplacian pyramid as a compact image code,”IEEE Trans. Commun. COM-31, 532–540 (1983).
[CrossRef]

Campbell, F. W.

F. W. Campbell, J. Nachmias, J. Jukes, “Spatial-frequency discrimination in human vision,”J. Opt. Soc. Am. 60, 555–559 (1970).
[CrossRef] [PubMed]

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,”J. Phys. London 197, 551–566 (1968).

Crick, F. H. C.

F. H. C. Crick, D. C. Marr, T. Poggio, “An information processing approach to understanding the visual cortex,” in The Organization of the Cerebral Cortex, S. G. Dennis, ed. (MIT U. Press, Cambridge, Mass., 1981).

Crowley, J. L.

J. L. Crowley, R. M. Stern, “Fast computation of the difference of low-pass transform,”IEEE Trans. Pattern Anal. Mach. Intelligence PAMI-6, 212–222 (1984).
[CrossRef]

Fable, M.

M. Fable, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. London Ser. B 213, 451–477 (1981).
[CrossRef]

Farrell, J.

A. B. Watson, A. J. Ahumada, J. Farrell, “The window of visibility: a psychophysical theory of fidelity in time-sampled visual motion displays,”NASA Tech. Pap. 2211, (1983).

Fourtes, M. G. F.

M. G. F. Fourtes, A. L. Hodgkin, “Changes in the time scale and sensitivity in the omatidia of limulus,”J. Physiol. London 172, 239–263 (1964).

Gibson, J. J.

J. J. Gibson, The Perception of the Visual World (Houghton Mifflin, Boston, Mass., 1950).

Graham, N.

N. Graham, “Spatial-frequency channels in the human visual system: effects of luminance and pattern drift rate,” Vision Res. 12, 53–68 (1972).
[CrossRef] [PubMed]

N. Graham, J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[CrossRef] [PubMed]

Hildreth, E. C.

E. C. Hildreth, The Measurement of Visual Motion (MIT U. Press, Cambridge, Mass., 1983).

Hodgkin, A. L.

M. G. F. Fourtes, A. L. Hodgkin, “Changes in the time scale and sensitivity in the omatidia of limulus,”J. Physiol. London 172, 239–263 (1964).

Jukes, J.

Kelly, D. H.

D. H. Kelly, “Motion and vision. II. Stabilized spatio-temporal theshold surface,”J. Opt. Soc. Am. 69, 1340–1349 (1979).
[CrossRef] [PubMed]

D. H. Kelly, “Adaptation effects on spatio-temporal sine-wave thresholds,” Vision Res. 12, 89–101 (1972).
[CrossRef] [PubMed]

Klein, S.

Klein, S. A.

Koenderink, J. J.

Kolers, P. A.

P. A. Kolers, Aspects of Apparent Motion (Pergamon, New York, 1972).

Kronauer, R. E.

Levinson, E.

E. Levinson, R. Sekuler, “Adaptation alters perceived direction of motion,” Vision Res. 16, 779–781 (1976).
[CrossRef] [PubMed]

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

E. Levinson, R. Sekuler, “Inhibition and disinhibition of direction-specific mechanisms in human vision,” Nature 254, 692–694 (1975).
[CrossRef] [PubMed]

Longuet-Higgins, H. C.

H. C. Longuet-Higgins, K. Prazdny, “The interpretation of moving retinal images,” Proc. R. Soc. London Ser. B 208, 385–387 (1980).
[CrossRef]

Madsen, J. C.

Mansfield, R. J. W.

R. J. W. Mansfield, J. Nachmias, “Perceived direction of motion under retinal image stabilization,” Vision Res. 21, 1423–1425 (1981).
[CrossRef] [PubMed]

Marcelja, S.

Marr, D.

D. Marr, S. Ullman, “Directional selectivity and its use in early visual processing,” Proc. R. Soc. London Ser. B 211, 151–180 (1981).
[CrossRef]

Marr, D. C.

F. H. C. Crick, D. C. Marr, T. Poggio, “An information processing approach to understanding the visual cortex,” in The Organization of the Cerebral Cortex, S. G. Dennis, ed. (MIT U. Press, Cambridge, Mass., 1981).

McKee, S. P.

S. P. McKee, “A local mechanism for differential velocity detection,” Vision Res. 21, 491–500 (1981).
[CrossRef] [PubMed]

Moik, J. G.

J. G. Moik, “Digital processing of remotely sensed images,”NASA Doc. SP-431 (1980).

Movshon, J. A.

E. H. Adelson, J. A. Movshon, “Phenomenal coherence of moving visual patterns,” Nature 300, 523–525 (1982).
[CrossRef] [PubMed]

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, 341–347 (1980).
[CrossRef] [PubMed]

Nachmias, J.

R. J. W. Mansfield, J. Nachmias, “Perceived direction of motion under retinal image stabilization,” Vision Res. 21, 1423–1425 (1981).
[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, 341–347 (1980).
[CrossRef] [PubMed]

N. Graham, J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[CrossRef] [PubMed]

F. W. Campbell, J. Nachmias, J. Jukes, “Spatial-frequency discrimination in human vision,”J. Opt. Soc. Am. 60, 555–559 (1970).
[CrossRef] [PubMed]

Nas, H.

Pantle, A.

A. Pantle, “Motion aftereffect magnitude as a measure of the spatiotemporal response properties of direction-sensitive analyzers,” Vision Res. 14, 1229–1236 (1974).
[CrossRef] [PubMed]

Pavlidis, T.

S. Tanimoto, T. Pavlidis, “A hierarchical data structure for picture processing,” Comput. Graphics Image Process. 4, 104–119 (1975).
[CrossRef]

Poggio, T.

M. Fable, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. London Ser. B 213, 451–477 (1981).
[CrossRef]

F. H. C. Crick, D. C. Marr, T. Poggio, “An information processing approach to understanding the visual cortex,” in The Organization of the Cerebral Cortex, S. G. Dennis, ed. (MIT U. Press, Cambridge, Mass., 1981).

Prazdny, K.

H. C. Longuet-Higgins, K. Prazdny, “The interpretation of moving retinal images,” Proc. R. Soc. London Ser. B 208, 385–387 (1980).
[CrossRef]

Robson, J. G.

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vision Res. 21, 1115–1122 (1981).
[CrossRef]

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,”J. Phys. London 197, 551–566 (1968).

J. G. Robson, “Spatial and temporal contrast-sensitivity functions of the visual system,”J. Opt. Soc. Am. 56, 1141–1142 (1966).
[CrossRef]

Ross, J.

D. C. Burr, J. Ross, “Contrast sensitivity at high velocities,” Vision Res. 22, 479–484 (1982).
[CrossRef] [PubMed]

Sakitt, B.

B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybern. 43, 97–108 (1982).
[CrossRef] [PubMed]

Sekuler, R.

E. Levinson, R. Sekuler, “Adaptation alters perceived direction of motion,” Vision Res. 16, 779–781 (1976).
[CrossRef] [PubMed]

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

E. Levinson, R. Sekuler, “Inhibition and disinhibition of direction-specific mechanisms in human vision,” Nature 254, 692–694 (1975).
[CrossRef] [PubMed]

Sperling, G.

G. Sperling, J. P. H. van Santen, “Temporal covariance model of human 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]

Stern, R. M.

J. L. Crowley, R. M. Stern, “Fast computation of the difference of low-pass transform,”IEEE Trans. Pattern Anal. Mach. Intelligence PAMI-6, 212–222 (1984).
[CrossRef]

Stromeyer, C. F.

Tanimoto, S.

S. Tanimoto, T. Pavlidis, “A hierarchical data structure for picture processing,” Comput. Graphics Image Process. 4, 104–119 (1975).
[CrossRef]

Thompson, P.

P. Thompson, “The coding of velocity of movement in the human visual system,” Vision Res. 24, 41–45 (1984).
[CrossRef] [PubMed]

P. Thompson, “Discrimination of moving gratings at and above detection threshold,” Vision Res. 23, 1533–1538 (1983).
[CrossRef] [PubMed]

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, 341–347 (1980).
[CrossRef] [PubMed]

Tolhurst, D. J.

D. J. Tolhurst, “Separate channels for the analysis of the shape and the movement of a moving visual stimulus,”J. Physiol. London 231, 385–402 (1973).

Ullman, S.

D. Marr, S. Ullman, “Directional selectivity and its use in early visual processing,” Proc. R. Soc. London Ser. B 211, 151–180 (1981).
[CrossRef]

S. Ullman, The Interpretation of Visual Motion (MIT U. Press, Cambridge, Mass., 1979).

van Doorn, A. J.

van Nes, F. L.

van Santen, J. P. H.

Watson, A. B.

A. B. Watson, A. J. Ahumada, “A linear motion sensor,” Perception 12, A17 (1983).

A. B. Watson, “Summation of grating patches indicates many types of detector at one retinal location,” Vision Res. 22, 17–25 (1982).
[CrossRef] [PubMed]

A. B. Watson, “Derivation of the impulse response: comments on the method of Roufs and Blommaert,” Vision Res. 22, 1335–1337 (1982).
[CrossRef] [PubMed]

A. B. Watson, A. J. Ahumada, “Sampling, filtering, and apparent motion,” Perception 11, A15 (1982).

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vision Res. 21, 1115–1122 (1981).
[CrossRef]

A. B. Watson, P. G. Thompson, B. J. Murphy, J. Nachmias, “Summation and discrimination of gratings moving in opposite directions,” Vision Res. 20, 341–347 (1980).
[CrossRef] [PubMed]

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

A. B. Watson, A. J. Ahumada, J. Farrell, “The window of visibility: a psychophysical theory of fidelity in time-sampled visual motion displays,”NASA Tech. Pap. 2211, (1983).

A. B. Watson, “Temporal Sensitivity,” in Handbook of Perception and Human Performance, J. Thomas, ed. (Wiley, New York, to be published).

A. B. Watson, “Detection and recognition of simple spatial forms,” in Physical and Biological Processing of Images, A. C. Slade, ed. (Springer-Verlag, Berlin, 1983).
[CrossRef]

Zeevi, Y. Y.

Behav. Res. Methods Instrum. (1)

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

Biol. Cybern. (1)

B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybern. 43, 97–108 (1982).
[CrossRef] [PubMed]

Comput. Graphics Image Process. (1)

S. Tanimoto, T. Pavlidis, “A hierarchical data structure for picture processing,” Comput. Graphics Image Process. 4, 104–119 (1975).
[CrossRef]

IEEE Trans. Commun. (1)

P. J. Burt, E. H. Adelson, “The Laplacian pyramid as a compact image code,”IEEE Trans. Commun. COM-31, 532–540 (1983).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intelligence (1)

J. L. Crowley, R. M. Stern, “Fast computation of the difference of low-pass transform,”IEEE Trans. Pattern Anal. Mach. Intelligence PAMI-6, 212–222 (1984).
[CrossRef]

Invest. Ophthalmol. Vis. Sci. Suppl. (1)

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

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (2)

J. Phys. London (1)

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,”J. Phys. London 197, 551–566 (1968).

J. Physiol. London (3)

D. J. Tolhurst, “Separate channels for the analysis of the shape and the movement of a moving visual stimulus,”J. Physiol. London 231, 385–402 (1973).

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

M. G. F. Fourtes, A. L. Hodgkin, “Changes in the time scale and sensitivity in the omatidia of limulus,”J. Physiol. London 172, 239–263 (1964).

NASA Doc. SP-431 (1)

J. G. Moik, “Digital processing of remotely sensed images,”NASA Doc. SP-431 (1980).

Nature (2)

E. H. Adelson, J. A. Movshon, “Phenomenal coherence of moving visual patterns,” Nature 300, 523–525 (1982).
[CrossRef] [PubMed]

E. Levinson, R. Sekuler, “Inhibition and disinhibition of direction-specific mechanisms in human vision,” Nature 254, 692–694 (1975).
[CrossRef] [PubMed]

Opt. Lett. (1)

Perception (2)

A. B. Watson, A. J. Ahumada, “Sampling, filtering, and apparent motion,” Perception 11, A15 (1982).

A. B. Watson, A. J. Ahumada, “A linear motion sensor,” Perception 12, A17 (1983).

Proc. R. Soc. London Ser. B (3)

H. C. Longuet-Higgins, K. Prazdny, “The interpretation of moving retinal images,” Proc. R. Soc. London Ser. B 208, 385–387 (1980).
[CrossRef]

M. Fable, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. London Ser. B 213, 451–477 (1981).
[CrossRef]

D. Marr, S. Ullman, “Directional selectivity and its use in early visual processing,” Proc. R. Soc. London Ser. B 211, 151–180 (1981).
[CrossRef]

Vision Res. (14)

A. B. Watson, “Derivation of the impulse response: comments on the method of Roufs and Blommaert,” Vision Res. 22, 1335–1337 (1982).
[CrossRef] [PubMed]

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vision Res. 21, 1115–1122 (1981).
[CrossRef]

P. Thompson, “Discrimination of moving gratings at and above detection threshold,” Vision Res. 23, 1533–1538 (1983).
[CrossRef] [PubMed]

D. C. Burr, J. Ross, “Contrast sensitivity at high velocities,” Vision Res. 22, 479–484 (1982).
[CrossRef] [PubMed]

D. H. Kelly, “Adaptation effects on spatio-temporal sine-wave thresholds,” Vision Res. 12, 89–101 (1972).
[CrossRef] [PubMed]

R. J. W. Mansfield, J. Nachmias, “Perceived direction of motion under retinal image stabilization,” Vision Res. 21, 1423–1425 (1981).
[CrossRef] [PubMed]

N. Graham, J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[CrossRef] [PubMed]

A. B. Watson, “Summation of grating patches indicates many types of detector at one retinal location,” Vision Res. 22, 17–25 (1982).
[CrossRef] [PubMed]

N. Graham, “Spatial-frequency channels in the human visual system: effects of luminance and pattern drift rate,” Vision Res. 12, 53–68 (1972).
[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, 341–347 (1980).
[CrossRef] [PubMed]

P. Thompson, “The coding of velocity of movement in the human visual system,” Vision Res. 24, 41–45 (1984).
[CrossRef] [PubMed]

S. P. McKee, “A local mechanism for differential velocity detection,” Vision Res. 21, 491–500 (1981).
[CrossRef] [PubMed]

A. Pantle, “Motion aftereffect magnitude as a measure of the spatiotemporal response properties of direction-sensitive analyzers,” Vision Res. 14, 1229–1236 (1974).
[CrossRef] [PubMed]

E. Levinson, R. Sekuler, “Adaptation alters perceived direction of motion,” Vision Res. 16, 779–781 (1976).
[CrossRef] [PubMed]

Other (10)

E. C. Hildreth, The Measurement of Visual Motion (MIT U. Press, Cambridge, Mass., 1983).

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

J. J. Gibson, The Perception of the Visual World (Houghton Mifflin, Boston, Mass., 1950).

S. Ullman, The Interpretation of Visual Motion (MIT U. Press, Cambridge, Mass., 1979).

P. A. Kolers, Aspects of Apparent Motion (Pergamon, New York, 1972).

A. B. Watson, A. J. Ahumada, J. Farrell, “The window of visibility: a psychophysical theory of fidelity in time-sampled visual motion displays,”NASA Tech. Pap. 2211, (1983).

The orientation of a 2D frequency component is the angle of a normal to the wavefront.

F. H. C. Crick, D. C. Marr, T. Poggio, “An information processing approach to understanding the visual cortex,” in The Organization of the Cerebral Cortex, S. G. Dennis, ed. (MIT U. Press, Cambridge, Mass., 1981).

A. B. Watson, “Detection and recognition of simple spatial forms,” in Physical and Biological Processing of Images, A. C. Slade, ed. (Springer-Verlag, Berlin, 1983).
[CrossRef]

A. B. Watson, “Temporal Sensitivity,” in Handbook of Perception and Human Performance, J. Thomas, ed. (Wiley, New York, to be published).

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Figures (26)

Fig. 1
Fig. 1

The effect of motion on the Fourier transform of a 2D space–time image. The effect is shown for a single representative component of spatial frequency u0. The open circles show the location of the components of the static image. The dotted circles show the locations when the image moves with speed r. Each transform point is sheared in the w dimension by an amount −ru, as indicated by the arrows.

Fig. 2
Fig. 2

The effect of motion on the transform of a 3D space–time image. The effect is shown both for a plane representing the full spectrum and for a single representative component of spatial frequency f and orientation α. The solid plane and filled circles show the location of the spectrum of the static image defined by w = 0. Motion at velocity r shears the spectrum into the plane w = r · b, as shown by the dashed–dotted plane and dotted circles. The arrows indicate the displacement of a single spatial-frequency component.

Fig. 3
Fig. 3

An example of direction ambiguity. The motion of the contour seen through the aperture is consistent with any of the velocities a or b or c (after Marr and Ullman). Note that all possible velocities have the same velocity component orthogonal to the contour.39

Fig. 4
Fig. 4

Mathematical structure of the scalar motion sensor.

Fig. 5
Fig. 5

A, Impulse response; B, amplitude response; C, phase response of the basic temporal filter. The symbols in the center panel are contrast-sensitivity measurements made by Robson21 for a 0.5-cycle/deg grating.

Fig. 6
Fig. 6

Spatial impulse responses of a, main and b, quadrature paths. The spatial impulse response of thee basic spatial filter is equivalent to that of the main path, a. The gray background indicates a value of 0.

Fig. 7
Fig. 7

The Hilbert filter. a, Impulse response. b, Transfer function. The dotted line indicates an imaginary value.

Fig. 8
Fig. 8

Temporal impulse responses of main (solid line) and quadrature (dotted line) paths of the scalar motion sensor.

Fig. 9
Fig. 9

a, Impulse response and b, amplitude response of a scalar sensor for leftward motion (direction = 0). In a, the axes are x (horizontal) and t (vertical). In b, they are u and w.

Fig. 10
Fig. 10

a, Impulse response and b amplitude response of a scalar sensor for motion in direction = 3240. In a, frames show successive time samples of 12.5 msec. In b, frames show successive temporal frequency samples of 5 Hz, with the origin in frame 9.

Fig. 11
Fig. 11

Sensor-response amplitude as a function of the spatial frequency of a moving sinusoidal grating. a, Constant temporal frequency. b, Constant velocity. The shape of the curve is determined by the product of speed and sensor spatial frequency. The three curves are for values of 1 (solid), 16 (dotted), and 32 Hz (dashed).

Fig. 12
Fig. 12

Normalized response amplitude of the scalar motion sensor as a function of the direction of a moving sinusoid [Eq. (38)] with ρ = 0.795.

Fig. 13
Fig. 13

The structure of the vector motion sensor.

Fig. 14
Fig. 14

The shrink algorithm. The original image of width W is denoted C0. Application of a fft results in C ˜ 0. It contains frequencies up to 2−1W. A square core, which contains frequencies up to 2−2W, is inverse transformed to yield a movie C1 that is half as large in each spatial dimension. The procedure can be repeated to yield C2, and so on. The algorithm can be generalized to other size ratios and to the time dimension. This procedure is similar to other pyramid schemes4749 but has the advantage of precisely band limiting the signal before subsampling.

Fig. 15
Fig. 15

Computation of Rk,l, the sensor response at scale k, direction l. Shaded objects are Fourier transforms.

Fig. 16
Fig. 16

Illustration of the frequency-subsampling operator Su. The array at each successive scale is obtained by sampling every other element in u and v dimensions.

Fig. 17
Fig. 17

3D Gaussian blob. The dimensions are W = 32, H = 32, L = 16. The speed is 2 pixels/frame, and the direction is 315°. The spatial spread is two pixels; the temporal spread is eight frames.

Fig. 18
Fig. 18

Amplitude spectrum of the Gaussian blob. The axes are as in Fig. 11b.

Fig. 19
Fig. 19

Response of the scalar sensors of frequency 4 cycles/width and direction −36°.

Fig. 20
Fig. 20

Response of the vector sensors at scale 0 to the Gaussian blob. Each arrow is an estimate of image velocity at the corresponding spatial frequency and location. The contrast of the arrow indicates the strength of the response.

Fig. 21
Fig. 21

Vector-sensor responses to the Gaussian blob at scale 1.

Fig. 22
Fig. 22

Simulated responses to the sum of two gratings of different spatial frequencies that move in different directions. The frequencies were 2 and 8 cycles/width, the directions were 90° and 180°, and the speeds were both 1 pixel/frame. a, Response at the scale of the lower frequency. b, Response at the scale of the higher frequency.

Fig. 23
Fig. 23

Simulation of apparent motion. The input was as shown in Fig. 17 with all but frames 6 and 8 blank. The output at a scale of 4 cycles/width is shown.

Fig. 24
Fig. 24

A sequence of natural images in which two, objects (the hands) move in different directions. The width is 32 pixels.

Fig. 25
Fig. 25

The scalar-sensor responses Rk,l to the input in Fig. 24. Scale, 8 cycles/width, direction, 108°. The first four frames are wrap-arounds from the end of the response.

Fig. 26
Fig. 26

Vector-sensor responses to the hand-waving sequence, a, Output at a scale of 8 cycles/width, b, Output at a scale of 4 cycles/width.

Equations (47)

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c 0 ( x , y , t ) = c 0 ( x , y , 0 )             for all t .
c r ( x , y , t ) = c ( x - r x t , y - r y t , t ) .
a = ( x t ) ,             b = ( u w ) ,
c ( a ) 2 c ˜ ( b ) ,
a = ( x - r t t ) = A a ,             A = [ 1 - r 0 1 ] .
c ( a ) 2 c ˜ [ ( A - 1 ) T b ] ,
( A - 1 ) T = [ 1 0 r 1 ] ,
c ( x - r t , t ) 2 c ˜ ( u , w + r u ) .
c ( x - r x t , y - r y t , t ) 3 c ˜ ( u , v , w + r x u + r y v ) .
w = - r · f = - ( r x u + r y v ) = - r f cos ( θ - α ) ,
r θ = r α cos ( θ - α ) .
f ( t ) = ξ [ f 1 ( t ) - ζ f 2 ( t ) ] ,
f i ( t ) = u ( t ) τ i ( n i - 1 ) ! ( t / τ i ) n i - 1 e - t / τ i
f ˜ ( w ) = ξ [ f 1 ( w ) - ζ f ˜ 2 ( w ) ] ,
f ˜ i ( w ) = ( i 2 π w τ i + 1 ) - n i .
g ( x , y ) = a ( x ) b ( y ) ,
a ( x ) = exp [ - ( x / λ ) 2 ] cos ( 2 π u s x ) ,
b ( y ) = exp [ - ( y / λ ) 2 ] .
g ˜ ( u , v ) = a ˜ ( u ) b ˜ ( v ) ,
a ˜ ( u ) = π λ / 2 ( exp { - [ π λ ( u - u s ) ] 2 } + exp { - [ π λ ( u + u s ) ] 2 } ) ,
b ˜ ( v ) = π λ exp [ - ( π λ v ) 2 ] .
λ = ρ / f .
h ( x ) = - 1 / π x .
h ˜ ( u ) = i sgn ( u ) .
[ h ( x ) * a ( x ) ] b ( y )
i sign ( u ) a ˜ ( u ) b ˜ ( v )
m ˜ m ( u , v , w ) = a ˜ ( u ) b ˜ ( v ) f ˜ ( w ) exp ( - i 2 π w τ ) ,
m ˜ q ( u , v , w ) = - m ˜ m ( u , v , w ) sign ( u ) sign ( w ) .
m ˜ r ( u , v , w ) = m ˜ m + m ˜ q = a ˜ ( u ) b ˜ ( v ) f ˜ ( w ) × exp ( - i 2 π w τ ) [ 1 - sign ( u ) sign ( w ) ] .
u = u cos θ s + v sin θ s ,             v = - u sin θ s + v cos θ s .
m ˜ s ( f , w ) = G { exp [ - ( π λ s - f ) 2 ] + exp [ - ( π λ s + f ) 2 ] } × f ˜ ( w ) exp ( - i 2 π τ w ) [ 1 - sgn ( s · f ) sgn ( w ) ] .
a ( x ) b ( y ) c ( t ) * [ δ ( x , y , t ) + δ ( y ) h ( x ) h ( t ) ] ,
r ( x , y , t ) = c ( x , y , t ) * m ( x , y , t )
r ( x , y , t ) 3 c ˜ ( u , v , w ) m ˜ ( u , v , w ) .
w s = s · r .
G exp [ - ( π λ s - f ) 2 ] f ˜ ( s f ) [ 1 - sign ( s · r ) ] .
G exp { - [ π λ ( f - f s ) ] 2 } f ˜ ( r f ) .
G exp ( - π 2 s 2 [ f s 2 + f 2 - 2 f s f cos ( θ - θ s ) ] ) f ˜ ( r f ) .
exp { - [ 2 π ρ sin ( θ - θ s 2 ) ] 2 } .
exp { - [ π ρ ( θ - θ s ) ] 2 } .
w = r f s cos ( θ - θ s ) ,
C k S h C k + 1 .
M 0 , l fft M ˜ 0 , l ,
M k , l S u M k + 1 , l .
C k fft C ˜ k ,
M ˜ k , l C ˜ k = R ˜ k , l ,
R ˜ k , l fft - 1 R k , l .

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