Abstract

If we filter an ensemble of natural images with a high-pass filter, the output is a random variable. The probability density for this variable will, under fairly general assumptions, be a Gaussian scale mixture. The kurtosis for this type of distribution is always positive.

© 2001 Optical Society of America

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References

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  1. A. Bell and T. J. Sejnowski, Vision Res. 37, 3327 (1997).
    [CrossRef]
  2. P. Comon, Signal Process. 36, 287 (1994).
    [CrossRef]
  3. B. A. Olshausen and D. Field, Nature 381, 607 (1996).
    [CrossRef] [PubMed]
  4. B. A. Olshausen and D. Field, Network 7, 333 (1996).
    [CrossRef] [PubMed]
  5. J. J. Heine, S. R. Deans, and L. P. Clarke, J. Opt. Soc. Am. A 16, 6 (1999).
    [CrossRef]
  6. M. J. Wainwright and E. P. Simoncelli, in Advances in Neural Information Processing Systems, S. A. Solla, T. K. Leen, and K. R. Muller, eds. (MIT, Cambridge, Mass., 2000), Vol. 12, pp. 855–861.
  7. E. Y. Lam and J. W. Goodman, IEEE Trans. Image Process. 9, 1661 (2000).
    [CrossRef]
  8. M. A. Bernstein, D. M. Thomasson, and W. H. Perman, Med. Phys. 16, 813 (1989).
    [CrossRef] [PubMed]
  9. D. L. Ruderman and W. Bialek, Phys. Rev. Lett. 73, 814 (1994).
    [CrossRef] [PubMed]
  10. J. J. Heine, S. R. Deans, R. P. Vethuizen, and L. P. Clarke, Med. Phys. 26, 2254 (1999).
    [CrossRef] [PubMed]
  11. D. F. Andrews and C. L. Mallows, J. R. Statist. Soc. Ser. B 36, 99 (1974).
  12. E. Jakeman and P. N. Pusey, IEEE Trans. Antennas Propag. 24, 806 (1976).
    [CrossRef]
  13. E. Jakeman and R. J. A. Tough, J. Opt. Soc. Am. A 4, 1764 (1987).
    [CrossRef]
  14. F. A. Pentini, A. Farina, and F. Zirilli, IEE Proc. F 139, 239 (1992).
  15. V. M. Narayanan and P. M. Shanker, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 41, 845 (1994).
    [CrossRef]
  16. P. M. Shankar, Phys. Med. Biol. 40, 1633 (1995).
    [CrossRef] [PubMed]
  17. P. M. Shankar, R. Molthen, V. M. Narayanan, J. M. Reid, V. Genis, F. Forsberg, C. W. Piccoli, A. E. Lindenmayer, and B. B. Goldberg, Ultrasound Med. Biol. 22, 873 (1996).
    [CrossRef]

2000 (1)

E. Y. Lam and J. W. Goodman, IEEE Trans. Image Process. 9, 1661 (2000).
[CrossRef]

1999 (2)

J. J. Heine, S. R. Deans, R. P. Vethuizen, and L. P. Clarke, Med. Phys. 26, 2254 (1999).
[CrossRef] [PubMed]

J. J. Heine, S. R. Deans, and L. P. Clarke, J. Opt. Soc. Am. A 16, 6 (1999).
[CrossRef]

1997 (1)

A. Bell and T. J. Sejnowski, Vision Res. 37, 3327 (1997).
[CrossRef]

1996 (3)

B. A. Olshausen and D. Field, Nature 381, 607 (1996).
[CrossRef] [PubMed]

B. A. Olshausen and D. Field, Network 7, 333 (1996).
[CrossRef] [PubMed]

P. M. Shankar, R. Molthen, V. M. Narayanan, J. M. Reid, V. Genis, F. Forsberg, C. W. Piccoli, A. E. Lindenmayer, and B. B. Goldberg, Ultrasound Med. Biol. 22, 873 (1996).
[CrossRef]

1995 (1)

P. M. Shankar, Phys. Med. Biol. 40, 1633 (1995).
[CrossRef] [PubMed]

1994 (3)

D. L. Ruderman and W. Bialek, Phys. Rev. Lett. 73, 814 (1994).
[CrossRef] [PubMed]

V. M. Narayanan and P. M. Shanker, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 41, 845 (1994).
[CrossRef]

P. Comon, Signal Process. 36, 287 (1994).
[CrossRef]

1992 (1)

F. A. Pentini, A. Farina, and F. Zirilli, IEE Proc. F 139, 239 (1992).

1989 (1)

M. A. Bernstein, D. M. Thomasson, and W. H. Perman, Med. Phys. 16, 813 (1989).
[CrossRef] [PubMed]

1987 (1)

1976 (1)

E. Jakeman and P. N. Pusey, IEEE Trans. Antennas Propag. 24, 806 (1976).
[CrossRef]

1974 (1)

D. F. Andrews and C. L. Mallows, J. R. Statist. Soc. Ser. B 36, 99 (1974).

Andrews, D. F.

D. F. Andrews and C. L. Mallows, J. R. Statist. Soc. Ser. B 36, 99 (1974).

Bell, A.

A. Bell and T. J. Sejnowski, Vision Res. 37, 3327 (1997).
[CrossRef]

Bernstein, M. A.

M. A. Bernstein, D. M. Thomasson, and W. H. Perman, Med. Phys. 16, 813 (1989).
[CrossRef] [PubMed]

Bialek, W.

D. L. Ruderman and W. Bialek, Phys. Rev. Lett. 73, 814 (1994).
[CrossRef] [PubMed]

Clarke, L. P.

J. J. Heine, S. R. Deans, and L. P. Clarke, J. Opt. Soc. Am. A 16, 6 (1999).
[CrossRef]

J. J. Heine, S. R. Deans, R. P. Vethuizen, and L. P. Clarke, Med. Phys. 26, 2254 (1999).
[CrossRef] [PubMed]

Comon, P.

P. Comon, Signal Process. 36, 287 (1994).
[CrossRef]

Deans, S. R.

J. J. Heine, S. R. Deans, and L. P. Clarke, J. Opt. Soc. Am. A 16, 6 (1999).
[CrossRef]

J. J. Heine, S. R. Deans, R. P. Vethuizen, and L. P. Clarke, Med. Phys. 26, 2254 (1999).
[CrossRef] [PubMed]

Farina, A.

F. A. Pentini, A. Farina, and F. Zirilli, IEE Proc. F 139, 239 (1992).

Field, D.

B. A. Olshausen and D. Field, Nature 381, 607 (1996).
[CrossRef] [PubMed]

B. A. Olshausen and D. Field, Network 7, 333 (1996).
[CrossRef] [PubMed]

Forsberg, F.

P. M. Shankar, R. Molthen, V. M. Narayanan, J. M. Reid, V. Genis, F. Forsberg, C. W. Piccoli, A. E. Lindenmayer, and B. B. Goldberg, Ultrasound Med. Biol. 22, 873 (1996).
[CrossRef]

Genis, V.

P. M. Shankar, R. Molthen, V. M. Narayanan, J. M. Reid, V. Genis, F. Forsberg, C. W. Piccoli, A. E. Lindenmayer, and B. B. Goldberg, Ultrasound Med. Biol. 22, 873 (1996).
[CrossRef]

Goldberg, B. B.

P. M. Shankar, R. Molthen, V. M. Narayanan, J. M. Reid, V. Genis, F. Forsberg, C. W. Piccoli, A. E. Lindenmayer, and B. B. Goldberg, Ultrasound Med. Biol. 22, 873 (1996).
[CrossRef]

Goodman, J. W.

E. Y. Lam and J. W. Goodman, IEEE Trans. Image Process. 9, 1661 (2000).
[CrossRef]

Heine, J. J.

J. J. Heine, S. R. Deans, R. P. Vethuizen, and L. P. Clarke, Med. Phys. 26, 2254 (1999).
[CrossRef] [PubMed]

J. J. Heine, S. R. Deans, and L. P. Clarke, J. Opt. Soc. Am. A 16, 6 (1999).
[CrossRef]

Jakeman, E.

E. Jakeman and R. J. A. Tough, J. Opt. Soc. Am. A 4, 1764 (1987).
[CrossRef]

E. Jakeman and P. N. Pusey, IEEE Trans. Antennas Propag. 24, 806 (1976).
[CrossRef]

Lam, E. Y.

E. Y. Lam and J. W. Goodman, IEEE Trans. Image Process. 9, 1661 (2000).
[CrossRef]

Lindenmayer, A. E.

P. M. Shankar, R. Molthen, V. M. Narayanan, J. M. Reid, V. Genis, F. Forsberg, C. W. Piccoli, A. E. Lindenmayer, and B. B. Goldberg, Ultrasound Med. Biol. 22, 873 (1996).
[CrossRef]

Mallows, C. L.

D. F. Andrews and C. L. Mallows, J. R. Statist. Soc. Ser. B 36, 99 (1974).

Molthen, R.

P. M. Shankar, R. Molthen, V. M. Narayanan, J. M. Reid, V. Genis, F. Forsberg, C. W. Piccoli, A. E. Lindenmayer, and B. B. Goldberg, Ultrasound Med. Biol. 22, 873 (1996).
[CrossRef]

Narayanan, V. M.

P. M. Shankar, R. Molthen, V. M. Narayanan, J. M. Reid, V. Genis, F. Forsberg, C. W. Piccoli, A. E. Lindenmayer, and B. B. Goldberg, Ultrasound Med. Biol. 22, 873 (1996).
[CrossRef]

V. M. Narayanan and P. M. Shanker, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 41, 845 (1994).
[CrossRef]

Olshausen, B. A.

B. A. Olshausen and D. Field, Network 7, 333 (1996).
[CrossRef] [PubMed]

B. A. Olshausen and D. Field, Nature 381, 607 (1996).
[CrossRef] [PubMed]

Pentini, F. A.

F. A. Pentini, A. Farina, and F. Zirilli, IEE Proc. F 139, 239 (1992).

Perman, W. H.

M. A. Bernstein, D. M. Thomasson, and W. H. Perman, Med. Phys. 16, 813 (1989).
[CrossRef] [PubMed]

Piccoli, C. W.

P. M. Shankar, R. Molthen, V. M. Narayanan, J. M. Reid, V. Genis, F. Forsberg, C. W. Piccoli, A. E. Lindenmayer, and B. B. Goldberg, Ultrasound Med. Biol. 22, 873 (1996).
[CrossRef]

Pusey, P. N.

E. Jakeman and P. N. Pusey, IEEE Trans. Antennas Propag. 24, 806 (1976).
[CrossRef]

Reid, J. M.

P. M. Shankar, R. Molthen, V. M. Narayanan, J. M. Reid, V. Genis, F. Forsberg, C. W. Piccoli, A. E. Lindenmayer, and B. B. Goldberg, Ultrasound Med. Biol. 22, 873 (1996).
[CrossRef]

Ruderman, D. L.

D. L. Ruderman and W. Bialek, Phys. Rev. Lett. 73, 814 (1994).
[CrossRef] [PubMed]

Sejnowski, T. J.

A. Bell and T. J. Sejnowski, Vision Res. 37, 3327 (1997).
[CrossRef]

Shankar, P. M.

P. M. Shankar, R. Molthen, V. M. Narayanan, J. M. Reid, V. Genis, F. Forsberg, C. W. Piccoli, A. E. Lindenmayer, and B. B. Goldberg, Ultrasound Med. Biol. 22, 873 (1996).
[CrossRef]

P. M. Shankar, Phys. Med. Biol. 40, 1633 (1995).
[CrossRef] [PubMed]

Shanker, P. M.

V. M. Narayanan and P. M. Shanker, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 41, 845 (1994).
[CrossRef]

Simoncelli, E. P.

M. J. Wainwright and E. P. Simoncelli, in Advances in Neural Information Processing Systems, S. A. Solla, T. K. Leen, and K. R. Muller, eds. (MIT, Cambridge, Mass., 2000), Vol. 12, pp. 855–861.

Thomasson, D. M.

M. A. Bernstein, D. M. Thomasson, and W. H. Perman, Med. Phys. 16, 813 (1989).
[CrossRef] [PubMed]

Tough, R. J. A.

Vethuizen, R. P.

J. J. Heine, S. R. Deans, R. P. Vethuizen, and L. P. Clarke, Med. Phys. 26, 2254 (1999).
[CrossRef] [PubMed]

Wainwright, M. J.

M. J. Wainwright and E. P. Simoncelli, in Advances in Neural Information Processing Systems, S. A. Solla, T. K. Leen, and K. R. Muller, eds. (MIT, Cambridge, Mass., 2000), Vol. 12, pp. 855–861.

Zirilli, F.

F. A. Pentini, A. Farina, and F. Zirilli, IEE Proc. F 139, 239 (1992).

IEE Proc. F (1)

F. A. Pentini, A. Farina, and F. Zirilli, IEE Proc. F 139, 239 (1992).

IEEE Trans. Antennas Propag. (1)

E. Jakeman and P. N. Pusey, IEEE Trans. Antennas Propag. 24, 806 (1976).
[CrossRef]

IEEE Trans. Image Process. (1)

E. Y. Lam and J. W. Goodman, IEEE Trans. Image Process. 9, 1661 (2000).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

V. M. Narayanan and P. M. Shanker, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 41, 845 (1994).
[CrossRef]

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

J. R. Statist. Soc. Ser. B (1)

D. F. Andrews and C. L. Mallows, J. R. Statist. Soc. Ser. B 36, 99 (1974).

Med. Phys. (2)

J. J. Heine, S. R. Deans, R. P. Vethuizen, and L. P. Clarke, Med. Phys. 26, 2254 (1999).
[CrossRef] [PubMed]

M. A. Bernstein, D. M. Thomasson, and W. H. Perman, Med. Phys. 16, 813 (1989).
[CrossRef] [PubMed]

Nature (1)

B. A. Olshausen and D. Field, Nature 381, 607 (1996).
[CrossRef] [PubMed]

Network (1)

B. A. Olshausen and D. Field, Network 7, 333 (1996).
[CrossRef] [PubMed]

Phys. Med. Biol. (1)

P. M. Shankar, Phys. Med. Biol. 40, 1633 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

D. L. Ruderman and W. Bialek, Phys. Rev. Lett. 73, 814 (1994).
[CrossRef] [PubMed]

Signal Process. (1)

P. Comon, Signal Process. 36, 287 (1994).
[CrossRef]

Ultrasound Med. Biol. (1)

P. M. Shankar, R. Molthen, V. M. Narayanan, J. M. Reid, V. Genis, F. Forsberg, C. W. Piccoli, A. E. Lindenmayer, and B. B. Goldberg, Ultrasound Med. Biol. 22, 873 (1996).
[CrossRef]

Vision Res. (1)

A. Bell and T. J. Sejnowski, Vision Res. 37, 3327 (1997).
[CrossRef]

Other (1)

M. J. Wainwright and E. P. Simoncelli, in Advances in Neural Information Processing Systems, S. A. Solla, T. K. Leen, and K. R. Muller, eds. (MIT, Cambridge, Mass., 2000), Vol. 12, pp. 855–861.

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Figures (1)

Fig. 1
Fig. 1

The probability density for z is the projection of the joint density for u,v along lines oriented at 45° to the u axis.

Equations (16)

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

z=zk,l=m,n=-MMhm,nfk-m,l-n.
h+m,n=hm,nhm,n>00otherwise,
h-m,n=-hm,nhm,n<00otherwise.
u=uk,l=m,n=-MMh+m,nfj-m,k-n,
v=vk,l=m,n=-MMh-m,nfj-m,k-n.
m,n=-MMh+m,n=m,n=-MMh-m,n.
xySnn-xSnnySnn=xySn-xSnySnn+xSnySnn-xSnnySnn.
pUV(u,v|a,b,c,s)=12πac-b2×exp-12u-sv-sTabbc-1u-sv-s.
pUVu,v=0ΩpUV(u,v|a,b,c,s)×pABCSa,b,c,sdadbdcds.
pZz=-pUVu,u-zdu.
pZz=0ΩpZ(z|a,b,c,s)×pABCSa,b,c,sdadbdcds,
pZ(z|a,b,c,s)=12πa+c-2b1/2×exp-z22a+c-2b.
w2=1/2a+c-2b,
pZz=12π0exp-z24w2pWwdww,
zm=2mΓm+1/2πwmm even0m odd.
kurtZ=z4-3z22z22=12w4-w22w22>0.

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