OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 18, Iss. 10 — Oct. 1, 2001
  • pp: 2404–2413

Perception of spatiotemporal random fractals: an extension of colorimetric methods to the study of dynamic texture

Vincent A. Billock, Douglas W. Cunningham, Paul R. Havig, and Brian H. Tsou  »View Author Affiliations

JOSA A, Vol. 18, Issue 10, pp. 2404-2413 (2001)

View Full Text Article

Acrobat PDF (316 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(100.6740) Image processing : Synthetic discrimination functions
(330.1730) Vision, color, and visual optics : Colorimetry
(330.5510) Vision, color, and visual optics : Psychophysics
(330.6100) Vision, color, and visual optics : Spatial discrimination
(330.6110) Vision, color, and visual optics : Spatial filtering
(330.6790) Vision, color, and visual optics : Temporal discrimination

Vincent A. Billock, Douglas W. Cunningham, Paul R. Havig, and Brian H. Tsou, "Perception of spatiotemporal random fractals: an extension of colorimetric methods to the study of dynamic texture," J. Opt. Soc. Am. A 18, 2404-2413 (2001)

Sort:  Author  |  Year  |  Journal  |  Reset


  1. D. B. Judd and G. Wyszecki, Color in Business, Science and Industry, 2nd ed. (Wiley, New York, 1963).
  2. G. Wyzecki and 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.
  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).
  5. J. E. Cutting and J. J. Garvin, “Fractal curves and complexity,” Percept. Psychophys. 42, 365–370 (1987).
  6. T. Kumar, P. Zhou, and 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).
  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).
  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).
  10. D. J. Tolhurst, Y. Tadmor, and T. Chou, “The amplitude spectrum of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
  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).
  12. B. E. Rogowitz and R. F. Voss, “Shape perception and low-dimensional fractal boundaries,” Human Vision and Electronic Imaging: Models, Methods, and Applications, J. P. Allebach and B. E. Rogowitz, eds., Proc. SPIE 1249, 387–394 (1990).
  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, and A. B. Watson, “Separability of spatiotemporal spectra of image sequences,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1210–1213 (1992).
  16. D. W. Dong and J. J. Atick, “Statistics of time varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
  17. J. H. van Hateren, “Processing of natural time series by the blowfly visual system,” Vision Res. 37, 3407–3416 (1997).
  18. V. A. Billock, G. C. De Guzman, and J. A. S. Kelso, “Fractal time and 1/f spectra in dynamic images and human vision,” Physica D 148, 136–146 (2001).
  19. M. Savilli, G. Lecoy, and 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).
  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).
  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, and L. F. Gatley, “The standard deviation of luminance as a metric for contrast in random-dot images,” Perception 19, 79–101 (1990).
  26. D. C. Knill, D. Field, and D. Kersten, “Human discrimination of fractal textures,” J. Opt. Soc. Am. A 7, 1113–1123 (1990).
  27. Y. Tadmor and D. J. Tolhurst, “Discrimination of changes in the second-order statistics of natural and synthetic images.” Vision Res. 34, 541–554 (1994).
  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 and J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Rev. 10, 422–437 (1968).
  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, and 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.
  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 and J. J. Koenderink, “Detection of noise-like luminance functions,” Percept. Psychophys. 55, 28–41 (1994).
  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. Watama-niuk, “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.
  32. W. D. Wright, “The graphical representation of small color differences,” J. Opt. Soc. Am. 33, 632–636 (1943).
  33. D. L. MacAdam, “Visual sensitivity to color differences in daylight,” J. Opt. Soc. Am. 32, 247–274 (1942).
  34. L. Silberstein and D. L. MacAdam, “The distribution of color matchings around a color center,” J. Opt. Soc. Am. 35, 32–39 (1945).
  35. W. R. J. Brown and D. L. MacAdam, “Visual sensitivities to combined chromaticity and luminance differences,” J. Opt. Soc. Am. 39, 808–834 (1949).
  36. R. F. Voss and J. Clarke, “1/f noise in music: music from 1/f noise,” J. Acoust. Soc. Am. 63, 258–263 (1978).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited