Abstract

Recent work establishes that static and dynamic natural images have fractal-like 1/fα spatiotemporal spectra. Artifical textures, with randomized phase spectra, and 1/fα amplitude spectra are also used in studies of texture and noise perception. Influenced by colorimetric principles and motivated by the ubiquity of 1/fα spatial and temporal image spectra, we treat the spatial and temporal frequency exponents as the dimensions characterizing a dynamic texture space, and we characterize two key attributes of this space, the spatiotemporal appearance map and the spatiotemporal discrimination function (a map of MacAdam-like just-noticeable-difference contours).

© 2001 Optical Society of America

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  1. D. B. Judd, G. Wyszecki, Color in Business, Science and Industry, 2nd ed. (Wiley, New York, 1963).
  2. G. Wyzecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).
  3. W. Richards, “Quantifying sensory channels: generalizing colorimetry to spatiotemporal texture, touch and tones,” Sens. Processes 3, 207–229 (1979). In general, Richards found that arbitrary textures could be matched by mixtures of four separated texture primaries and that temporal modulations could be matched by mixtures of three separated flicker primaries.
    [PubMed]
  4. The analogy to colorimetry cannot be pushed too far here. Brill, in an interesting gedanken experiment, showed that under some circumstances Grassman’s additivity law is violated for some textures: M. H. Brill, “Formalizing Grassman’s laws in a generalized colorimetry,” Sens. Processes 3, 370–372 (1979).
    [PubMed]
  5. J. E. Cutting, J. J. Garvin, “Fractal curves and complexity,” Percept. Psychophys. 42, 365–370 (1987).
    [CrossRef] [PubMed]
  6. T. Kumar, P. Zhou, D. A. Glaser, “Comparison of human performance with algorithms for estimating fractal dimension of fractional Brownian statistics,” J. Opt. Soc. Am. A 10, 1136–1146 (1993).
    [CrossRef] [PubMed]
  7. A. Pentland, “Fractal-based description of surfaces,” in Natural Computation, W. Richards, ed. (MIT Press, Cambridge, Mass., 1988), pp. 279–299.
  8. A. P. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661–674 (1984).
    [CrossRef]
  9. D. J. Field, “Relations between the statistics of natural images and the responses of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
    [CrossRef] [PubMed]
  10. D. J. Tolhurst, Y. Tadmor, T. Chou, “The amplitude spectrum of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
    [CrossRef] [PubMed]
  11. V. A. Billock, “Neural acclimation to 1/f spatial frequency spectra in natural images transduced by the human visual system,” Physica D 137, 379–391 (2000).
    [CrossRef]
  12. B. E. Rogowitz, R. F. Voss, “Shape perception and low-dimensional fractal boundaries,” Human Vision and Electronic Imaging: Models, Methods, and Applications, J. P. Allebach, B. E. Rogowitz, eds., Proc. SPIE1249, 387–394 (1990).
    [CrossRef]
  13. R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms for Computer Graphics, R. A. Earnshaw, ed. (Springer, Berlin, 1985), pp. 805–835.
  14. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983).
  15. M. P. Eckert, G. Buchsbaum, A. B. Watson, “Separability of spatiotemporal spectra of image sequences,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1210–1213 (1992).
    [CrossRef]
  16. D. W. Dong, J. J. Atick, “Statistics of time varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
    [CrossRef]
  17. J. H. van Hateren, “Processing of natural time series by the blowfly visual system,” Vision Res. 37, 3407–3416 (1997).
    [CrossRef]
  18. V. A. Billock, G. C. De Guzman, J. A. S. Kelso, “Fractal time and 1/f spectra in dynamic images and human vision,” Physica D 148, 136–146 (2001).
    [CrossRef]
  19. M. Savilli, G. Lecoy, J. P. Nougier, Noise in Physical Systems and 1/f Noise (Elsevier, New York, 1983).
  20. M. S. Keshner, “1/f noise,” Proc. IEEE 70, 212–218 (1982).
    [CrossRef]
  21. Technically, fractals have infinite spectral bandwidths. Purists would designate textures physically obtainable on displays as prefractals or pseudofractals.
  22. K. L. Kelly, “Color designations for lights,” J. Opt. Soc. Am. 33, 627–632 (1943).
    [CrossRef]
  23. P. Keller, “1976-UCS chromaticity diagram with color boundaries,” Proc. Soc. Inf. Disp. 24, 317–321 (1983).
  24. Color appearance maps of Kelly’s CIE 1931 and Keller’s 1976 spaces are available from Photo Research, 3000 N. Hollywood Way, Burbank, California 91505.
  25. B. Moulden, F. Kingdom, L. F. Gatley, “The standard deviation of luminance as a metric for contrast in random-dot images,” Perception 19, 79–101 (1990).
    [CrossRef] [PubMed]
  26. D. C. Knill, D. Field, D. Kersten, “Human discrimination of fractal textures,” J. Opt. Soc. Am. A 7, 1113–1123 (1990).
    [CrossRef] [PubMed]
  27. Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the second-order statistics of natural and synthetic images.” Vision Res. 34, 541–554 (1994).
    [CrossRef] [PubMed]
  28. In the theory of fractional (biased) Brownian motion, the motion bias is quantified by the distance that a biased walker moves from the origin in unit time. A Brownian (random) walker’s distance over unbroken ground is proportional to t and has an amplitude spectrum exponent of 1.0. Greater distance than this is covered if the motion is persistently biased (if bias is perfect, distance is proportional to time), less distance is covered if the bias is antipersistent. B. B. Mandelbrot, J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10, 422–437 (1968).
    [CrossRef]
  29. Over broken ground, a random walker covers less distance from the origin. The case closest to our stimuli that has been studied in statistical physics is Brownian motion on a fractal fractured surface. The temporal frequency spectrum exponent α for this case is 0.5. W. Lehr, J. Machta, M. Nelkin, “Current noise and long time tails in biased disordered random walks,” J. Stat. Phys. 36, 15–29 (1984). This is in accord with our finding that stimuli with exponents below 0.5 appear to move in a jittery fashion, whereas exponents above 0.5 move smoothly.
    [CrossRef]
  30. In an interesting experiment Snippe and Koenderink studied human ability to perceive correlations between members of a row of light sources. Positive correlations between two lights were detected as apparent motion, but the corresponding anticorrelation between the same two lights was undetectable as motion. H. P. Snipe, J. J. Koenderink, “Detection of noise-like luminance functions,” Percept. Psychophys. 55, 28–41 (1994).
    [CrossRef]
  31. One oddity of random motion is that the visual system does not seem to be able to compensate for motion blur for stimuli undergoing Brownian motion. S. N. J. Watamaniuk, “Visual persistence is reduced by fixed-trajectory motion but not random motion,” Perception 21, 791–802 (1992). No one has yet studied deblurring for anticorrelated (antipersistent) motion.
    [CrossRef]
  32. W. D. Wright, “The graphical representation of small color differences,” J. Opt. Soc. Am. 33, 632–636 (1943).
    [CrossRef]
  33. D. L. MacAdam, “Visual sensitivity to color differences in daylight,” J. Opt. Soc. Am. 32, 247–274 (1942).
    [CrossRef]
  34. L. Silberstein, D. L. MacAdam, “The distribution of color matchings around a color center,” J. Opt. Soc. Am. 35, 32–39 (1945).
    [CrossRef]
  35. W. R. J. Brown, D. L. MacAdam, “Visual sensitivities to combined chromaticity and luminance differences,” J. Opt. Soc. Am. 39, 808–834 (1949).
    [CrossRef] [PubMed]
  36. R. F. Voss, J. Clarke, “1/f noise in music: music from 1/f noise,” J. Acoust. Soc. Am. 63, 258–263 (1978).
    [CrossRef]

2001 (1)

V. A. Billock, G. C. De Guzman, J. A. S. Kelso, “Fractal time and 1/f spectra in dynamic images and human vision,” Physica D 148, 136–146 (2001).
[CrossRef]

2000 (1)

V. A. Billock, “Neural acclimation to 1/f spatial frequency spectra in natural images transduced by the human visual system,” Physica D 137, 379–391 (2000).
[CrossRef]

1997 (1)

J. H. van Hateren, “Processing of natural time series by the blowfly visual system,” Vision Res. 37, 3407–3416 (1997).
[CrossRef]

1995 (1)

D. W. Dong, J. J. Atick, “Statistics of time varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
[CrossRef]

1994 (2)

Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the second-order statistics of natural and synthetic images.” Vision Res. 34, 541–554 (1994).
[CrossRef] [PubMed]

In an interesting experiment Snippe and Koenderink studied human ability to perceive correlations between members of a row of light sources. Positive correlations between two lights were detected as apparent motion, but the corresponding anticorrelation between the same two lights was undetectable as motion. H. P. Snipe, J. J. Koenderink, “Detection of noise-like luminance functions,” Percept. Psychophys. 55, 28–41 (1994).
[CrossRef]

1993 (1)

1992 (3)

One oddity of random motion is that the visual system does not seem to be able to compensate for motion blur for stimuli undergoing Brownian motion. S. N. J. Watamaniuk, “Visual persistence is reduced by fixed-trajectory motion but not random motion,” Perception 21, 791–802 (1992). No one has yet studied deblurring for anticorrelated (antipersistent) motion.
[CrossRef]

D. J. Tolhurst, Y. Tadmor, T. Chou, “The amplitude spectrum of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

M. P. Eckert, G. Buchsbaum, A. B. Watson, “Separability of spatiotemporal spectra of image sequences,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1210–1213 (1992).
[CrossRef]

1990 (2)

B. Moulden, F. Kingdom, L. F. Gatley, “The standard deviation of luminance as a metric for contrast in random-dot images,” Perception 19, 79–101 (1990).
[CrossRef] [PubMed]

D. C. Knill, D. Field, D. Kersten, “Human discrimination of fractal textures,” J. Opt. Soc. Am. A 7, 1113–1123 (1990).
[CrossRef] [PubMed]

1987 (2)

1984 (2)

A. P. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661–674 (1984).
[CrossRef]

Over broken ground, a random walker covers less distance from the origin. The case closest to our stimuli that has been studied in statistical physics is Brownian motion on a fractal fractured surface. The temporal frequency spectrum exponent α for this case is 0.5. W. Lehr, J. Machta, M. Nelkin, “Current noise and long time tails in biased disordered random walks,” J. Stat. Phys. 36, 15–29 (1984). This is in accord with our finding that stimuli with exponents below 0.5 appear to move in a jittery fashion, whereas exponents above 0.5 move smoothly.
[CrossRef]

1983 (1)

P. Keller, “1976-UCS chromaticity diagram with color boundaries,” Proc. Soc. Inf. Disp. 24, 317–321 (1983).

1982 (1)

M. S. Keshner, “1/f noise,” Proc. IEEE 70, 212–218 (1982).
[CrossRef]

1979 (2)

W. Richards, “Quantifying sensory channels: generalizing colorimetry to spatiotemporal texture, touch and tones,” Sens. Processes 3, 207–229 (1979). In general, Richards found that arbitrary textures could be matched by mixtures of four separated texture primaries and that temporal modulations could be matched by mixtures of three separated flicker primaries.
[PubMed]

The analogy to colorimetry cannot be pushed too far here. Brill, in an interesting gedanken experiment, showed that under some circumstances Grassman’s additivity law is violated for some textures: M. H. Brill, “Formalizing Grassman’s laws in a generalized colorimetry,” Sens. Processes 3, 370–372 (1979).
[PubMed]

1978 (1)

R. F. Voss, J. Clarke, “1/f noise in music: music from 1/f noise,” J. Acoust. Soc. Am. 63, 258–263 (1978).
[CrossRef]

1968 (1)

In the theory of fractional (biased) Brownian motion, the motion bias is quantified by the distance that a biased walker moves from the origin in unit time. A Brownian (random) walker’s distance over unbroken ground is proportional to t and has an amplitude spectrum exponent of 1.0. Greater distance than this is covered if the motion is persistently biased (if bias is perfect, distance is proportional to time), less distance is covered if the bias is antipersistent. B. B. Mandelbrot, J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10, 422–437 (1968).
[CrossRef]

1949 (1)

1945 (1)

1943 (2)

1942 (1)

Atick, J. J.

D. W. Dong, J. J. Atick, “Statistics of time varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
[CrossRef]

Billock, V. A.

V. A. Billock, G. C. De Guzman, J. A. S. Kelso, “Fractal time and 1/f spectra in dynamic images and human vision,” Physica D 148, 136–146 (2001).
[CrossRef]

V. A. Billock, “Neural acclimation to 1/f spatial frequency spectra in natural images transduced by the human visual system,” Physica D 137, 379–391 (2000).
[CrossRef]

Brill, M. H.

The analogy to colorimetry cannot be pushed too far here. Brill, in an interesting gedanken experiment, showed that under some circumstances Grassman’s additivity law is violated for some textures: M. H. Brill, “Formalizing Grassman’s laws in a generalized colorimetry,” Sens. Processes 3, 370–372 (1979).
[PubMed]

Brown, W. R. J.

Buchsbaum, G.

M. P. Eckert, G. Buchsbaum, A. B. Watson, “Separability of spatiotemporal spectra of image sequences,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1210–1213 (1992).
[CrossRef]

Chou, T.

D. J. Tolhurst, Y. Tadmor, T. Chou, “The amplitude spectrum of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

Clarke, J.

R. F. Voss, J. Clarke, “1/f noise in music: music from 1/f noise,” J. Acoust. Soc. Am. 63, 258–263 (1978).
[CrossRef]

Cutting, J. E.

J. E. Cutting, J. J. Garvin, “Fractal curves and complexity,” Percept. Psychophys. 42, 365–370 (1987).
[CrossRef] [PubMed]

De Guzman, G. C.

V. A. Billock, G. C. De Guzman, J. A. S. Kelso, “Fractal time and 1/f spectra in dynamic images and human vision,” Physica D 148, 136–146 (2001).
[CrossRef]

Dong, D. W.

D. W. Dong, J. J. Atick, “Statistics of time varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
[CrossRef]

Eckert, M. P.

M. P. Eckert, G. Buchsbaum, A. B. Watson, “Separability of spatiotemporal spectra of image sequences,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1210–1213 (1992).
[CrossRef]

Field, D.

Field, D. J.

Garvin, J. J.

J. E. Cutting, J. J. Garvin, “Fractal curves and complexity,” Percept. Psychophys. 42, 365–370 (1987).
[CrossRef] [PubMed]

Gatley, L. F.

B. Moulden, F. Kingdom, L. F. Gatley, “The standard deviation of luminance as a metric for contrast in random-dot images,” Perception 19, 79–101 (1990).
[CrossRef] [PubMed]

Glaser, D. A.

Judd, D. B.

D. B. Judd, G. Wyszecki, Color in Business, Science and Industry, 2nd ed. (Wiley, New York, 1963).

Keller, P.

P. Keller, “1976-UCS chromaticity diagram with color boundaries,” Proc. Soc. Inf. Disp. 24, 317–321 (1983).

Kelly, K. L.

Kelso, J. A. S.

V. A. Billock, G. C. De Guzman, J. A. S. Kelso, “Fractal time and 1/f spectra in dynamic images and human vision,” Physica D 148, 136–146 (2001).
[CrossRef]

Kersten, D.

Keshner, M. S.

M. S. Keshner, “1/f noise,” Proc. IEEE 70, 212–218 (1982).
[CrossRef]

Kingdom, F.

B. Moulden, F. Kingdom, L. F. Gatley, “The standard deviation of luminance as a metric for contrast in random-dot images,” Perception 19, 79–101 (1990).
[CrossRef] [PubMed]

Knill, D. C.

Koenderink, J. J.

In an interesting experiment Snippe and Koenderink studied human ability to perceive correlations between members of a row of light sources. Positive correlations between two lights were detected as apparent motion, but the corresponding anticorrelation between the same two lights was undetectable as motion. H. P. Snipe, J. J. Koenderink, “Detection of noise-like luminance functions,” Percept. Psychophys. 55, 28–41 (1994).
[CrossRef]

Kumar, T.

Lecoy, G.

M. Savilli, G. Lecoy, J. P. Nougier, Noise in Physical Systems and 1/f Noise (Elsevier, New York, 1983).

Lehr, W.

Over broken ground, a random walker covers less distance from the origin. The case closest to our stimuli that has been studied in statistical physics is Brownian motion on a fractal fractured surface. The temporal frequency spectrum exponent α for this case is 0.5. W. Lehr, J. Machta, M. Nelkin, “Current noise and long time tails in biased disordered random walks,” J. Stat. Phys. 36, 15–29 (1984). This is in accord with our finding that stimuli with exponents below 0.5 appear to move in a jittery fashion, whereas exponents above 0.5 move smoothly.
[CrossRef]

MacAdam, D. L.

Machta, J.

Over broken ground, a random walker covers less distance from the origin. The case closest to our stimuli that has been studied in statistical physics is Brownian motion on a fractal fractured surface. The temporal frequency spectrum exponent α for this case is 0.5. W. Lehr, J. Machta, M. Nelkin, “Current noise and long time tails in biased disordered random walks,” J. Stat. Phys. 36, 15–29 (1984). This is in accord with our finding that stimuli with exponents below 0.5 appear to move in a jittery fashion, whereas exponents above 0.5 move smoothly.
[CrossRef]

Mandelbrot, B. B.

In the theory of fractional (biased) Brownian motion, the motion bias is quantified by the distance that a biased walker moves from the origin in unit time. A Brownian (random) walker’s distance over unbroken ground is proportional to t and has an amplitude spectrum exponent of 1.0. Greater distance than this is covered if the motion is persistently biased (if bias is perfect, distance is proportional to time), less distance is covered if the bias is antipersistent. B. B. Mandelbrot, J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10, 422–437 (1968).
[CrossRef]

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983).

Moulden, B.

B. Moulden, F. Kingdom, L. F. Gatley, “The standard deviation of luminance as a metric for contrast in random-dot images,” Perception 19, 79–101 (1990).
[CrossRef] [PubMed]

Nelkin, M.

Over broken ground, a random walker covers less distance from the origin. The case closest to our stimuli that has been studied in statistical physics is Brownian motion on a fractal fractured surface. The temporal frequency spectrum exponent α for this case is 0.5. W. Lehr, J. Machta, M. Nelkin, “Current noise and long time tails in biased disordered random walks,” J. Stat. Phys. 36, 15–29 (1984). This is in accord with our finding that stimuli with exponents below 0.5 appear to move in a jittery fashion, whereas exponents above 0.5 move smoothly.
[CrossRef]

Nougier, J. P.

M. Savilli, G. Lecoy, J. P. Nougier, Noise in Physical Systems and 1/f Noise (Elsevier, New York, 1983).

Pentland, A.

A. Pentland, “Fractal-based description of surfaces,” in Natural Computation, W. Richards, ed. (MIT Press, Cambridge, Mass., 1988), pp. 279–299.

Pentland, A. P.

A. P. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661–674 (1984).
[CrossRef]

Richards, W.

W. Richards, “Quantifying sensory channels: generalizing colorimetry to spatiotemporal texture, touch and tones,” Sens. Processes 3, 207–229 (1979). In general, Richards found that arbitrary textures could be matched by mixtures of four separated texture primaries and that temporal modulations could be matched by mixtures of three separated flicker primaries.
[PubMed]

Rogowitz, B. E.

B. E. Rogowitz, R. F. Voss, “Shape perception and low-dimensional fractal boundaries,” Human Vision and Electronic Imaging: Models, Methods, and Applications, J. P. Allebach, B. E. Rogowitz, eds., Proc. SPIE1249, 387–394 (1990).
[CrossRef]

Savilli, M.

M. Savilli, G. Lecoy, J. P. Nougier, Noise in Physical Systems and 1/f Noise (Elsevier, New York, 1983).

Silberstein, L.

Snipe, H. P.

In an interesting experiment Snippe and Koenderink studied human ability to perceive correlations between members of a row of light sources. Positive correlations between two lights were detected as apparent motion, but the corresponding anticorrelation between the same two lights was undetectable as motion. H. P. Snipe, J. J. Koenderink, “Detection of noise-like luminance functions,” Percept. Psychophys. 55, 28–41 (1994).
[CrossRef]

Stiles, W. S.

G. Wyzecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

Tadmor, Y.

Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the second-order statistics of natural and synthetic images.” Vision Res. 34, 541–554 (1994).
[CrossRef] [PubMed]

D. J. Tolhurst, Y. Tadmor, T. Chou, “The amplitude spectrum of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

Tolhurst, D. J.

Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the second-order statistics of natural and synthetic images.” Vision Res. 34, 541–554 (1994).
[CrossRef] [PubMed]

D. J. Tolhurst, Y. Tadmor, T. Chou, “The amplitude spectrum of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

van Hateren, J. H.

J. H. van Hateren, “Processing of natural time series by the blowfly visual system,” Vision Res. 37, 3407–3416 (1997).
[CrossRef]

van Ness, J. W.

In the theory of fractional (biased) Brownian motion, the motion bias is quantified by the distance that a biased walker moves from the origin in unit time. A Brownian (random) walker’s distance over unbroken ground is proportional to t and has an amplitude spectrum exponent of 1.0. Greater distance than this is covered if the motion is persistently biased (if bias is perfect, distance is proportional to time), less distance is covered if the bias is antipersistent. B. B. Mandelbrot, J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10, 422–437 (1968).
[CrossRef]

Voss, R. F.

R. F. Voss, J. Clarke, “1/f noise in music: music from 1/f noise,” J. Acoust. Soc. Am. 63, 258–263 (1978).
[CrossRef]

B. E. Rogowitz, R. F. Voss, “Shape perception and low-dimensional fractal boundaries,” Human Vision and Electronic Imaging: Models, Methods, and Applications, J. P. Allebach, B. E. Rogowitz, eds., Proc. SPIE1249, 387–394 (1990).
[CrossRef]

R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms for Computer Graphics, R. A. Earnshaw, ed. (Springer, Berlin, 1985), pp. 805–835.

Watamaniuk, S. N. J.

One oddity of random motion is that the visual system does not seem to be able to compensate for motion blur for stimuli undergoing Brownian motion. S. N. J. Watamaniuk, “Visual persistence is reduced by fixed-trajectory motion but not random motion,” Perception 21, 791–802 (1992). No one has yet studied deblurring for anticorrelated (antipersistent) motion.
[CrossRef]

Watson, A. B.

M. P. Eckert, G. Buchsbaum, A. B. Watson, “Separability of spatiotemporal spectra of image sequences,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1210–1213 (1992).
[CrossRef]

Wright, W. D.

Wyszecki, G.

D. B. Judd, G. Wyszecki, Color in Business, Science and Industry, 2nd ed. (Wiley, New York, 1963).

Wyzecki, G.

G. Wyzecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

Zhou, P.

IEEE Trans. Pattern Anal. Mach. Intell. (2)

M. P. Eckert, G. Buchsbaum, A. B. Watson, “Separability of spatiotemporal spectra of image sequences,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1210–1213 (1992).
[CrossRef]

A. P. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661–674 (1984).
[CrossRef]

J. Acoust. Soc. Am. (1)

R. F. Voss, J. Clarke, “1/f noise in music: music from 1/f noise,” J. Acoust. Soc. Am. 63, 258–263 (1978).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

J. Stat. Phys. (1)

Over broken ground, a random walker covers less distance from the origin. The case closest to our stimuli that has been studied in statistical physics is Brownian motion on a fractal fractured surface. The temporal frequency spectrum exponent α for this case is 0.5. W. Lehr, J. Machta, M. Nelkin, “Current noise and long time tails in biased disordered random walks,” J. Stat. Phys. 36, 15–29 (1984). This is in accord with our finding that stimuli with exponents below 0.5 appear to move in a jittery fashion, whereas exponents above 0.5 move smoothly.
[CrossRef]

Network Comput. Neural Syst. (1)

D. W. Dong, J. J. Atick, “Statistics of time varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
[CrossRef]

Ophthalmic Physiol. Opt. (1)

D. J. Tolhurst, Y. Tadmor, T. Chou, “The amplitude spectrum of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

Percept. Psychophys. (2)

J. E. Cutting, J. J. Garvin, “Fractal curves and complexity,” Percept. Psychophys. 42, 365–370 (1987).
[CrossRef] [PubMed]

In an interesting experiment Snippe and Koenderink studied human ability to perceive correlations between members of a row of light sources. Positive correlations between two lights were detected as apparent motion, but the corresponding anticorrelation between the same two lights was undetectable as motion. H. P. Snipe, J. J. Koenderink, “Detection of noise-like luminance functions,” Percept. Psychophys. 55, 28–41 (1994).
[CrossRef]

Perception (2)

One oddity of random motion is that the visual system does not seem to be able to compensate for motion blur for stimuli undergoing Brownian motion. S. N. J. Watamaniuk, “Visual persistence is reduced by fixed-trajectory motion but not random motion,” Perception 21, 791–802 (1992). No one has yet studied deblurring for anticorrelated (antipersistent) motion.
[CrossRef]

B. Moulden, F. Kingdom, L. F. Gatley, “The standard deviation of luminance as a metric for contrast in random-dot images,” Perception 19, 79–101 (1990).
[CrossRef] [PubMed]

Physica D (2)

V. A. Billock, “Neural acclimation to 1/f spatial frequency spectra in natural images transduced by the human visual system,” Physica D 137, 379–391 (2000).
[CrossRef]

V. A. Billock, G. C. De Guzman, J. A. S. Kelso, “Fractal time and 1/f spectra in dynamic images and human vision,” Physica D 148, 136–146 (2001).
[CrossRef]

Proc. IEEE (1)

M. S. Keshner, “1/f noise,” Proc. IEEE 70, 212–218 (1982).
[CrossRef]

Proc. Soc. Inf. Disp. (1)

P. Keller, “1976-UCS chromaticity diagram with color boundaries,” Proc. Soc. Inf. Disp. 24, 317–321 (1983).

Sens. Processes (2)

W. Richards, “Quantifying sensory channels: generalizing colorimetry to spatiotemporal texture, touch and tones,” Sens. Processes 3, 207–229 (1979). In general, Richards found that arbitrary textures could be matched by mixtures of four separated texture primaries and that temporal modulations could be matched by mixtures of three separated flicker primaries.
[PubMed]

The analogy to colorimetry cannot be pushed too far here. Brill, in an interesting gedanken experiment, showed that under some circumstances Grassman’s additivity law is violated for some textures: M. H. Brill, “Formalizing Grassman’s laws in a generalized colorimetry,” Sens. Processes 3, 370–372 (1979).
[PubMed]

SIAM Rev. (1)

In the theory of fractional (biased) Brownian motion, the motion bias is quantified by the distance that a biased walker moves from the origin in unit time. A Brownian (random) walker’s distance over unbroken ground is proportional to t and has an amplitude spectrum exponent of 1.0. Greater distance than this is covered if the motion is persistently biased (if bias is perfect, distance is proportional to time), less distance is covered if the bias is antipersistent. B. B. Mandelbrot, J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10, 422–437 (1968).
[CrossRef]

Vision Res. (2)

Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the second-order statistics of natural and synthetic images.” Vision Res. 34, 541–554 (1994).
[CrossRef] [PubMed]

J. H. van Hateren, “Processing of natural time series by the blowfly visual system,” Vision Res. 37, 3407–3416 (1997).
[CrossRef]

Other (9)

M. Savilli, G. Lecoy, J. P. Nougier, Noise in Physical Systems and 1/f Noise (Elsevier, New York, 1983).

B. E. Rogowitz, R. F. Voss, “Shape perception and low-dimensional fractal boundaries,” Human Vision and Electronic Imaging: Models, Methods, and Applications, J. P. Allebach, B. E. Rogowitz, eds., Proc. SPIE1249, 387–394 (1990).
[CrossRef]

R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms for Computer Graphics, R. A. Earnshaw, ed. (Springer, Berlin, 1985), pp. 805–835.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983).

Color appearance maps of Kelly’s CIE 1931 and Keller’s 1976 spaces are available from Photo Research, 3000 N. Hollywood Way, Burbank, California 91505.

Technically, fractals have infinite spectral bandwidths. Purists would designate textures physically obtainable on displays as prefractals or pseudofractals.

D. B. Judd, G. Wyszecki, Color in Business, Science and Industry, 2nd ed. (Wiley, New York, 1963).

G. Wyzecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

A. Pentland, “Fractal-based description of surfaces,” in Natural Computation, W. Richards, ed. (MIT Press, Cambridge, Mass., 1988), pp. 279–299.

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Figures (9)

Fig. 1
Fig. 1

Appearance map of CIE 1931 color space. Reproduced from Ref. 22 by permission of the Optical Society of America.

Fig. 2
Fig. 2

Appearance map of spatiotemporal fractal space. The appearance map is based on the responses of four observers. To help render the chart more readable to a diverse audience, the real-world descriptions provided by the subjects (which were highly individualistic and culturally based) have been replaced where possible with stimulus-based descriptions provided by the other subjects.

Fig. 3
Fig. 3

Static, random phase, fractal-like textures produced by 1/fβ spatial-frequency filtering of random white noise. Static snapshots of five of the ten fractal exponents (for one seed value of the random number generator) are shown here. From top to bottom, the values of the exponent β are 0.4, 0.8, 1.2, 1.6, 2.0. Note that as the spatial exponent increases, the apparent coarseness of the texture also increases.

Fig. 4
Fig. 4

MacAdam’s ellipses.33 Discrimination thresholds (JNDs) for various directions in CIE 1931 color space. Ellipses are for MacAdam’s observer PGN and are shown at ten times their actual size. Reproduced by permission of the Optical Society of America.

Fig. 5
Fig. 5

Wright–MacAdam-like contours for discrimination thresholds in four directions in a spatiotemporal fractal space. Average of four observers. The contours are simple quarter-circles fitted to the JNDs that span their quadrants. These contours are not theoretically motivated and are used to reduce the confusion that occurs for overlapping JND crosses. Note that over a wide range of the space the discrimination thresholds vary only slightly; e.g., the space approaches the ideal of a uniform colorimetric space (such as the CIE 1976 uniform chromaticity space).1,2

Fig. 6
Fig. 6

Spatial JND’s collapsed across temporal exponents. Here the spatial discriminations are plotted as a function of spatial exponent. Each subfigure is for an observer named by his initials.

Fig. 7
Fig. 7

Spatial JND’s collapsed across spatial exponents. Here the spatial discriminations are plotted as a function of temporal exponent. The temporal exponent labeled “0” is a place holder for the static condition. Observers are identified by initials.

Fig. 8
Fig. 8

Temporal JND’s collapsed across spatial exponents. Here the temporal discriminations are plotted as a function of temporal exponent. Each subfigure is for an observer named by his initials.

Fig. 9
Fig. 9

Temporal JND’s collapsed across temporal exponents. Here the temporal discriminations are plotted as a function of spatial exponent. Each subfigure is for an observer named by his initials.

Equations (1)

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A(fs, ft)=Kfs-βft-α.

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