Abstract

We propose a model of direction-sensitive units in human vision. It is a modified and elaborated version of a model by Reichardt [ Z. Naturforsch. Teil B 12, 447 ( 1957)]. The model is applied to threshold experiments in which subjects view adjacent vertical bars with independently (typically sinusoidally), temporally modulated luminances. The subject must report whether the patterns moved to the left or to the right. According to the model, a basic motion-detecting unit consists of two subunits tuned to opposite directions. Each performs a spatial and temporal linear filtering of its input; outputs of the filters are multiplied, and the multiplied output is integrated (for a time that is long relative to the modulation period). The model’s output consists of the difference between the subunit outputs. Direction of movement is indicated by the sign of the model output. Mathematical analysis of the model yielded several predictions that were confirmed experimentally. Specifically, we found that (1) performance with complex patterns can be predicted by spatiotemporal Fourier analysis that results in the segregation and linear addition in the output for different temporal frequencies; (2) under special conditions, performance depends on the product of adjacent bar amplitudes, offering strong support for the multiplication principle; (3) performance is unaffected by addition of stationary patterns; and (4) addition of homogeneous flicker normally produces no effect but under special conditions reverses perceived direction. These and other results confirm our model and reject several other models, including Reichardt’s original model.

© 1984 Optical Society of America

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  1. O. Braddick, “A short-range process in apparent motion,” Vis. Res. 14, 519–529 (1974); G. Westheimer, “The spatial sense of the eye,” Invest. Ophthalmol. Vis. Sci. 18, 893–912 (1979).
    [Crossref] [PubMed]
  2. S. M. Anstis, “Apparent movement,” in Handbook of Sensory Physiology, Vol. VIII: Perception, R. Held, H. W. Leibowitz, H.-L. Teuber, eds. (Springer-Verlag, New York, 1977).
  3. A. J. Pantle, L. Picciano, “A multi-stable movement display: evidence for two separate motion systems in humans,” Science 193, 500–502 (1976).
    [Crossref] [PubMed]
  4. O. J. Braddick, “Low-level and high-level processes in apparent motion,” Philos. Trans. R. Soc. London Sect. B 290, 137–151 (1980).
    [Crossref]
  5. F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,”J. Physiol. (London) 197, 551–566 (1968).
  6. H. R. Wilson, J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–33 (1979).
    [Crossref] [PubMed]
  7. E. Levinson, R. Sekuler, “The independence of channels in human vision selective for direction of movement,”J. Physiol. (London) 250, 347–366 (1975).
  8. 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]
  9. E. Adelson, “Some new motion illusions, and some old ones, analyzed in terms of their Fourier components,” presented at the meetings of the Association for Research in Vision and Ophthalmology, Sarasota, Florida, 1982.
  10. W. Reichardt, “Autokorrelationsauswertung als Funktionsprinzip des Zentralnervensystems,” Z. Naturforsch. Teil B 12, 447–457 (1957); W. Reichardt, D. Varju, “Uebertragungseigenschaften im Auswertesystem fuer das Bewegungssehen,”Z. Naturforsch. Teil B 14, 674–689 (1959); W. Reichardt, “Autocorrelation, a principle for the evaluation of sensory information by the central nervous system,” in Sensory Communication, W. A. Rosenblith, ed. (Wiley, New York, 1961).
  11. Equation (10) implies the separability hypothesis [H. R. Wilson, “Spatiotemporal characterization of a transient mechanism in the human vision system,” Vision Res. 20, 443–452 (1980)] according to which the spatiotemporal frequency response of a detector is the product of the spatial- and the temporal-frequency responses. However, Eqs. (11)–(22) can still be derived if we replace Eq. (10) by a weaker formyH,0=Σn=0∞∫rnω(x)Ln(x,t) dx,, which does not imply separability.
    [Crossref]
  12. D. Gabor, “Theory of communication,”J. Inst. Electr. Eng. 93, 429–457 (1946).
  13. S. Marcelja, “Mathematical description of the responses of simple cortical cells,”J. Opt. Soc. Am. 70, 1297–1300 (1980).
    [Crossref] [PubMed]
  14. For example, J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 837–846 (1980); C. R. Carlson, R. W. Klopfenstein, C. H. Anderson, “Spatially inhomogeneous scaled transforms for vision and pattern recognition,” Opt. Lett. 6, 386–388 (1981).
    [Crossref] [PubMed]
  15. D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
    [Crossref] [PubMed]
  16. In fact, Eq. (15) holds for any pair of symmetric receptive fields, including ones that are completely low-pass. This implies that the detector can have band-pass characteristics even when its input receptive fields are low-pass and that, thus, a psychophysical band-pass response, to the extent that it is based on Reichardt-type detectors, does not imply the existence of single cells with band-pass linear receptive fields.
  17. The critical property needed for the specific voting rule V is that, when detector output zi is of the form zi= xξi(where ξi is a factor specific to detector Di), then V is monotonic in x. We need this property to test the multiplicative law [Eq. (19)], where the role of x is played by the detector-independent term moddmevenand the role of ξi is played by the detector-dependent term Σk=1,3,5,…F-1Σj=1F-k p(ϑj+k-ϑj)Ajk. The voting rule used to generate predictions isV(z1,…,zM)=q [∑i=1Mk sign(zi)∣zi∣p],where q is a function that ranges between 0 and 1, is antisymmetric around 0.50 [i.e., q(z) −0.50 = 0.50 − q(−z)], and is strictly increasing; k and p are arbitrary positive constants. This rule includes both the additive case (for p= k= 1), in which the response depends on the sum of the detector outputs, and the maximum case [for p→ ∞, k= 1, and q(z) = z1/p],in which the response depends on the maximum of the detector outputs (as in threshold models).
  18. J. A. Leese, C. S. Novak, V. R. Taylor, “The determination of cloud pattern motions from geosynchronous satellite image data,” Pattern Recognition 2, 279–292 (1970).
    [Crossref]
  19. E. A. Smith, D. R. Phillips, “Automated cloud tracking using precisely aligned digital ATS pictures,”IEEE Trans. Comput. C-21, 715–729 (1972).
    [Crossref]
  20. R. C. Lo, J. A. Parikh, “A study of the application of Fourier transforms to cloud movement estimation from satellite photographs,” (University of Maryland, College Park, Md., 1973).
  21. S. M. Anstis, B. J. Rogers, “Illusory reversal of visual depth and movement during changes of contrast,” Vision Res. 15, 957–961 (1975).
    [Crossref] [PubMed]
  22. S. Ullman, “Analysis of vision motion by biological and computer systems,” Computer 14, 57–69 (1981).
    [Crossref]
  23. J. O. Limb, J. A. Murphy, “Estimating the velocity of moving images in television signals,” Comput. Graphics 4, 311–327 (1975).
  24. C. L. Fennema, W. B. Thompson, “Velocity determination in scenes containing several moving objects,” Comput. Graphics 9, 301–315 (1979).
  25. D. Marr, S. Ullman, “Directional selectivity and its use in early visual processing,” Proc. R. Soc. London Sect. B 211, 151–180 (1981).
    [Crossref]
  26. D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Sect. B 207, 187–217 (1980).
    [Crossref]
  27. J. F. Schouten, “Subjective stroboscopy and a model of visual movement detection,” in Proceedings of the Symposium on Models of the Perception of Speech and Visual Form (MIT, Cambridge, Mass., 1967).
  28. D. H. Foster, “A model of the human visual system in its response to certain classes of moving stimuli,” Kybernetik 8, 69–84 (1971).
    [Crossref] [PubMed]
  29. G. Sperling, “Movement perception in computer-driven visual displays,” Behav. Res. Methods Instrum. 8, 144–151 (1976).
    [Crossref]
  30. A. Korte, “Kinematoscopische Untersuchungen,”Z. Psychol. 72, 193–296 (1915).
  31. D. H. Kelly, “Motion and vision. II. Stabilized spatiotemporal threshold surface,”J. Opt. Soc. Am. 69, 1340–1349 (1979).
    [Crossref] [PubMed]
  32. D. C. Burr, J. Ross, “Contrast sensitivity at high velocities,” Vision Res. 22, 479–484 (1982).
    [Crossref] [PubMed]
  33. These simulations consisted of calculating the value of Σj=13Aj2, with xc(location of detector center relative to display center), xright–xleft(receptive-field center distance), and scale parameter σ factorially taking values between 0.05 and 105.2 by steps of a factor of 2. We did this for rH(x) = exp[−(x− xH)2/σ2], exp(−|x− xH|/σ) uniform with width parameter σ, triangular, and semicircular.
  34. R. E. Barlow, D. J. Bartholomew, J. M. Bremner, H. D. Brunk, Statistical Inference under Order Restrictions; The Theory and Application of Isotonic Regression (Wiley, New York, 1972).
  35. B. J. Murphy, “Pattern thresholds for moving and stationary gratings,” Vision Res. 18, 521–530 (1978).
    [Crossref]
  36. J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat’s striate cortex,”J. Physiol. 283, 79–99 (1978).
  37. J. Thorson, “Small-signal analysis of a visual reflex in the locust. II. Frequency dependence,” Kybernetik 3, 53–66 (1966).
    [Crossref] [PubMed]
  38. A. B. Watson, A. J. Ahumada, “A Look at Motion in the Frequency Domain,”NASA Tech. Mem. 84352 (National Technical Information Service, Springfield, Va., 1983).

1982 (1)

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

1981 (3)

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

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

S. Ullman, “Analysis of vision motion by biological and computer systems,” Computer 14, 57–69 (1981).
[Crossref]

1980 (6)

For example, J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 837–846 (1980); C. R. Carlson, R. W. Klopfenstein, C. H. Anderson, “Spatially inhomogeneous scaled transforms for vision and pattern recognition,” Opt. Lett. 6, 386–388 (1981).
[Crossref] [PubMed]

Equation (10) implies the separability hypothesis [H. R. Wilson, “Spatiotemporal characterization of a transient mechanism in the human vision system,” Vision Res. 20, 443–452 (1980)] according to which the spatiotemporal frequency response of a detector is the product of the spatial- and the temporal-frequency responses. However, Eqs. (11)–(22) can still be derived if we replace Eq. (10) by a weaker formyH,0=Σn=0∞∫rnω(x)Ln(x,t) dx,, which does not imply separability.
[Crossref]

O. J. Braddick, “Low-level and high-level processes in apparent motion,” Philos. Trans. R. Soc. London Sect. B 290, 137–151 (1980).
[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]

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Sect. B 207, 187–217 (1980).
[Crossref]

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

1979 (3)

D. H. Kelly, “Motion and vision. II. Stabilized spatiotemporal threshold surface,”J. Opt. Soc. Am. 69, 1340–1349 (1979).
[Crossref] [PubMed]

C. L. Fennema, W. B. Thompson, “Velocity determination in scenes containing several moving objects,” Comput. Graphics 9, 301–315 (1979).

H. R. Wilson, J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–33 (1979).
[Crossref] [PubMed]

1978 (2)

B. J. Murphy, “Pattern thresholds for moving and stationary gratings,” Vision Res. 18, 521–530 (1978).
[Crossref]

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat’s striate cortex,”J. Physiol. 283, 79–99 (1978).

1976 (2)

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

A. J. Pantle, L. Picciano, “A multi-stable movement display: evidence for two separate motion systems in humans,” Science 193, 500–502 (1976).
[Crossref] [PubMed]

1975 (3)

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

J. O. Limb, J. A. Murphy, “Estimating the velocity of moving images in television signals,” Comput. Graphics 4, 311–327 (1975).

S. M. Anstis, B. J. Rogers, “Illusory reversal of visual depth and movement during changes of contrast,” Vision Res. 15, 957–961 (1975).
[Crossref] [PubMed]

1974 (1)

O. Braddick, “A short-range process in apparent motion,” Vis. Res. 14, 519–529 (1974); G. Westheimer, “The spatial sense of the eye,” Invest. Ophthalmol. Vis. Sci. 18, 893–912 (1979).
[Crossref] [PubMed]

1972 (1)

E. A. Smith, D. R. Phillips, “Automated cloud tracking using precisely aligned digital ATS pictures,”IEEE Trans. Comput. C-21, 715–729 (1972).
[Crossref]

1971 (1)

D. H. Foster, “A model of the human visual system in its response to certain classes of moving stimuli,” Kybernetik 8, 69–84 (1971).
[Crossref] [PubMed]

1970 (1)

J. A. Leese, C. S. Novak, V. R. Taylor, “The determination of cloud pattern motions from geosynchronous satellite image data,” Pattern Recognition 2, 279–292 (1970).
[Crossref]

1968 (1)

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

1966 (1)

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

1957 (1)

W. Reichardt, “Autokorrelationsauswertung als Funktionsprinzip des Zentralnervensystems,” Z. Naturforsch. Teil B 12, 447–457 (1957); W. Reichardt, D. Varju, “Uebertragungseigenschaften im Auswertesystem fuer das Bewegungssehen,”Z. Naturforsch. Teil B 14, 674–689 (1959); W. Reichardt, “Autocorrelation, a principle for the evaluation of sensory information by the central nervous system,” in Sensory Communication, W. A. Rosenblith, ed. (Wiley, New York, 1961).

1946 (1)

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

1915 (1)

A. Korte, “Kinematoscopische Untersuchungen,”Z. Psychol. 72, 193–296 (1915).

Adelson, E.

E. Adelson, “Some new motion illusions, and some old ones, analyzed in terms of their Fourier components,” presented at the meetings of the Association for Research in Vision and Ophthalmology, Sarasota, Florida, 1982.

Ahumada, A. J.

A. B. Watson, A. J. Ahumada, “A Look at Motion in the Frequency Domain,”NASA Tech. Mem. 84352 (National Technical Information Service, Springfield, Va., 1983).

Anstis, S. M.

S. M. Anstis, B. J. Rogers, “Illusory reversal of visual depth and movement during changes of contrast,” Vision Res. 15, 957–961 (1975).
[Crossref] [PubMed]

S. M. Anstis, “Apparent movement,” in Handbook of Sensory Physiology, Vol. VIII: Perception, R. Held, H. W. Leibowitz, H.-L. Teuber, eds. (Springer-Verlag, New York, 1977).

Barlow, R. E.

R. E. Barlow, D. J. Bartholomew, J. M. Bremner, H. D. Brunk, Statistical Inference under Order Restrictions; The Theory and Application of Isotonic Regression (Wiley, New York, 1972).

Bartholomew, D. J.

R. E. Barlow, D. J. Bartholomew, J. M. Bremner, H. D. Brunk, Statistical Inference under Order Restrictions; The Theory and Application of Isotonic Regression (Wiley, New York, 1972).

Bergen, J. R.

H. R. Wilson, J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–33 (1979).
[Crossref] [PubMed]

Braddick, O.

O. Braddick, “A short-range process in apparent motion,” Vis. Res. 14, 519–529 (1974); G. Westheimer, “The spatial sense of the eye,” Invest. Ophthalmol. Vis. Sci. 18, 893–912 (1979).
[Crossref] [PubMed]

Braddick, O. J.

O. J. Braddick, “Low-level and high-level processes in apparent motion,” Philos. Trans. R. Soc. London Sect. B 290, 137–151 (1980).
[Crossref]

Bremner, J. M.

R. E. Barlow, D. J. Bartholomew, J. M. Bremner, H. D. Brunk, Statistical Inference under Order Restrictions; The Theory and Application of Isotonic Regression (Wiley, New York, 1972).

Brunk, H. D.

R. E. Barlow, D. J. Bartholomew, J. M. Bremner, H. D. Brunk, Statistical Inference under Order Restrictions; The Theory and Application of Isotonic Regression (Wiley, New York, 1972).

Burr, D. C.

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

Campbell, F. W.

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

Daugman, J. G.

For example, J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 837–846 (1980); C. R. Carlson, R. W. Klopfenstein, C. H. Anderson, “Spatially inhomogeneous scaled transforms for vision and pattern recognition,” Opt. Lett. 6, 386–388 (1981).
[Crossref] [PubMed]

Fennema, C. L.

C. L. Fennema, W. B. Thompson, “Velocity determination in scenes containing several moving objects,” Comput. Graphics 9, 301–315 (1979).

Foster, D. H.

D. H. Foster, “A model of the human visual system in its response to certain classes of moving stimuli,” Kybernetik 8, 69–84 (1971).
[Crossref] [PubMed]

Gabor, D.

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

Hildreth, E.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Sect. B 207, 187–217 (1980).
[Crossref]

Kelly, D. H.

Korte, A.

A. Korte, “Kinematoscopische Untersuchungen,”Z. Psychol. 72, 193–296 (1915).

Leese, J. A.

J. A. Leese, C. S. Novak, V. R. Taylor, “The determination of cloud pattern motions from geosynchronous satellite image data,” Pattern Recognition 2, 279–292 (1970).
[Crossref]

Levinson, E.

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

Limb, J. O.

J. O. Limb, J. A. Murphy, “Estimating the velocity of moving images in television signals,” Comput. Graphics 4, 311–327 (1975).

Lo, R. C.

R. C. Lo, J. A. Parikh, “A study of the application of Fourier transforms to cloud movement estimation from satellite photographs,” (University of Maryland, College Park, Md., 1973).

Marcelja, S.

Marr, D.

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

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Sect. B 207, 187–217 (1980).
[Crossref]

Movshon, J. A.

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat’s striate cortex,”J. Physiol. 283, 79–99 (1978).

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]

B. J. Murphy, “Pattern thresholds for moving and stationary gratings,” Vision Res. 18, 521–530 (1978).
[Crossref]

Murphy, J. A.

J. O. Limb, J. A. Murphy, “Estimating the velocity of moving images in television signals,” Comput. Graphics 4, 311–327 (1975).

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, 341–347 (1980).
[Crossref] [PubMed]

Novak, C. S.

J. A. Leese, C. S. Novak, V. R. Taylor, “The determination of cloud pattern motions from geosynchronous satellite image data,” Pattern Recognition 2, 279–292 (1970).
[Crossref]

Pantle, A. J.

A. J. Pantle, L. Picciano, “A multi-stable movement display: evidence for two separate motion systems in humans,” Science 193, 500–502 (1976).
[Crossref] [PubMed]

Parikh, J. A.

R. C. Lo, J. A. Parikh, “A study of the application of Fourier transforms to cloud movement estimation from satellite photographs,” (University of Maryland, College Park, Md., 1973).

Phillips, D. R.

E. A. Smith, D. R. Phillips, “Automated cloud tracking using precisely aligned digital ATS pictures,”IEEE Trans. Comput. C-21, 715–729 (1972).
[Crossref]

Picciano, L.

A. J. Pantle, L. Picciano, “A multi-stable movement display: evidence for two separate motion systems in humans,” Science 193, 500–502 (1976).
[Crossref] [PubMed]

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]

Reichardt, W.

W. Reichardt, “Autokorrelationsauswertung als Funktionsprinzip des Zentralnervensystems,” Z. Naturforsch. Teil B 12, 447–457 (1957); W. Reichardt, D. Varju, “Uebertragungseigenschaften im Auswertesystem fuer das Bewegungssehen,”Z. Naturforsch. Teil B 14, 674–689 (1959); W. Reichardt, “Autocorrelation, a principle for the evaluation of sensory information by the central nervous system,” in Sensory Communication, W. A. Rosenblith, ed. (Wiley, New York, 1961).

Robson, J. G.

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

Rogers, B. J.

S. M. Anstis, B. J. Rogers, “Illusory reversal of visual depth and movement during changes of contrast,” Vision Res. 15, 957–961 (1975).
[Crossref] [PubMed]

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]

Ross, J.

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

Schouten, J. F.

J. F. Schouten, “Subjective stroboscopy and a model of visual movement detection,” in Proceedings of the Symposium on Models of the Perception of Speech and Visual Form (MIT, Cambridge, Mass., 1967).

Sekuler, R.

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

Smith, E. A.

E. A. Smith, D. R. Phillips, “Automated cloud tracking using precisely aligned digital ATS pictures,”IEEE Trans. Comput. C-21, 715–729 (1972).
[Crossref]

Sperling, G.

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

Taylor, V. R.

J. A. Leese, C. S. Novak, V. R. Taylor, “The determination of cloud pattern motions from geosynchronous satellite image data,” Pattern Recognition 2, 279–292 (1970).
[Crossref]

Thompson, I. D.

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat’s striate cortex,”J. Physiol. 283, 79–99 (1978).

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]

Thompson, W. B.

C. L. Fennema, W. B. Thompson, “Velocity determination in scenes containing several moving objects,” Comput. Graphics 9, 301–315 (1979).

Thorson, J.

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

Tolhurst, D. J.

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat’s striate cortex,”J. Physiol. 283, 79–99 (1978).

Ullman, S.

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

S. Ullman, “Analysis of vision motion by biological and computer systems,” Computer 14, 57–69 (1981).
[Crossref]

Watson, A. B.

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 (National Technical Information Service, Springfield, Va., 1983).

Wilson, H. R.

Equation (10) implies the separability hypothesis [H. R. Wilson, “Spatiotemporal characterization of a transient mechanism in the human vision system,” Vision Res. 20, 443–452 (1980)] according to which the spatiotemporal frequency response of a detector is the product of the spatial- and the temporal-frequency responses. However, Eqs. (11)–(22) can still be derived if we replace Eq. (10) by a weaker formyH,0=Σn=0∞∫rnω(x)Ln(x,t) dx,, which does not imply separability.
[Crossref]

H. R. Wilson, J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–33 (1979).
[Crossref] [PubMed]

Behav. Res. Methods Instrum. (1)

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

Comput. Graphics (2)

J. O. Limb, J. A. Murphy, “Estimating the velocity of moving images in television signals,” Comput. Graphics 4, 311–327 (1975).

C. L. Fennema, W. B. Thompson, “Velocity determination in scenes containing several moving objects,” Comput. Graphics 9, 301–315 (1979).

Computer (1)

S. Ullman, “Analysis of vision motion by biological and computer systems,” Computer 14, 57–69 (1981).
[Crossref]

IEEE Trans. Comput. (1)

E. A. Smith, D. R. Phillips, “Automated cloud tracking using precisely aligned digital ATS pictures,”IEEE Trans. Comput. C-21, 715–729 (1972).
[Crossref]

J. Inst. Electr. Eng. (1)

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

J. Opt. Soc. Am. (2)

J. Physiol. (1)

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat’s striate cortex,”J. Physiol. 283, 79–99 (1978).

J. Physiol. (London) (2)

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

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

Kybernetik (2)

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

D. H. Foster, “A model of the human visual system in its response to certain classes of moving stimuli,” Kybernetik 8, 69–84 (1971).
[Crossref] [PubMed]

Pattern Recognition (1)

J. A. Leese, C. S. Novak, V. R. Taylor, “The determination of cloud pattern motions from geosynchronous satellite image data,” Pattern Recognition 2, 279–292 (1970).
[Crossref]

Philos. Trans. R. Soc. London Sect. B (1)

O. J. Braddick, “Low-level and high-level processes in apparent motion,” Philos. Trans. R. Soc. London Sect. B 290, 137–151 (1980).
[Crossref]

Proc. R. Soc. London Sect. B (2)

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

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Sect. B 207, 187–217 (1980).
[Crossref]

Science (2)

A. J. Pantle, L. Picciano, “A multi-stable movement display: evidence for two separate motion systems in humans,” Science 193, 500–502 (1976).
[Crossref] [PubMed]

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

Vis. Res. (1)

O. Braddick, “A short-range process in apparent motion,” Vis. Res. 14, 519–529 (1974); G. Westheimer, “The spatial sense of the eye,” Invest. Ophthalmol. Vis. Sci. 18, 893–912 (1979).
[Crossref] [PubMed]

Vision Res. (7)

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. R. Wilson, J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–33 (1979).
[Crossref] [PubMed]

For example, J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 837–846 (1980); C. R. Carlson, R. W. Klopfenstein, C. H. Anderson, “Spatially inhomogeneous scaled transforms for vision and pattern recognition,” Opt. Lett. 6, 386–388 (1981).
[Crossref] [PubMed]

Equation (10) implies the separability hypothesis [H. R. Wilson, “Spatiotemporal characterization of a transient mechanism in the human vision system,” Vision Res. 20, 443–452 (1980)] according to which the spatiotemporal frequency response of a detector is the product of the spatial- and the temporal-frequency responses. However, Eqs. (11)–(22) can still be derived if we replace Eq. (10) by a weaker formyH,0=Σn=0∞∫rnω(x)Ln(x,t) dx,, which does not imply separability.
[Crossref]

S. M. Anstis, B. J. Rogers, “Illusory reversal of visual depth and movement during changes of contrast,” Vision Res. 15, 957–961 (1975).
[Crossref] [PubMed]

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

B. J. Murphy, “Pattern thresholds for moving and stationary gratings,” Vision Res. 18, 521–530 (1978).
[Crossref]

Z. Naturforsch. Teil B (1)

W. Reichardt, “Autokorrelationsauswertung als Funktionsprinzip des Zentralnervensystems,” Z. Naturforsch. Teil B 12, 447–457 (1957); W. Reichardt, D. Varju, “Uebertragungseigenschaften im Auswertesystem fuer das Bewegungssehen,”Z. Naturforsch. Teil B 14, 674–689 (1959); W. Reichardt, “Autocorrelation, a principle for the evaluation of sensory information by the central nervous system,” in Sensory Communication, W. A. Rosenblith, ed. (Wiley, New York, 1961).

Z. Psychol. (1)

A. Korte, “Kinematoscopische Untersuchungen,”Z. Psychol. 72, 193–296 (1915).

Other (9)

A. B. Watson, A. J. Ahumada, “A Look at Motion in the Frequency Domain,”NASA Tech. Mem. 84352 (National Technical Information Service, Springfield, Va., 1983).

These simulations consisted of calculating the value of Σj=13Aj2, with xc(location of detector center relative to display center), xright–xleft(receptive-field center distance), and scale parameter σ factorially taking values between 0.05 and 105.2 by steps of a factor of 2. We did this for rH(x) = exp[−(x− xH)2/σ2], exp(−|x− xH|/σ) uniform with width parameter σ, triangular, and semicircular.

R. E. Barlow, D. J. Bartholomew, J. M. Bremner, H. D. Brunk, Statistical Inference under Order Restrictions; The Theory and Application of Isotonic Regression (Wiley, New York, 1972).

J. F. Schouten, “Subjective stroboscopy and a model of visual movement detection,” in Proceedings of the Symposium on Models of the Perception of Speech and Visual Form (MIT, Cambridge, Mass., 1967).

In fact, Eq. (15) holds for any pair of symmetric receptive fields, including ones that are completely low-pass. This implies that the detector can have band-pass characteristics even when its input receptive fields are low-pass and that, thus, a psychophysical band-pass response, to the extent that it is based on Reichardt-type detectors, does not imply the existence of single cells with band-pass linear receptive fields.

The critical property needed for the specific voting rule V is that, when detector output zi is of the form zi= xξi(where ξi is a factor specific to detector Di), then V is monotonic in x. We need this property to test the multiplicative law [Eq. (19)], where the role of x is played by the detector-independent term moddmevenand the role of ξi is played by the detector-dependent term Σk=1,3,5,…F-1Σj=1F-k p(ϑj+k-ϑj)Ajk. The voting rule used to generate predictions isV(z1,…,zM)=q [∑i=1Mk sign(zi)∣zi∣p],where q is a function that ranges between 0 and 1, is antisymmetric around 0.50 [i.e., q(z) −0.50 = 0.50 − q(−z)], and is strictly increasing; k and p are arbitrary positive constants. This rule includes both the additive case (for p= k= 1), in which the response depends on the sum of the detector outputs, and the maximum case [for p→ ∞, k= 1, and q(z) = z1/p],in which the response depends on the maximum of the detector outputs (as in threshold models).

R. C. Lo, J. A. Parikh, “A study of the application of Fourier transforms to cloud movement estimation from satellite photographs,” (University of Maryland, College Park, Md., 1973).

E. Adelson, “Some new motion illusions, and some old ones, analyzed in terms of their Fourier components,” presented at the meetings of the Association for Research in Vision and Ophthalmology, Sarasota, Florida, 1982.

S. M. Anstis, “Apparent movement,” in Handbook of Sensory Physiology, Vol. VIII: Perception, R. Held, H. W. Leibowitz, H.-L. Teuber, eds. (Springer-Verlag, New York, 1977).

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Figures (16)

Fig. 1
Fig. 1

Representation of a five-field display. A, luminance modulations in each of five fields, Lj(t) = L0j + mjh(tϑj), is here represented as Lj(t)/Lref, where Lref = 51 cd/m2 for most displays. The modulation function h(t) is an 8-point approximation to a sinusoid, which is indicated by a continuous curve for the leftmost field. Between-field asynchrony in this example is 128 msec (¼ cycle, π/2 rad). The temporal phase line (dashed line) interconnects the peak luminances of the dominant Fourier component in each bar. B, the five-field display. Overall height and width are 0.44 and 0.22 deg, respectively.

Fig. 2
Fig. 2

A, the Reichardt model. Input consists of the stimulus L(x, t) sampled at locations xleft and xright;yi,H represents the signal at the various stages i for the left and right subunits (H = left, right). TF indicates a linear, time-invariant filter with attenuation βω, and phase shift δω; × indicates a multiplication unit; TA indicates a time averaging unit, and + indicates a unit that adds its (negative and positive) inputs. B, proposed modification of the Reichardt model in which the point input assumption is generalized to the input of the entire stimulus L(x, t) through a linear spatial filter, denoted SF.

Fig. 3
Fig. 3

Candidates for spatial input filters (SF’s) in elaborated Reichardt model. A, symmetric–antisymmetric receptive-field arrangement [Eq. (12)]. B, completely symmetric receptive-field arrangement [Eq. (14)].

Fig. 4
Fig. 4

Comparison of difference of Gaussian curves (dashed curve) and best-fitting (in terms of maximal squared deviation) curves of the type W(x)cos(fx) (solid curve). Here, W(x) = Aexp(−x2/σ2), where σ = 2.83, A = 1.05, and f = 1.76.

Fig. 5
Fig. 5

Displays used in Experiment 1. Only modulations in the first two (of five) fields are depicted. A, same sinusoidal modulation function as in Fig. 1, 1.95 Hz, with asynchrony ϑj+1ϑj of 128 msec (hence phase difference φ of π/2 rad). B, same sinusoidal modulation function as in Fig. 1, but with asynchrony of 32 msec (φ = π/8 rad). C, modulation function that is a permutation of the function used in A and B. Between-field asynchrony is 128 msec, but the dominant fourth harmonics are in phase and hence convey no directional information. D, same modulation function as in C, but asynchrony is 32 msec (π/2 rad in terms of the dominant fourth harmonic).

Fig. 6
Fig. 6

Display used in Experiment 2. A, modulation function consists of pulses. Phase difference is 120 msec for 2.08-Hz display; 30 msec for 8.33-Hz display (both π/2 rad). B, same as in A, except that modulation functions in even fields are reversed in sign. Note that in B, fields containing same-sign pulses are either in phase or in counterphase and thus do not contain direction information.

Fig. 7
Fig. 7

Results of Experiment 2. The ordinate indicates the proportion of direction responses that are consistent with the objective direction of the dominant temporal-frequency component. The abscissa indicates modulation depth. Displays with sign-reversed pulses in even fields (Δ) yield performance roughly equivalent to average performance with positive pulses (□) and negative pulses (○).

Fig. 8
Fig. 8

Display used in Experiment 3. Modulation function is sinusoidal with asynchrony of 140 msec for 1.8-Hz display (π/2 rad) and 20 msec for 12.5-Hz display (π/2 rad). All fields have the same average luminance, but even-field modulation is four times larger than odd-field modulation. Note that fields with large modulations are in counterphase with each other and hence do not contain directional information.

Fig. 9
Fig. 9

Results from Experiment 3, the effects of odd- and even-field modulation. The abscissa is modd; the curve parameter is meven; the ordinate is the percent of directionally correct responses. Separate panels are shown for 1.8 and 12.5 Hz for both subjects, NB and JvS.

Fig. 10
Fig. 10

Results from Experiment 3. Percent correct left-right responses as a function of the product of modd and meven. (○), Data points; solid curve indicates isotonic regression.

Fig. 11
Fig. 11

Seventy-five-percent threshold modulations (m/L0) for displays with sinusoidal modulation functions, as a function of temporal frequency. Phase differences are 0.25π rad (○), 0.5π rad (□), or 0.75π rad (Δ), corresponding to spatial frequencies of 2.84 (○), 5.68 (□), and 8.52 (Δ) cpd.

Fig. 12
Fig. 12

One of the three types of display (1&2) used in Experiment 5. Modulation function is sinusoidal, with same modulation mj but different average luminance L0j across fields. Other displays (1) and (2) in this experiment have the same average luminances across fields. Inset: Alternative representation of displays (1), (1&2), and (2) used in Experiment 5. Arrows indicate modulation mj, and height of vertical bars indicates average luminance L0j.

Fig. 13
Fig. 13

Effects on the spatial phase path of a moving sine wave when a stationary sine wave with the same spatial frequency is added. Luminance of a given location in a frame is indicated by the vertical deviation from the horizontal dashed null line for the frame. Vertical dashed lines connect locations having the same spatial phase in successive frames. Left-hand panel: six successive frames of sin(x + t), where x denotes location and t(= t1, t2,…, t6) time. Right-hand panel: six successive frames of sin(x + t) + 2 sin(x).

Fig. 14
Fig. 14

Results of Experiment 6. Effects of adding a stationary sinusoid having the same spatial frequency ω and modulation mj as a moving sinusoid. Error bars indicate 95% confidence intervals. Inset; Alternative representation of displays (1), (1&2), and (2) used in Experiment 6.

Fig. 15
Fig. 15

Displays used in Experiment 7. A, sinusoidal modulation function with a synchrony of π/6 rad. B, sinusoidal modulation function with asynchrony of 0 rad (homogeneousflicker) and same temporal frequency ω but twice the modulation m used in A. C, result of adding flicker in B to display in A. Note that asynchronies are opposite in sign to those in A, whereas modulation mj varies across fields.

Fig. 16
Fig. 16

Results from Experiment 7. Effects of adding homogeneous flicker. Dotted horizontal lines indicate performance without added flicker. The remaining lines indicate performance with added flicker. The flicker frequency is 2ω(○), ω(□), or ω/2(Δ), where ω is the temporal frequency of the display without added flicker.

Equations (27)

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L n ( x , t ) = ζ n ( x ) sin [ 2 π n ω t - η n ( x ) ] .
L j ( t ) = L 0 j + m j h ( t - ϑ j ) .
h ( t ) = n = 0 ɛ n sin ( 2 π n ω t - κ n ) .
y H , 0 ( t ) = L ( x H , t ) = n = 0 α H , n sin ( 2 π n ω t - γ H , n ) .
y left , 0 ( t ) = α H , 0 + n = 1 α H , n sin ( 2 π n ω t - γ H , n ) .
g H ( t ) = n = 1 α H , n β n ω sin ( 2 π n ω t - γ H , n - δ n ω ) .
y left , 2 ( t ) = β 0 α left , 0 α right , 0 + α left , 0 m = 1 α right , m β m ω sin ( 2 π m ω t - γ right , m - δ m ω ) + α right , 0 β 0 m = 1 α left , n sin ( 2 π n ω t - γ left , n ) + n = 1 m = 1 1 2 α left , n α right , m β m ω × { cos [ 2 π ( n - ω ) ω t - γ left , m + γ right , m + δ m ω ] - cos [ 2 π ( n + m ) ω t - γ left , m - γ right , m - δ m ω ] } .
y left , 3 ( t ) = β 0 α left , 0 α right , 0 + n = 1 1 2 α left , n α right , n β n ω × cos [ δ n ω - ( γ left , n - γ right , n ) ] .
y 4 ( L ) = n = 1 α right , n α left , n β n ω sin δ n ω sin ( γ right , n - γ left , n ) .
y 4 ( L n ) = α right , n α left , n β n ω sin δ n ω sin ( γ right , n - γ left , n ) .
S ( m , d , f , ω ) = L 0 + m sin ( 2 π d f x + 2 π ω t + Φ ) ,
y H , 0 = r H ( x ) L ( x , t ) d x ,             H = left , right .
y 4 [ S ( m , d , f , ω ) ] = m 2 d β ω sin δ ω P left ( f ) P right ( f ) D ( f ) .
r left ( x ) = W ( x - x c ) cos [ f 0 ( x - x c ) ] , r right ( x ) = W ( x - x c ) sin [ f 0 ( x - x c ) ] .
m 2 d β ω sin δ ω P left ( f ) P right ( f ) .
r H ( x ) = W ( x - x H ) cos [ f 0 ( x - x H ) ] ,             H = left , right .
m d β ω sin δ ω P left ( f ) sin [ 2 π f ( x right - x left ) ] .
A j k = a left , j a right , j + k - a left , j a right , j .
y 4 ( L ) = k = 1 F - 1 j = 1 F - k m j m j + k p ( ϑ j + k - ϑ j ) A j k .
y 4 ( φ = π / 2 ) = k = 1 , 3 , 5 , F - 1 j = 1 F - k m j m j + k p ( ϑ j + k - ϑ j ) A j k .
y 4 ( φ = π / 2 ) = m odd m even k = 1 , 3 , 5 , .. F - 1 j = 1 F - k p ( φ j + k - ϑ j ) A j k .
L j ( t ) = L 0 j + m sin ( 2 π ω t - ϑ j ) .
y 4 ( sin ) = β ω sin δ ω m 2 k = 1 F - 1 j = 1 F - k sin ( ϑ j + k - ϑ j ) A j k .
y 4 ( sin ) = β ω sin δ ω m 2 k = 1 F - 1 j = 1 F - k sin ( k φ ) A j k .
y 4 ( L ) = j = 1 F - 1 m j m j + 1 p ( ϑ j + 1 - ϑ j ) .
sin ( ω t + x ) + K sin ( ω t + π ) = A ( x ) sin [ ω t + B ( x ) ] .
V(z1,,zM)=q[i=1Mksign(zi)zip],

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