Abstract

Several studies have demonstrated that the power spectra of natural image ensembles scale as f-α. A stronger claim that has been made is that the power spectra of single natural images typically also scale as f-α. Results are presented that challenge this latter claim. The results are based on a method for estimating large-scale structure in single images that compares aliasing artifacts produced by image windows of different shape. Failures of f-α scaling are found at large scales in many natural images. These failures cannot be accounted for by f-α scaling models such as a linear superposition model or a model based on two-dimensional occlusions in the image plane. The results imply that claims about f-α scaling in single natural images have been exaggerated. The results also offer insight into why such failures of f-α scaling occur.

© 2000 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]

1998 (2)

J. H. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 359–366 (1998).
[CrossRef]

C. A. Parrago, G. Brelstaff, T. Troscianko, “Color and luminance information in natural scenes,” J. Opt. Soc. Am. A 15, 563–569 (1998).
[CrossRef]

1997 (2)

D. L. Ruderman, “Origins of scaling in natural images,” Vision Res. 37, 3385–3398 (1997).
[CrossRef]

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3383 (1997).
[CrossRef]

1996 (1)

A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

1994 (2)

D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

D. L. Ruderman, “The statistics of natural images,” Network 5, 517–548 (1994).
[CrossRef]

1992 (3)

J. H. van Hateren, “Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation,” J. Comp. Physiol. A 171, 157–170 (1992).

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

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

1987 (2)

1982 (1)

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]

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] [PubMed]

Atick, J. J.

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

Bialek, W.

D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

Brady, N.

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3383 (1997).
[CrossRef]

Brelstaff, G.

Burton, G. J.

Chao, T.

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[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]

Field, D. J.

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3383 (1997).
[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]

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]

Lythgoe, J. N.

J. N. Lythgoe, The Ecology of Vision (Clarendon, Oxford, UK, 1979).

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]

Moorhead, I. R.

Parrago, C. A.

Redlich, A. N.

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

Ruderman, D. L.

D. L. Ruderman, “Origins of scaling in natural images,” Vision Res. 37, 3385–3398 (1997).
[CrossRef]

D. L. Ruderman, “The statistics of natural images,” Network 5, 517–548 (1994).
[CrossRef]

D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

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]

Tadmor, Y.

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

Tolhurst, D. J.

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

Troscianko, T.

van der Schaaf, A.

J. H. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 359–366 (1998).
[CrossRef]

A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

van Hateren, J. H.

J. H. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 359–366 (1998).
[CrossRef]

A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

J. H. van Hateren, “Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation,” J. Comp. Physiol. A 171, 157–170 (1992).

Appl. Opt. (1)

J. Comp. Physiol. A (1)

J. H. van Hateren, “Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation,” J. Comp. Physiol. A 171, 157–170 (1992).

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

Network (1)

D. L. Ruderman, “The statistics of natural images,” Network 5, 517–548 (1994).
[CrossRef]

Neural Comput. (1)

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

Ophthalmic Physiol. Opt. (1)

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

D. L. Ruderman, W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. B (2)

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]

J. H. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 359–366 (1998).
[CrossRef]

Vision Res. (4)

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]

D. L. Ruderman, “Origins of scaling in natural images,” Vision Res. 37, 3385–3398 (1997).
[CrossRef]

A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3383 (1997).
[CrossRef]

Other (1)

J. N. Lythgoe, The Ecology of Vision (Clarendon, Oxford, UK, 1979).

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Figures (3)

Fig. 1
Fig. 1

Mesh plots of the + bias from a square window and the × bias from a diamond window, obtained from image ensembles defined by Eqs. (1) and (2) and for various values of (c, α).

Fig. 2
Fig. 2

Mean power spectrum of the low-β (dashed curves) and high-β (solid curves) classes for the (a) van Hateren, (b) Gaussian, and (c) Ruderman ensembles.

Fig. 3
Fig. 3

Two windowed images are shown (M=256). These are composed of overlapping squares of widths in the ranges (a) 64–128 pixels and (b) 16–32 pixels. (c) The mean power spectra of 100 such images are shown for the larger square images in (a) (solid curve) and the smaller square images in (b) (dashed curve).

Tables (1)

Tables Icon

Table 1 Mean Values of α for the Images in the Low-β and High-β Classes and for Each of the Three Ensembles Tested

Equations (5)

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

bias(f)log2R(f)*|W(f)|2R(f).
R(f)(c+|f|)-α.
Aw(f)=xw(x)[I(x)-μ]exp(if·x).
μxI(x)w(x)xw(x),
βf=f1f2log2A×(f, f)A×(f, -f)A+(f, f)A+(f, -f).

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