Abstract

In order to transmit information in images efficiently, the visual system should be tuned to the statistical structure of the ensemble of images that it sees. Several authors have suggested that the ensemble of natural images exhibits fractal behavior and, therefore, has a power spectrum that drops off proportionally to 1/fβ(2 < β < 4). In this paper we investigate the question of which value of the exponent β describes the power spectrum of the ensemble of images to which the visual system is optimally tuned. An experiment in which subjects were asked to discriminate randomly generated noise textures based on their spectral drop-off was used. Whereas the discrimination-threshold function of an ideal observer was flat for different spectral drop-offs, human observers showed a broad peak in sensitivity for 2.8 < β < 3.6. The results are consistent with, but do not provide direct evidence for, the theory that the visual system is tuned to an ensemble of images with Markov statistics.

© 1990 Optical Society of America

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  1. F. Attneave, “Informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
    [CrossRef] [PubMed]
  2. H. B. Barlow, “Sensory mechanisms, the reduction of redundancy and intelligence,” NPL Symposium on the Mechanization of Thought Processes, No. 10 (H. M. Stationary Office, London, 1959), pp. 535–539.
  3. H. B. Barlow, “The coding of sensory messages,” in Current Problems in Animal Behavior, W. H. Thorpe, O. L. Zangwill, eds. (Cambridge U. Press, Cambridge, 1961), pp. 331–360.
  4. H. B. Barlow, “The Ferrier lecture: critical limiting factors in the design of the eye and visual cortex,” Proc. R. Soc. London Ser. B 212, 1–34 (1981).
    [CrossRef]
  5. M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London Ser. B 216, 427–459 (1982).
    [CrossRef]
  6. T. Bossomaier, A. W. Synder, “Why spatial frequency processing in the visual cortex?” Vision Res. 26, 1307–1309 (1986).
    [CrossRef] [PubMed]
  7. D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
    [CrossRef] [PubMed]
  8. R. Linsker, “Self-organization in a perceptual network,” IEEE Trans. Comput. 21, 105–117 (1988).
  9. D. M. Kammen, A. L. Yuille, “Spontaneous symmetry-breaking energy functions and the emergence of orientation selective cortical cells,” Biol. Cybern. 59, 23–31 (1988).
    [CrossRef] [PubMed]
  10. H. B. Barlow, P. Z. Foldiak, “Adaptation and decorrelation in the cortex,” in The Computing Neuron, R. C. Miall, R. M. Durbin, G. J. Mitchison, eds. (Addison-Wesley, Reading, Mass., 1989).
  11. C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Champaign, Ill, 1949).
  12. Stationarity implies that the statistics of a random field are invariant over translations of the coordinate system on which it is defined. Isotropy implies that they are also invariant over rotations of the coordinate system. One result of these two assumptions is that the autocorrelation function can be expressed as a function of Euclidean distance. The assumption of stationarity is intuitively attractive, as it can result from viewing scenes from a range of positions, so that shifted versions of any given image are equally likely. A similar argument, however, cannot be made for the isotropy assumption, as we generally view scenes with our heads perpendicular to the ground. In a study related to the question of isotropy, Switkes et al.13 found more power at horizontal and vertical orientations in images of both natural and synthetic scenes. The assumption does, however, simplify our investigation by allowing us to look at the correlational structure of images as a function only of distance between points.
  13. E. Switkes, M. J. Mayer, J. A. Sloan, “Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis,” Vision Res. 18, 1393–1399 (1978).
    [CrossRef] [PubMed]
  14. The autocorrelation function of a random field I is given byRI((xi,yi),(xj,yj))=E[I(xi,yi)I(xj,yj)],where E[·] is the expectation operator. Since we are assuming that I is stationary and isotropic, we can write this as a function of the Euclidean distance between points:RI(Δr)=E[I(x,y)I(x+Δrcosθ,y+Δrsinθ)],where r= (Δx2+ Δy2)1/2and θ is the angle between the points. The second-order statistical structure of a stationary ensemble of images, given by the autocorrelation function in space, is given by the power spectrum in the frequency domain. The power spectrum is the Fourier transform of the autocorrelation functionPI(fx,fy)=∫0∞∫0∞RI(x,y)exp[i2π(fx+fy)]dxdy.The power at a given frequency is twice the variance of the corresponding Fourier coefficients (real and imaginary) of images in an ensemble. The real and imaginary parts of the Fourier coefficient at a given frequency are uncorrelated and have equal variance. The power at fx = fy = 0 is the squared mean of the ensemble. For an isotropic ensemble, the power spectrum may be written as a function of radial spatial frequency, PI(fr), where fr= (fx2 + fy2)1/2.
  15. B. Julesz, “Spatial frequency channels in one-, two- and three-dimensional vision: variations on an auditory theme by Bekesy,” in Vision Coding and Adaptability, C. S. Harris, ed. (Erlbaum, Hillside, N.J., 1980).
  16. B. B. Mandelbrot, Fractals: Form, Chance, and Dimension (Freeman, San Francisco, Calif., 1977).
  17. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).
  18. A. P. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Anal. Mech. Intell. PAMI-6, 661–673 (1984).
    [CrossRef]
  19. The power spectrum of an ensemble with an exponential auto-correlation function clearly shows the effect of the scale constant k. For an isotropic ensemble, the spectrum is given byPI(fr)∝1(k+fr3/2)2.At frequencies much lower than k, the spectrum approximates white noise; that is, points in the image separated by a distance much greater than 1/k are effectively uncorrelated. For frequencies much greater than k, the spectrum falls off according to the power law 1/fr3. The images of this ensemble exhibit qualitatively different statistical behavior at different scales.
  20. B. Julesz, “Visual pattern discrimination,” IRE Trans. Inf. Theory IT-8, 84–92 (1962).
    [CrossRef]
  21. B. Julesz, E. Gilbert, L. Shepp, H. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics revisited,” Perception 2, 391–405 (1973).
    [CrossRef]
  22. B. Julesz, J. Bergen, “Textons, the fundamental elements in preattentive vision and perception of texture,” Bell Syst. Tech. J. 62, 619–1645 (1983).
  23. W. K. Pratt, O. D. Faugeras, A. Gagalowicz, “Visual discrimination of stochastic texture fields,” IEEE Trans. Syst. Man Cybern. SMC-8, 796–804 (1978).
    [CrossRef]
  24. R. A. Rensink, On the Visual Discrimination of Self-Similar Random Textures, Department of Computer Science Tech. Rep. 86-16 (University of British Columbia, Vancouver, B.C., Canada, 1986).
  25. Rensink generated line textures by using one-dimensional power spectra. Peak performance was found to be at a spectral drop-off of β1D= 3 for these textures. The equivalent two-dimensional spectral drop-off is given by β2D= β1D + 1 = 4.26
  26. R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms for Computer Science, R. A. Earnshaw, ed. (Springer-Verlag, Berlin, 1985), pp. 805–829.
    [CrossRef]
  27. H. B. Barlow, “The efficiency of detecting changes of density in random dot patterns,” Vision Res. 18, 637–650 (1978).
    [CrossRef] [PubMed]
  28. H. B. Barlow, “Measurements of the quantum efficiency of discrimination in human scotopic vision,” J. Physiol. 160, 169–188 (1962).
    [PubMed]
  29. A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature (London) 302, 419–422 (1983).
    [CrossRef]
  30. A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
    [CrossRef] [PubMed]
  31. D. Kersten, “Spatial summation in visual noise,” Vision Res. 24, 1977–1990 (1984).
    [CrossRef] [PubMed]
  32. We corrected for the nonlinearity by raising the entries of the lookup table to an exponent of 0.375 (1/2.67) and rescaling them to give a range of 256 gray levels. The equation used for correcting the lookup-table entries wasl[i]=i0.375*250.00.625,0≤i<256,where l[i] is the ith entry in the lookup table.
  33. The resulting image statistics had toroidal symmetry, reflecting the symmetry of the fast Fourier-transform algorithm.
  34. A. B. Watson, D. G. Pelli, “quest: a Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
    [CrossRef] [PubMed]
  35. W. A. Weibull, “A statistical distribution function of wide applicability,” J. Appl. Mech. 18, 292–297 (1951).
  36. D. Kersten, “Statistical efficiency for the detection of visual noise,” Vision Res. 24, 1977–1990 (1984);Vision Res. 27, 1029–1040 (1987).
    [CrossRef]
  37. H. J. Larson, B. O. Shubert, Probabilistic Models in Engineering Sciences, Vol. 1: Random Variables and Stochastic Processes (Wiley, New York, 1979).
  38. D. Kersten, “Predictability and redundancy of natural images,” J. Opt. Soc. Am. A 4, 2395–2400 (1987).
    [CrossRef] [PubMed]
  39. H. O. Peitgen, P. H. Richter, The Beauty of Fractals (Springers-Verlag, Berlin, 1986).
    [CrossRef]

1988

R. Linsker, “Self-organization in a perceptual network,” IEEE Trans. Comput. 21, 105–117 (1988).

D. M. Kammen, A. L. Yuille, “Spontaneous symmetry-breaking energy functions and the emergence of orientation selective cortical cells,” Biol. Cybern. 59, 23–31 (1988).
[CrossRef] [PubMed]

1987

1986

T. Bossomaier, A. W. Synder, “Why spatial frequency processing in the visual cortex?” Vision Res. 26, 1307–1309 (1986).
[CrossRef] [PubMed]

1984

A. P. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Anal. Mech. Intell. PAMI-6, 661–673 (1984).
[CrossRef]

D. Kersten, “Spatial summation in visual noise,” Vision Res. 24, 1977–1990 (1984).
[CrossRef] [PubMed]

D. Kersten, “Statistical efficiency for the detection of visual noise,” Vision Res. 24, 1977–1990 (1984);Vision Res. 27, 1029–1040 (1987).
[CrossRef]

1983

A. B. Watson, D. G. Pelli, “quest: a Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
[CrossRef] [PubMed]

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature (London) 302, 419–422 (1983).
[CrossRef]

B. Julesz, J. Bergen, “Textons, the fundamental elements in preattentive vision and perception of texture,” Bell Syst. Tech. J. 62, 619–1645 (1983).

1982

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London Ser. B 216, 427–459 (1982).
[CrossRef]

1981

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

H. B. Barlow, “The Ferrier lecture: critical limiting factors in the design of the eye and visual cortex,” Proc. R. Soc. London Ser. B 212, 1–34 (1981).
[CrossRef]

1978

W. K. Pratt, O. D. Faugeras, A. Gagalowicz, “Visual discrimination of stochastic texture fields,” IEEE Trans. Syst. Man Cybern. SMC-8, 796–804 (1978).
[CrossRef]

H. B. Barlow, “The efficiency of detecting changes of density in random dot patterns,” Vision Res. 18, 637–650 (1978).
[CrossRef] [PubMed]

E. Switkes, M. J. Mayer, J. A. Sloan, “Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis,” Vision Res. 18, 1393–1399 (1978).
[CrossRef] [PubMed]

1973

B. Julesz, E. Gilbert, L. Shepp, H. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics revisited,” Perception 2, 391–405 (1973).
[CrossRef]

1962

B. Julesz, “Visual pattern discrimination,” IRE Trans. Inf. Theory IT-8, 84–92 (1962).
[CrossRef]

H. B. Barlow, “Measurements of the quantum efficiency of discrimination in human scotopic vision,” J. Physiol. 160, 169–188 (1962).
[PubMed]

1954

F. Attneave, “Informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

1951

W. A. Weibull, “A statistical distribution function of wide applicability,” J. Appl. Mech. 18, 292–297 (1951).

Attneave, F.

F. Attneave, “Informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

Barlow, H. B.

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature (London) 302, 419–422 (1983).
[CrossRef]

H. B. Barlow, “The Ferrier lecture: critical limiting factors in the design of the eye and visual cortex,” Proc. R. Soc. London Ser. B 212, 1–34 (1981).
[CrossRef]

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

H. B. Barlow, “The efficiency of detecting changes of density in random dot patterns,” Vision Res. 18, 637–650 (1978).
[CrossRef] [PubMed]

H. B. Barlow, “Measurements of the quantum efficiency of discrimination in human scotopic vision,” J. Physiol. 160, 169–188 (1962).
[PubMed]

H. B. Barlow, “The coding of sensory messages,” in Current Problems in Animal Behavior, W. H. Thorpe, O. L. Zangwill, eds. (Cambridge U. Press, Cambridge, 1961), pp. 331–360.

H. B. Barlow, P. Z. Foldiak, “Adaptation and decorrelation in the cortex,” in The Computing Neuron, R. C. Miall, R. M. Durbin, G. J. Mitchison, eds. (Addison-Wesley, Reading, Mass., 1989).

H. B. Barlow, “Sensory mechanisms, the reduction of redundancy and intelligence,” NPL Symposium on the Mechanization of Thought Processes, No. 10 (H. M. Stationary Office, London, 1959), pp. 535–539.

Bergen, J.

B. Julesz, J. Bergen, “Textons, the fundamental elements in preattentive vision and perception of texture,” Bell Syst. Tech. J. 62, 619–1645 (1983).

Bossomaier, T.

T. Bossomaier, A. W. Synder, “Why spatial frequency processing in the visual cortex?” Vision Res. 26, 1307–1309 (1986).
[CrossRef] [PubMed]

Burgess, A. E.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Dubs, A.

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London Ser. B 216, 427–459 (1982).
[CrossRef]

Faugeras, O. D.

W. K. Pratt, O. D. Faugeras, A. Gagalowicz, “Visual discrimination of stochastic texture fields,” IEEE Trans. Syst. Man Cybern. SMC-8, 796–804 (1978).
[CrossRef]

Field, D. J.

Foldiak, P. Z.

H. B. Barlow, P. Z. Foldiak, “Adaptation and decorrelation in the cortex,” in The Computing Neuron, R. C. Miall, R. M. Durbin, G. J. Mitchison, eds. (Addison-Wesley, Reading, Mass., 1989).

Frisch, H.

B. Julesz, E. Gilbert, L. Shepp, H. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics revisited,” Perception 2, 391–405 (1973).
[CrossRef]

Gagalowicz, A.

W. K. Pratt, O. D. Faugeras, A. Gagalowicz, “Visual discrimination of stochastic texture fields,” IEEE Trans. Syst. Man Cybern. SMC-8, 796–804 (1978).
[CrossRef]

Gilbert, E.

B. Julesz, E. Gilbert, L. Shepp, H. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics revisited,” Perception 2, 391–405 (1973).
[CrossRef]

Jennings, R. J.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Julesz, B.

B. Julesz, J. Bergen, “Textons, the fundamental elements in preattentive vision and perception of texture,” Bell Syst. Tech. J. 62, 619–1645 (1983).

B. Julesz, E. Gilbert, L. Shepp, H. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics revisited,” Perception 2, 391–405 (1973).
[CrossRef]

B. Julesz, “Visual pattern discrimination,” IRE Trans. Inf. Theory IT-8, 84–92 (1962).
[CrossRef]

B. Julesz, “Spatial frequency channels in one-, two- and three-dimensional vision: variations on an auditory theme by Bekesy,” in Vision Coding and Adaptability, C. S. Harris, ed. (Erlbaum, Hillside, N.J., 1980).

Kammen, D. M.

D. M. Kammen, A. L. Yuille, “Spontaneous symmetry-breaking energy functions and the emergence of orientation selective cortical cells,” Biol. Cybern. 59, 23–31 (1988).
[CrossRef] [PubMed]

Kersten, D.

D. Kersten, “Predictability and redundancy of natural images,” J. Opt. Soc. Am. A 4, 2395–2400 (1987).
[CrossRef] [PubMed]

D. Kersten, “Spatial summation in visual noise,” Vision Res. 24, 1977–1990 (1984).
[CrossRef] [PubMed]

D. Kersten, “Statistical efficiency for the detection of visual noise,” Vision Res. 24, 1977–1990 (1984);Vision Res. 27, 1029–1040 (1987).
[CrossRef]

Larson, H. J.

H. J. Larson, B. O. Shubert, Probabilistic Models in Engineering Sciences, Vol. 1: Random Variables and Stochastic Processes (Wiley, New York, 1979).

Laughlin, S. B.

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London Ser. B 216, 427–459 (1982).
[CrossRef]

Linsker, R.

R. Linsker, “Self-organization in a perceptual network,” IEEE Trans. Comput. 21, 105–117 (1988).

Mandelbrot, B. B.

B. B. Mandelbrot, Fractals: Form, Chance, and Dimension (Freeman, San Francisco, Calif., 1977).

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).

Mayer, M. J.

E. Switkes, M. J. Mayer, J. A. Sloan, “Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis,” Vision Res. 18, 1393–1399 (1978).
[CrossRef] [PubMed]

Peitgen, H. O.

H. O. Peitgen, P. H. Richter, The Beauty of Fractals (Springers-Verlag, Berlin, 1986).
[CrossRef]

Pelli, D. G.

A. B. Watson, D. G. Pelli, “quest: a Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
[CrossRef] [PubMed]

Pentland, A. P.

A. P. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Anal. Mech. Intell. PAMI-6, 661–673 (1984).
[CrossRef]

Pratt, W. K.

W. K. Pratt, O. D. Faugeras, A. Gagalowicz, “Visual discrimination of stochastic texture fields,” IEEE Trans. Syst. Man Cybern. SMC-8, 796–804 (1978).
[CrossRef]

Rensink, R. A.

R. A. Rensink, On the Visual Discrimination of Self-Similar Random Textures, Department of Computer Science Tech. Rep. 86-16 (University of British Columbia, Vancouver, B.C., Canada, 1986).

Richter, P. H.

H. O. Peitgen, P. H. Richter, The Beauty of Fractals (Springers-Verlag, Berlin, 1986).
[CrossRef]

Robson, J. G.

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature (London) 302, 419–422 (1983).
[CrossRef]

Shannon, C. E.

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Champaign, Ill, 1949).

Shepp, L.

B. Julesz, E. Gilbert, L. Shepp, H. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics revisited,” Perception 2, 391–405 (1973).
[CrossRef]

Shubert, B. O.

H. J. Larson, B. O. Shubert, Probabilistic Models in Engineering Sciences, Vol. 1: Random Variables and Stochastic Processes (Wiley, New York, 1979).

Sloan, J. A.

E. Switkes, M. J. Mayer, J. A. Sloan, “Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis,” Vision Res. 18, 1393–1399 (1978).
[CrossRef] [PubMed]

Srinivasan, M. V.

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London Ser. B 216, 427–459 (1982).
[CrossRef]

Switkes, E.

E. Switkes, M. J. Mayer, J. A. Sloan, “Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis,” Vision Res. 18, 1393–1399 (1978).
[CrossRef] [PubMed]

Synder, A. W.

T. Bossomaier, A. W. Synder, “Why spatial frequency processing in the visual cortex?” Vision Res. 26, 1307–1309 (1986).
[CrossRef] [PubMed]

Voss, R. F.

R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms for Computer Science, R. A. Earnshaw, ed. (Springer-Verlag, Berlin, 1985), pp. 805–829.
[CrossRef]

Wagner, R. F.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Watson, A. B.

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature (London) 302, 419–422 (1983).
[CrossRef]

A. B. Watson, D. G. Pelli, “quest: a Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
[CrossRef] [PubMed]

Weaver, W.

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Champaign, Ill, 1949).

Weibull, W. A.

W. A. Weibull, “A statistical distribution function of wide applicability,” J. Appl. Mech. 18, 292–297 (1951).

Yuille, A. L.

D. M. Kammen, A. L. Yuille, “Spontaneous symmetry-breaking energy functions and the emergence of orientation selective cortical cells,” Biol. Cybern. 59, 23–31 (1988).
[CrossRef] [PubMed]

Bell Syst. Tech. J.

B. Julesz, J. Bergen, “Textons, the fundamental elements in preattentive vision and perception of texture,” Bell Syst. Tech. J. 62, 619–1645 (1983).

Biol. Cybern.

D. M. Kammen, A. L. Yuille, “Spontaneous symmetry-breaking energy functions and the emergence of orientation selective cortical cells,” Biol. Cybern. 59, 23–31 (1988).
[CrossRef] [PubMed]

IEEE Trans. Comput.

R. Linsker, “Self-organization in a perceptual network,” IEEE Trans. Comput. 21, 105–117 (1988).

IEEE Trans. Pattern Anal. Mech. Intell.

A. P. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Anal. Mech. Intell. PAMI-6, 661–673 (1984).
[CrossRef]

IEEE Trans. Syst. Man Cybern.

W. K. Pratt, O. D. Faugeras, A. Gagalowicz, “Visual discrimination of stochastic texture fields,” IEEE Trans. Syst. Man Cybern. SMC-8, 796–804 (1978).
[CrossRef]

IRE Trans. Inf. Theory

B. Julesz, “Visual pattern discrimination,” IRE Trans. Inf. Theory IT-8, 84–92 (1962).
[CrossRef]

J. Appl. Mech.

W. A. Weibull, “A statistical distribution function of wide applicability,” J. Appl. Mech. 18, 292–297 (1951).

J. Opt. Soc. Am. A

J. Physiol.

H. B. Barlow, “Measurements of the quantum efficiency of discrimination in human scotopic vision,” J. Physiol. 160, 169–188 (1962).
[PubMed]

Nature (London)

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature (London) 302, 419–422 (1983).
[CrossRef]

Percept. Psychophys.

A. B. Watson, D. G. Pelli, “quest: a Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
[CrossRef] [PubMed]

Perception

B. Julesz, E. Gilbert, L. Shepp, H. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics revisited,” Perception 2, 391–405 (1973).
[CrossRef]

Proc. R. Soc. London Ser. B

H. B. Barlow, “The Ferrier lecture: critical limiting factors in the design of the eye and visual cortex,” Proc. R. Soc. London Ser. B 212, 1–34 (1981).
[CrossRef]

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London Ser. B 216, 427–459 (1982).
[CrossRef]

Psychol. Rev.

F. Attneave, “Informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

Science

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Vision Res.

D. Kersten, “Spatial summation in visual noise,” Vision Res. 24, 1977–1990 (1984).
[CrossRef] [PubMed]

D. Kersten, “Statistical efficiency for the detection of visual noise,” Vision Res. 24, 1977–1990 (1984);Vision Res. 27, 1029–1040 (1987).
[CrossRef]

H. B. Barlow, “The efficiency of detecting changes of density in random dot patterns,” Vision Res. 18, 637–650 (1978).
[CrossRef] [PubMed]

T. Bossomaier, A. W. Synder, “Why spatial frequency processing in the visual cortex?” Vision Res. 26, 1307–1309 (1986).
[CrossRef] [PubMed]

E. Switkes, M. J. Mayer, J. A. Sloan, “Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis,” Vision Res. 18, 1393–1399 (1978).
[CrossRef] [PubMed]

Other

The autocorrelation function of a random field I is given byRI((xi,yi),(xj,yj))=E[I(xi,yi)I(xj,yj)],where E[·] is the expectation operator. Since we are assuming that I is stationary and isotropic, we can write this as a function of the Euclidean distance between points:RI(Δr)=E[I(x,y)I(x+Δrcosθ,y+Δrsinθ)],where r= (Δx2+ Δy2)1/2and θ is the angle between the points. The second-order statistical structure of a stationary ensemble of images, given by the autocorrelation function in space, is given by the power spectrum in the frequency domain. The power spectrum is the Fourier transform of the autocorrelation functionPI(fx,fy)=∫0∞∫0∞RI(x,y)exp[i2π(fx+fy)]dxdy.The power at a given frequency is twice the variance of the corresponding Fourier coefficients (real and imaginary) of images in an ensemble. The real and imaginary parts of the Fourier coefficient at a given frequency are uncorrelated and have equal variance. The power at fx = fy = 0 is the squared mean of the ensemble. For an isotropic ensemble, the power spectrum may be written as a function of radial spatial frequency, PI(fr), where fr= (fx2 + fy2)1/2.

B. Julesz, “Spatial frequency channels in one-, two- and three-dimensional vision: variations on an auditory theme by Bekesy,” in Vision Coding and Adaptability, C. S. Harris, ed. (Erlbaum, Hillside, N.J., 1980).

B. B. Mandelbrot, Fractals: Form, Chance, and Dimension (Freeman, San Francisco, Calif., 1977).

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).

H. B. Barlow, P. Z. Foldiak, “Adaptation and decorrelation in the cortex,” in The Computing Neuron, R. C. Miall, R. M. Durbin, G. J. Mitchison, eds. (Addison-Wesley, Reading, Mass., 1989).

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Champaign, Ill, 1949).

Stationarity implies that the statistics of a random field are invariant over translations of the coordinate system on which it is defined. Isotropy implies that they are also invariant over rotations of the coordinate system. One result of these two assumptions is that the autocorrelation function can be expressed as a function of Euclidean distance. The assumption of stationarity is intuitively attractive, as it can result from viewing scenes from a range of positions, so that shifted versions of any given image are equally likely. A similar argument, however, cannot be made for the isotropy assumption, as we generally view scenes with our heads perpendicular to the ground. In a study related to the question of isotropy, Switkes et al.13 found more power at horizontal and vertical orientations in images of both natural and synthetic scenes. The assumption does, however, simplify our investigation by allowing us to look at the correlational structure of images as a function only of distance between points.

The power spectrum of an ensemble with an exponential auto-correlation function clearly shows the effect of the scale constant k. For an isotropic ensemble, the spectrum is given byPI(fr)∝1(k+fr3/2)2.At frequencies much lower than k, the spectrum approximates white noise; that is, points in the image separated by a distance much greater than 1/k are effectively uncorrelated. For frequencies much greater than k, the spectrum falls off according to the power law 1/fr3. The images of this ensemble exhibit qualitatively different statistical behavior at different scales.

R. A. Rensink, On the Visual Discrimination of Self-Similar Random Textures, Department of Computer Science Tech. Rep. 86-16 (University of British Columbia, Vancouver, B.C., Canada, 1986).

Rensink generated line textures by using one-dimensional power spectra. Peak performance was found to be at a spectral drop-off of β1D= 3 for these textures. The equivalent two-dimensional spectral drop-off is given by β2D= β1D + 1 = 4.26

R. F. Voss, “Random fractal forgeries,” in Fundamental Algorithms for Computer Science, R. A. Earnshaw, ed. (Springer-Verlag, Berlin, 1985), pp. 805–829.
[CrossRef]

H. O. Peitgen, P. H. Richter, The Beauty of Fractals (Springers-Verlag, Berlin, 1986).
[CrossRef]

H. B. Barlow, “Sensory mechanisms, the reduction of redundancy and intelligence,” NPL Symposium on the Mechanization of Thought Processes, No. 10 (H. M. Stationary Office, London, 1959), pp. 535–539.

H. B. Barlow, “The coding of sensory messages,” in Current Problems in Animal Behavior, W. H. Thorpe, O. L. Zangwill, eds. (Cambridge U. Press, Cambridge, 1961), pp. 331–360.

H. J. Larson, B. O. Shubert, Probabilistic Models in Engineering Sciences, Vol. 1: Random Variables and Stochastic Processes (Wiley, New York, 1979).

We corrected for the nonlinearity by raising the entries of the lookup table to an exponent of 0.375 (1/2.67) and rescaling them to give a range of 256 gray levels. The equation used for correcting the lookup-table entries wasl[i]=i0.375*250.00.625,0≤i<256,where l[i] is the ith entry in the lookup table.

The resulting image statistics had toroidal symmetry, reflecting the symmetry of the fast Fourier-transform algorithm.

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Figures (4)

Fig. 1
Fig. 1

One-dimensional slices through the power spectra of two fractional Brownian motion ensembles, one with fractal dimension D = 2.5 (β = 3.0) and the other with fractal dimension D = 3.0 (β = 2.0). On a log–log plot, the spectra fall off linearly, with a slope given by exponent β. The scatter plots show the actual spectra of sample images drawn from these two ensembles, with the power along different orientations collapsed onto one dimension.

Fig. 2
Fig. 2

Three examples of images randomly drawn from ensembles of fractional Brownian motion. The images are drawn from ensembles with spectral drop-offs of (a) 2.0, (b) 3.0, and (c) 4.0. The corresponding fractal dimensions are (a) 3.0, (b) 2.5, and (c) 2.0.

Fig. 3
Fig. 3

Discrimination threshold curves for subjects (a) DCK, (b) DR, and (c) DF. The thresholds are plotted as decibel increments versus reference spectral drop-off. A plot of the threshold function for the low-pass rms contrast model is shown as a bold line on each graph. The average standard error is shown on the right-hand side of each graph. An increment of approximately −3 dB is the lowest measured threshold. This value corresponds to a real increment of 0.14.

Fig. 4
Fig. 4

Discrimination threshold curves with standard-error bars for each subject averaged across the three conditions. Only thresholds for spectral drop-offs that are greater than or equal to 2.0 are shown.

Tables (2)

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Table 1 Ideal Observer’s Thresholds for the Ten Reference Spectral Drop-offs

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Table 2 Ideal Observer’s Thresholds for Four Different Image Sizes Using β = 3.2 as the Reference Spectral Drop-off

Equations (36)

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R I ( Δ r ) e k Δ r ,
P I ( f r ) 1 f r β
β = 8 2 D .
C = [ P ( f x , f y ) d f x d f y μ 2 μ 2 ] 1 / 2 ,
C = { 2 π [ 0 f r P ( f r ) d f r μ 2 ] μ 2 } 1 / 2 ,
p ( T i | x i ) = p ( x i | T i ) p ( T i ) p ( x i ) ,
p ( T i | x i ) = K * p ( x i | T i ) ,
K = 1.0 0 p ( x i | T i ) d T i .
D = Ĉ 1 Ĉ 2 = { μ 2 2 [ 0 f c f r P ̂ 1 ( f r ) d f r μ 1 2 ] μ 1 2 [ 0 f c f r P ̂ 2 ( f r ) d f r μ 2 2 ] } 1 / 2 .
D = Ĉ 1 Ĉ 2 = { μ 2 2 [ f c f r P ̂ 1 ( f r ) d f r μ 1 2 ] μ 1 2 [ f c f r P ̂ 2 ( f r ) d f r μ 2 2 ] } 1 / 2 .
E = Δ β i 2 Δ β s 2 .
N = n D ,
D = log N log n .
p [ I ( x + Δ x ) I ( x ) | Δ x | H < y ] = f ( y ) ,
D = E + 1 H ,
β = 1 + 2 H .
β = 1 + 2 H + ( E 1 ) = E + 2 H .
β = 8 2 D .
( f x , f y ) = X r ( f x , f y ) + i X i ( f x , f y ) .
σ ( f x , f y ) 2 = E [ X r ( f x , f y ) 2 ] = E [ X i ( f x , f y ) 2 ] = ½ P ( f x , f y ) .
P s ( f x , f y ) = X r ( f x , f y ) 2 + X i ( f x , f y ) 2
f [ P s ( f x , f y ) ] = 1 2 σ ( f x , f y ) 2 exp [ P s ( f x , f y ) 2 σ ( f x , f y ) 2 ] .
P ( f x , f y ) = E [ P s ( f x , f y ) ] = 2 σ ( f x , f y ) 2 ,
P ( f r ) = k f r β .
p [ P s ( f r ) | β , k ] = k 1 f r β exp [ P s ( f r ) k 1 f r β ] .
p ( P s | β , k ) = k 1 f r β exp [ P s ( f r ) k 1 f r β ] ,
L ( P s | β , k ) = log ( k ) + β log ( f r ) k 1 f r β P s ( f r ) .
L ( P s | β , k ) = N log ( k ) + β log ( f r ) k 1 f r β P s ( f r ) .
k = 1 N f r β P s ( f r ) .
log ( f r ) = N f r β log ( f r ) P s ( f r ) f r β P s ( f r ) .
log [ P ( f r ) ] = log ( k ) β log ( f r ) .
RI((xi,yi),(xj,yj))=E[I(xi,yi)I(xj,yj)],
RI(Δr)=E[I(x,y)I(x+Δrcosθ,y+Δrsinθ)],
PI(fx,fy)=00RI(x,y)exp[i2π(fx+fy)]dxdy.
PI(fr)1(k+fr3/2)2.
l[i]=i0.375*250.00.625,0i<256,

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