Abstract

Many visual displays, such as movies and television, rely on sampling in the time domain. We derive the spatiotemporal-frequency spectra for some simple moving images and illustrate how these spectra are altered by sampling in the time domain. We construct a simple model of the human perceiver that predicts the critical sample rate required to render sampled and continuous moving images indistinguishable. The rate is shown to depend on the spatial and the temporal acuity of the observer and on the velocity and spatial-frequency content of the image. Several predictions of this model are tested and confirmed. The model is offered as an explanation of many of the phenomena known as apparent motion. Finally, the implications of the model for computer-generated imagery are discussed.

© 1986 Optical Society of America

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References

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  1. S. Exner, “Ueber das Sehen von Bewegungen und die Theorie des zusammengesetzen Auges,” Sitzungber. Akad. Wiss. Wien 72, 156–160 (1875).
  2. O. Braddick, “A short-range process in apparent motion,” Vision Res. 14, 519–527 (1974).
    [CrossRef] [PubMed]
  3. P. A. Kolers, Aspects of Apparent Motion (Pergamon, New York, 1972).
  4. M. J. Morgan, “Perception of continuity in stroboscopic motion: a temporal frequency analysis,” Vision Res. 19, 491–500 (1979).
    [CrossRef] [PubMed]
  5. M. J. Morgan, “Analogue models of motion perception,” Phil. Trans. R. Soc. London Ser. B 290, 117–135 (1980).
    [CrossRef]
  6. M. J. Morgan, “Spatiotemporal filtering and the interpolation effect in apparent motion,” Perception 9, 161–174 (1980).
    [CrossRef] [PubMed]
  7. M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. London Ser. B 213, 451–477 (1981).
    [CrossRef]
  8. O. H. Shade, “Optical and photoelectric analog of the eye,”J. Opt. Soc. Am. 46, 721–739 (1956).
    [CrossRef]
  9. H. de Lange, “Relationship between critical flicker frequency and a set of low frequency characteristics of the eye,”J. Opt. Soc. Am. 44, 380–389 (1954).
    [CrossRef]
  10. J. G. Robson, “Spatial and temporal contrast sensitivity functions of the visual system,”J. Opt. Soc. Am. 56, 1141–1142 (1966).
    [CrossRef]
  11. J. J. Koenderink, A. J. van Doorn, “Spatiotemporal contrast detection threshold surface is bimodal,” Opt. Lett. 4, 32–34 (1979).
    [CrossRef] [PubMed]
  12. A. B. Watson, “Summation of grating patches indicates many types of detector at one retinal location,” Vision Res. 22, 17–25 (1982).
    [CrossRef] [PubMed]
  13. 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]
  14. A. J. Ahumada, A. B. Watson, “Equivalent noise model for contrast detection and discrimination,” J. Opt. Soc. Am. A 2, 1133–1139 (1985).
    [CrossRef] [PubMed]
  15. K. R. K. Nielsen, A. B. Watson, A. J. Ahumada, “Application of a computable model of human spatial vision to phase discrimination,” J. Opt. Soc. Am. A 2, 1600–1606 (1985).
    [CrossRef] [PubMed]
  16. A. B. Watson, A. J. Ahumada, “Model of human visualmotion sensing,” J. Opt. Soc. Am. A 2, 322–342 (1985).
    [CrossRef] [PubMed]
  17. A. B. Watson, “Temporal sensitivity,” in Handbook of Perception and Human Performance, J. Thomas, ed. (Wiley, New York, to be published).
  18. G. Sperling, “Movement perception in computer-driven visual displays,” Behavior Res. Methods Instrum. 8, 144–151 (1976).
    [CrossRef]
  19. A. B. Watson, A. J. Ahumada, “A theory of apparently real motion,” Invest. Ophthalmol. Visual Sci. Suppl. 22, 143 (1982).
  20. A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,”NASA Tech. Memo. 84352 (NASA, Washington, D.C., 1983).
  21. J. P. H. van Santen, G. Sperling, “Elaborated Reichardt detectors,” J. Opt. Soc. Am. A 2, 300–321 (1985).
    [CrossRef] [PubMed]
  22. E. H. Adelson, J. R. Bergen, “Spatiotemporal energy models for the perception of motion,” J. Opt. Soc. Am. A 2, 284–299 (1985).
    [CrossRef] [PubMed]
  23. M. Potmesil, I. Chakravarty, “Modeling motion blur in computer generated images,” Comput. Graphics 17, 389–399 (1983).
    [CrossRef]
  24. J. Korein, N. Badler, “Temporal anti-aliasing in computer generated animation,” Comput. Graphics 17, 377–388 (1983).
    [CrossRef]
  25. R. Cook, T. Porter, L. Carpenter, “Distributed ray tracing,” Comput. Graphics 18, 137–145 (1984).
    [CrossRef]
  26. M. A. Z. Dippe, E. H. Wold, “Antialiasing through stochastic sampling,” Comput. Graphics 19, 69–78 (1985).
    [CrossRef]
  27. 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. Paper 2211 (NASA, Washington, D.C., 1983).

1985 (6)

1984 (1)

R. Cook, T. Porter, L. Carpenter, “Distributed ray tracing,” Comput. Graphics 18, 137–145 (1984).
[CrossRef]

1983 (2)

M. Potmesil, I. Chakravarty, “Modeling motion blur in computer generated images,” Comput. Graphics 17, 389–399 (1983).
[CrossRef]

J. Korein, N. Badler, “Temporal anti-aliasing in computer generated animation,” Comput. Graphics 17, 377–388 (1983).
[CrossRef]

1982 (2)

A. B. Watson, A. J. Ahumada, “A theory of apparently real motion,” Invest. Ophthalmol. Visual Sci. Suppl. 22, 143 (1982).

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

1981 (1)

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

1980 (2)

M. J. Morgan, “Analogue models of motion perception,” Phil. Trans. R. Soc. London Ser. B 290, 117–135 (1980).
[CrossRef]

M. J. Morgan, “Spatiotemporal filtering and the interpolation effect in apparent motion,” Perception 9, 161–174 (1980).
[CrossRef] [PubMed]

1979 (2)

M. J. Morgan, “Perception of continuity in stroboscopic motion: a temporal frequency analysis,” Vision Res. 19, 491–500 (1979).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, “Spatiotemporal contrast detection threshold surface is bimodal,” Opt. Lett. 4, 32–34 (1979).
[CrossRef] [PubMed]

1976 (1)

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

1974 (1)

O. Braddick, “A short-range process in apparent motion,” Vision Res. 14, 519–527 (1974).
[CrossRef] [PubMed]

1966 (1)

1956 (1)

1954 (1)

1875 (1)

S. Exner, “Ueber das Sehen von Bewegungen und die Theorie des zusammengesetzen Auges,” Sitzungber. Akad. Wiss. Wien 72, 156–160 (1875).

Adelson, E. H.

Ahumada, A. J.

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

A. J. Ahumada, A. B. Watson, “Equivalent noise model for contrast detection and discrimination,” J. Opt. Soc. Am. A 2, 1133–1139 (1985).
[CrossRef] [PubMed]

K. R. K. Nielsen, A. B. Watson, A. J. Ahumada, “Application of a computable model of human spatial vision to phase discrimination,” J. Opt. Soc. Am. A 2, 1600–1606 (1985).
[CrossRef] [PubMed]

A. B. Watson, A. J. Ahumada, “A theory of apparently real motion,” Invest. Ophthalmol. Visual Sci. Suppl. 22, 143 (1982).

A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,”NASA Tech. Memo. 84352 (NASA, Washington, D.C., 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. Paper 2211 (NASA, Washington, D.C., 1983).

Badler, N.

J. Korein, N. Badler, “Temporal anti-aliasing in computer generated animation,” Comput. Graphics 17, 377–388 (1983).
[CrossRef]

Bergen, J. R.

Braddick, O.

O. Braddick, “A short-range process in apparent motion,” Vision Res. 14, 519–527 (1974).
[CrossRef] [PubMed]

Carpenter, L.

R. Cook, T. Porter, L. Carpenter, “Distributed ray tracing,” Comput. Graphics 18, 137–145 (1984).
[CrossRef]

Chakravarty, I.

M. Potmesil, I. Chakravarty, “Modeling motion blur in computer generated images,” Comput. Graphics 17, 389–399 (1983).
[CrossRef]

Cook, R.

R. Cook, T. Porter, L. Carpenter, “Distributed ray tracing,” Comput. Graphics 18, 137–145 (1984).
[CrossRef]

de Lange, H.

Dippe, M. A. Z.

M. A. Z. Dippe, E. H. Wold, “Antialiasing through stochastic sampling,” Comput. Graphics 19, 69–78 (1985).
[CrossRef]

Exner, S.

S. Exner, “Ueber das Sehen von Bewegungen und die Theorie des zusammengesetzen Auges,” Sitzungber. Akad. Wiss. Wien 72, 156–160 (1875).

Fahle, M.

M. Fahle, 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. Paper 2211 (NASA, Washington, D.C., 1983).

Koenderink, J. J.

Kolers, P. A.

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

Korein, J.

J. Korein, N. Badler, “Temporal anti-aliasing in computer generated animation,” Comput. Graphics 17, 377–388 (1983).
[CrossRef]

Morgan, M. J.

M. J. Morgan, “Analogue models of motion perception,” Phil. Trans. R. Soc. London Ser. B 290, 117–135 (1980).
[CrossRef]

M. J. Morgan, “Spatiotemporal filtering and the interpolation effect in apparent motion,” Perception 9, 161–174 (1980).
[CrossRef] [PubMed]

M. J. Morgan, “Perception of continuity in stroboscopic motion: a temporal frequency analysis,” Vision Res. 19, 491–500 (1979).
[CrossRef] [PubMed]

Nielsen, K. R. K.

Poggio, T.

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

Porter, T.

R. Cook, T. Porter, L. Carpenter, “Distributed ray tracing,” Comput. Graphics 18, 137–145 (1984).
[CrossRef]

Potmesil, M.

M. Potmesil, I. Chakravarty, “Modeling motion blur in computer generated images,” Comput. Graphics 17, 389–399 (1983).
[CrossRef]

Robson, J. G.

Shade, O. H.

Sperling, G.

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

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

van Doorn, A. J.

van Santen, J. P. H.

Watson, A. B.

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

K. R. K. Nielsen, A. B. Watson, A. J. Ahumada, “Application of a computable model of human spatial vision to phase discrimination,” J. Opt. Soc. Am. A 2, 1600–1606 (1985).
[CrossRef] [PubMed]

A. J. Ahumada, A. B. Watson, “Equivalent noise model for contrast detection and discrimination,” J. Opt. Soc. Am. A 2, 1133–1139 (1985).
[CrossRef] [PubMed]

A. B. Watson, A. J. Ahumada, “A theory of apparently real motion,” Invest. Ophthalmol. Visual Sci. Suppl. 22, 143 (1982).

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, “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).

A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,”NASA Tech. Memo. 84352 (NASA, Washington, D.C., 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. Paper 2211 (NASA, Washington, D.C., 1983).

Wold, E. H.

M. A. Z. Dippe, E. H. Wold, “Antialiasing through stochastic sampling,” Comput. Graphics 19, 69–78 (1985).
[CrossRef]

Behavior Res. Methods Instrum. (1)

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

Comput. Graphics (4)

M. Potmesil, I. Chakravarty, “Modeling motion blur in computer generated images,” Comput. Graphics 17, 389–399 (1983).
[CrossRef]

J. Korein, N. Badler, “Temporal anti-aliasing in computer generated animation,” Comput. Graphics 17, 377–388 (1983).
[CrossRef]

R. Cook, T. Porter, L. Carpenter, “Distributed ray tracing,” Comput. Graphics 18, 137–145 (1984).
[CrossRef]

M. A. Z. Dippe, E. H. Wold, “Antialiasing through stochastic sampling,” Comput. Graphics 19, 69–78 (1985).
[CrossRef]

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

A. B. Watson, A. J. Ahumada, “A theory of apparently real motion,” Invest. Ophthalmol. Visual Sci. Suppl. 22, 143 (1982).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Opt. Lett. (1)

Perception (1)

M. J. Morgan, “Spatiotemporal filtering and the interpolation effect in apparent motion,” Perception 9, 161–174 (1980).
[CrossRef] [PubMed]

Phil. Trans. R. Soc. London Ser. B (1)

M. J. Morgan, “Analogue models of motion perception,” Phil. Trans. R. Soc. London Ser. B 290, 117–135 (1980).
[CrossRef]

Proc. R. Soc. London Ser. B (1)

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

Sitzungber. Akad. Wiss. Wien (1)

S. Exner, “Ueber das Sehen von Bewegungen und die Theorie des zusammengesetzen Auges,” Sitzungber. Akad. Wiss. Wien 72, 156–160 (1875).

Vision Res. (3)

O. Braddick, “A short-range process in apparent motion,” Vision Res. 14, 519–527 (1974).
[CrossRef] [PubMed]

M. J. Morgan, “Perception of continuity in stroboscopic motion: a temporal frequency analysis,” Vision Res. 19, 491–500 (1979).
[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]

Other (5)

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]

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

A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,”NASA Tech. Memo. 84352 (NASA, Washington, D.C., 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, A. J. Ahumada, J. Farrell, “The window of visibility: a psychophysical theory of fidelity in time-sampled visual motion displays,”NASA Tech. Paper 2211 (NASA, Washington, D.C., 1983).

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Figures (12)

Fig. 1
Fig. 1

Graphs and spectra of smooth and sampled lines. Points and lines should be viewed as impulses and line impulses projecting out from the page. (a) The distribution of contrast over space and time of a line moving smoothly to the right at a velocity r deg/sec. The distribution is δ(xrt), where δ is the impulse function. (b) Contrast distribution of a sampled version of the moving line. The points indicate the times and positions at which the sample are presented. The distribution is Δ t δ ( x - r t ) Σ n = - δ ( t - n Δ t ). (c) The spatiotemporal-frequency spectrum of the smoothly moving line. To create a smoothly moving line from sinusoidal components we require that all spatial frequencies and their temporal frequencies increase in proportion to the spatial frequency. The spectrum is δ(w + ur), where w is temporal frequency in hertz and u is spatial frequency in cycles per degree. (d) The spectrum of the time sampled moving line is identical to the spectrum in (c), except for the addition of parallel replicas at intervals of ws. The spectrum is Σ n = - δ ( w + u r - n w s ). A similar analysis of spectra of smooth and sampled motion has been provided by. Fahle and Poggio.7

Fig. 2
Fig. 2

Line constructed from sinusoids. (a) Five sinusoids whose peaks superimpose at a point. (b) The result of adding the five sinusoids together and dividing by five.

Fig. 3
Fig. 3

Window of visibility. The shaded region contains combinations of spatial and temporal frequency that are invisible to the human eye. The window is bounded by ul and wl, the limits of spatial and temporal resolution.

Fig. 4
Fig. 4

Boundary condition for identical appearance of smooth and stroboscopic motion. For clarity, only the first spectral replica on the right is shown. It is just touching the corner of the window of visiblity.

Fig. 5
Fig. 5

Critical sampling frequency for stroboscopic motion as a function of velocity for two observers. The straight lines are fitted by eye. The slope (ul) and the intercept (wl) of each line are indicated.

Fig. 6
Fig. 6

Derivation of the frequency spectrum of staircase motion. (a) The stair function is a unit pulse in t multiplied by an impulse in x. (b) The contrast distribution of staircase motion is the convolution of the stair function with the stroboscopic-motion function pictured in Fig. 1(b). (c) Frequency spectrum of the stroboscopic-motion function. (d) Frequency spectrum of the stair function, a sinc function with its first zero at ws.

Fig. 7
Fig. 7

Frequency spectrum of staircase motion, Lz(u, w). The modulus of the spectrum is shown for clarity.

Fig. 8
Fig. 8

Windowed spectra for stroboscopic and staircase motion when the sampling frequency is given by ws = wL + ruL. The stippled region indicates the difference between the two. The support plane is commensurate with the window of visibility.

Fig. 9
Fig. 9

Critical sampling frequency as a function of velocity for staircase and stroboscopic motion. The dashed line is a least-squares fit to the stroboscopic data (Observer ABW: intercept, 33.2 Hz; slope, 17.0 cycles/deg. Observer JEF: intercept, 46.2 Hz; slope, 11.0 cycles/deg.).

Fig. 10
Fig. 10

Boundary conditions for a moving stimulus spatially band limited to below u0 cycles/deg. The slope of the spectrum is −r−1. The first replica is just touching the window of visibility at the point wLu0.

Fig. 11
Fig. 11

Critical sampling frequency for stroboscopically moving gratings. Data are for observer DW.

Fig. 12
Fig. 12

The effect of temporal filtering on the spectrum of a moving image. The band limit of the temporal filter is wf. The temporal filter removes all spatial frequencies above wf/r.

Equations (13)

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

l ( x , t ) = δ ( x - r t ) ,
s ( t ) = Δ t n = - δ ( t - n Δ t ) .
l s ( x , t ) = l ( x , t ) s ( t ) = Δ t δ ( x - r t ) n = - δ ( t - n Δ t ) .
L ( u , w ) = F T x , t [ l ( x , t ) ] = F T x , t [ δ ( x - r t ) ] = F T t exp ( - i 2 π r t u ) = δ ( w + r u ) ,
L s ( u , w ) = F T x , t [ l s ( x , t ) ] = F T x , t [ s ( t ) l ( x , t ) ] = S ( w ) * L ( u , w ) = δ ( w + r u ) * n = - δ ( w - n / Δ t ) = n = - δ ( w + r u - n w s ) .
w c = w l + r u l .
z ( x , t ) = w s u ( t w s ) δ ( x ) ,
l z ( x , t ) = l s ( x , t ) * z ( x , t ) .
L z ( x , t ) = l s ( x , t ) * z ( x , t ) .
Z ( u , w ) = F T x , t [ w s u ( w s T ) δ ( x ) ] = sinc ( w / w s ) .
w c = w l + r u 0 .
w c = w l + r ( w f / r ) = w l + w f .
w c = w l + r min ( u 0 , u l , w f / r ) .

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