Abstract

Real-world visual scenes display consistent first- and second-order statistical regularities to which visual neural representations may be perceptually matched, but these lower-order regularities stem from constraints on image power spectra, which appear to carry much less perceptual information than image phase spectra. Natural scenes are shown also to display consistent higher-order statistical regularities, and an analysis of these regularities in terms of fourth-order spectra shows that they are strongly dependent on spatial frequency. These findings have important consequences for the design of a visual system that aims to maximize sparseness in neural representations.

© 1999 Optical Society of America

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References

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  1. 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]
  2. L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
    [Crossref] [PubMed]
  3. The term “conventional nth-order statistics” is used here to describe statistics derived from standard nth-order correlation functions; these are not the same as the statistics defined by Julesz and colleagues.4 This study deals only with the former, although the two are closely related.
  4. G. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics revisited,” Perception 2, 391–405 (1973).
    [Crossref]
  5. H. B. Barlow, “What is the computational goal of the neocortex?” in Large-Scale Neuronal Theories of the Brain, C. Koch, ed. (MIT Press, Cambridge, Mass., 1994).
  6. D. J. Field, “What is the goal of sensory coding?” Neural Comput. 6, 559–601 (1994).
    [Crossref]
  7. R. J. Baddeley, “Searching for filters with ‘interesting’ output distributions: an uninteresting direction to explore?” Network Comput. Neural Syst. 7, 409–421 (1996).
    [Crossref]
  8. R. J. Baddeley, “An efficient code in V1?” Nature (London) 381, 560–561 (1996).
    [Crossref]
  9. M. G. A. Thomson, D. H. Foster, “The role of second- and third-order statistics in the discriminability of natural images,” J. Opt. Soc. Am. A 14, 2081–2090 (1997).
    [Crossref]
  10. Arguments are often made for incorporating some sort of spectral windowing into the whitening process such that low-amplitude components near the Nyquist limit are given relatively less power (for example, to avoid the effects of rectangular sampling in the frequency domain or to avoid boosting noise at high frequency). In analyses of coherence measures, however, the value of such windowing is unclear; windowing was therefore avoided here.
  11. C. Nikias, A. Petropolu, Higher-Order Spectra Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1996).
  12. C. Zetzsche, E. Barth, B. Wegmann, “The importance of intrinsically two-dimensional image features in biological vision and picture coding,” in Digital Images and Human Vision, A. B. Watson, ed. (MIT Press, Cambridge, Mass., 1993).
  13. J. J. Atick, N. A. Redlich, “What does the retina know about natural scenes?” Neural Comput. 4, 196–210 (1992).
    [Crossref]
  14. 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]
  15. M. C. Morrone, D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London, Ser. B 235, 221–24 (1988).
    [Crossref]
  16. B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive-field properties by learning a sparse code for natural scenes,” Nature (London) 381, 607–609 (1996).
    [Crossref]

1997 (1)

1996 (3)

R. J. Baddeley, “Searching for filters with ‘interesting’ output distributions: an uninteresting direction to explore?” Network Comput. Neural Syst. 7, 409–421 (1996).
[Crossref]

R. J. Baddeley, “An efficient code in V1?” Nature (London) 381, 560–561 (1996).
[Crossref]

B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive-field properties by learning a sparse code for natural scenes,” Nature (London) 381, 607–609 (1996).
[Crossref]

1994 (1)

D. J. Field, “What is the goal of sensory coding?” Neural Comput. 6, 559–601 (1994).
[Crossref]

1992 (1)

J. J. Atick, N. A. Redlich, “What does the retina know about natural scenes?” Neural Comput. 4, 196–210 (1992).
[Crossref]

1988 (1)

M. C. Morrone, D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London, Ser. B 235, 221–24 (1988).
[Crossref]

1987 (1)

1982 (1)

L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[Crossref] [PubMed]

1978 (1)

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]

1973 (1)

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

Atick, J. J.

J. J. Atick, N. A. Redlich, “What does the retina know about natural scenes?” Neural Comput. 4, 196–210 (1992).
[Crossref]

Baddeley, R. J.

R. J. Baddeley, “Searching for filters with ‘interesting’ output distributions: an uninteresting direction to explore?” Network Comput. Neural Syst. 7, 409–421 (1996).
[Crossref]

R. J. Baddeley, “An efficient code in V1?” Nature (London) 381, 560–561 (1996).
[Crossref]

Barlow, H. B.

H. B. Barlow, “What is the computational goal of the neocortex?” in Large-Scale Neuronal Theories of the Brain, C. Koch, ed. (MIT Press, Cambridge, Mass., 1994).

Barth, E.

C. Zetzsche, E. Barth, B. Wegmann, “The importance of intrinsically two-dimensional image features in biological vision and picture coding,” in Digital Images and Human Vision, A. B. Watson, ed. (MIT Press, Cambridge, Mass., 1993).

Burr, D. C.

M. C. Morrone, D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London, Ser. B 235, 221–24 (1988).
[Crossref]

Campbell, F. W.

L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[Crossref] [PubMed]

Field, D. J.

B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive-field properties by learning a sparse code for natural scenes,” Nature (London) 381, 607–609 (1996).
[Crossref]

D. J. Field, “What is the goal of sensory coding?” Neural Comput. 6, 559–601 (1994).
[Crossref]

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]

Foster, D. H.

Frisch, H. L.

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

Gilbert, E. N.

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

Julesz, G.

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

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]

Morrone, M. C.

M. C. Morrone, D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London, Ser. B 235, 221–24 (1988).
[Crossref]

Nikias, C.

C. Nikias, A. Petropolu, Higher-Order Spectra Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1996).

Olshausen, B. A.

B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive-field properties by learning a sparse code for natural scenes,” Nature (London) 381, 607–609 (1996).
[Crossref]

Petropolu, A.

C. Nikias, A. Petropolu, Higher-Order Spectra Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1996).

Piotrowski, L. N.

L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[Crossref] [PubMed]

Redlich, N. A.

J. J. Atick, N. A. Redlich, “What does the retina know about natural scenes?” Neural Comput. 4, 196–210 (1992).
[Crossref]

Shepp, L. A.

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

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]

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]

Thomson, M. G. A.

Wegmann, B.

C. Zetzsche, E. Barth, B. Wegmann, “The importance of intrinsically two-dimensional image features in biological vision and picture coding,” in Digital Images and Human Vision, A. B. Watson, ed. (MIT Press, Cambridge, Mass., 1993).

Zetzsche, C.

C. Zetzsche, E. Barth, B. Wegmann, “The importance of intrinsically two-dimensional image features in biological vision and picture coding,” in Digital Images and Human Vision, A. B. Watson, ed. (MIT Press, Cambridge, Mass., 1993).

J. Opt. Soc. Am. A (2)

Nature (London) (2)

R. J. Baddeley, “An efficient code in V1?” Nature (London) 381, 560–561 (1996).
[Crossref]

B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive-field properties by learning a sparse code for natural scenes,” Nature (London) 381, 607–609 (1996).
[Crossref]

Network Comput. Neural Syst. (1)

R. J. Baddeley, “Searching for filters with ‘interesting’ output distributions: an uninteresting direction to explore?” Network Comput. Neural Syst. 7, 409–421 (1996).
[Crossref]

Neural Comput. (2)

D. J. Field, “What is the goal of sensory coding?” Neural Comput. 6, 559–601 (1994).
[Crossref]

J. J. Atick, N. A. Redlich, “What does the retina know about natural scenes?” Neural Comput. 4, 196–210 (1992).
[Crossref]

Perception (2)

L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[Crossref] [PubMed]

G. Julesz, E. N. Gilbert, L. A. Shepp, H. L. 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 (1)

M. C. Morrone, D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. London, Ser. B 235, 221–24 (1988).
[Crossref]

Vision Res. (1)

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]

Other (5)

H. B. Barlow, “What is the computational goal of the neocortex?” in Large-Scale Neuronal Theories of the Brain, C. Koch, ed. (MIT Press, Cambridge, Mass., 1994).

The term “conventional nth-order statistics” is used here to describe statistics derived from standard nth-order correlation functions; these are not the same as the statistics defined by Julesz and colleagues.4 This study deals only with the former, although the two are closely related.

Arguments are often made for incorporating some sort of spectral windowing into the whitening process such that low-amplitude components near the Nyquist limit are given relatively less power (for example, to avoid the effects of rectangular sampling in the frequency domain or to avoid boosting noise at high frequency). In analyses of coherence measures, however, the value of such windowing is unclear; windowing was therefore avoided here.

C. Nikias, A. Petropolu, Higher-Order Spectra Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1996).

C. Zetzsche, E. Barth, B. Wegmann, “The importance of intrinsically two-dimensional image features in biological vision and picture coding,” in Digital Images and Human Vision, A. B. Watson, ed. (MIT Press, Cambridge, Mass., 1993).

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Figures (3)

Fig. 1
Fig. 1

Plots of whitened kurtosis (γw) values for an ensemble of 85 natural images. Notice (top plot) that although kurtosis is in general a signed quantity, whitened natural images have strictly positive kurtosis. The bottom plot shows the distribution of γw values for the same image ensemble after phase randomization: The values are generally signed and are several orders of magnitude smaller than the corresponding values in the top plot (in theory, if these random-phase images were infinitely large, γw would be zero).

Fig. 2
Fig. 2

Spatial-frequency-specific contributions to γw for a 64×64-pixel realization of a test function (whitened square-wave plaid in white noise). The structure of the plot is as expected: The odd harmonics’ contributions to γw form a perfectly flat lattice situated at some positive nonzero value on the z axis, whereas the contributions of the Gaussian noise are close to zero everywhere [the plaid data (top) and noise data (bottom) have been gridded separately for clarity].

Fig. 3
Fig. 3

Spatial-frequency-specific contributions to γw averaged over 85 natural images. K(ωx, ωy) was computed from each whitened image; the resulting data were then averaged across the ensemble to produce a mean K(ωx, ωy). For display purposes, this mean K(ωx, ωy) has been averaged down to 64×64 samples and transformed from a Cartesian to a polar raster by grid interpolation: Data are thus plotted as a function of radial spatial frequency ωr and orientation θ.

Equations (4)

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

γ=E{[g(x, y)]4}E{[g(x, y)]2}2-3.
C(ω1, ω2, ω3)=F(ω1)F(ω2)F(ω3)F*(ω1+ω2+ω3).
T(ωx, ωy)=ω2x=-LLω2y=-LLω3x=-LLω3y=-LLC(ω1x, ω1y, ω2x, ω2y, ω3x, ω3y).
K(ωx, ωy)=TRe(ωx, ωy)-(3E{g(x, y)2}H(ωx, ωy))E{g(x, y)2}2.

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