Abstract

Models for the probability density functions of the Fourier amplitude of images are derived. The densities are based on a simple model of an image made up of independent objects and incorporates the observed behavior of the circularly averaged power spectrum versus spatial frequency. The density function over all spatial frequencies gives a good fit to spectral amplitude data from a variety of images.

© 2006 Optical Society of America

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References

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  1. W. K. Pratt, Digital Image Processing (Wiley, 1978).
  2. R. L. Joshi and T. R. Fischer, "Comparison of generalized Gaussian and Laplacian modeling in DCT image-coding," IEEE Signal Process. Lett. 2, 81-82 (1995).
    [CrossRef]
  3. D. L. Richards, "Statistical properties of speech," Proc. Inst. Electr. Eng. 111, 941-949 (1964).
    [CrossRef]
  4. S. Gazor and W. Zhang, "Speech probability distri-bution," IEEE Signal Process. Lett. 10, 204-207 (2003).
    [CrossRef]
  5. A. Mohammad-Djafari, "From deterministic to probabilistic approaches to solve inverse problems," in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE 3459, 2-11 (1998).
    [CrossRef]
  6. J. Ye, K. Webb, C. Bouman, and R. Millane, "Optical diffusion tomography by iterative-coordinate-descent optimization in a Bayesian framework," J. Opt. Soc. Am. A 16, 2400-2412 (1999).
    [CrossRef]
  7. S. Baskaran and R. Millane, "Bayesian image reconstruction from partial image and aliased spectral intensity data," IEEE Trans. Image Process. 8, 1420-1434 (1999).
    [CrossRef]
  8. F. Muller, "Distribution shape of 2-dimensional DCT coefficients of natural images," Electron. Lett. 29, 1935-1936 (1993).
    [CrossRef]
  9. J. H. Chang, "Image probability distribution based on generalized Gamma function," IEEE Signal Process. Lett. 12, 325-328 (2005).
    [CrossRef]
  10. J. W. Shin, J. H. Chang, and N. Kim, "Statistical modeling of speech signals based on generalized Gamma distribution," IEEE Signal Process. Lett. 12, 258-261 (2005).
    [CrossRef]
  11. J. K. Jao, "Amplitude distribution of composite terrain radar clutter and the k-distribution," IEEE Trans. Antennas Propag. AP-32, 1049-1062 (1984).
  12. E. Conte and M. Longo, "Characterization of radar clutter as a spherically invariant random process," IEE Proc. F, Radar Signal Process. 134, 191-196 (1987).
  13. R. P. Millane, "Phase retrieval in crystallography and optics," J. Opt. Soc. Am. A 7, 394-411 (1990).
    [CrossRef]
  14. W. H. Hsiao and R. P. Millane, "Effects of Fourier amplitude and phase errors on image reconstruction," in Image Reconstruction from Incomplete Data III, P. J. Bones, M. A. Fiddy, and R. P. Millane, eds., Proc. SPIE 5562, 27-37 (2004).
  15. D. L. Ruderman, "Origins of scaling in natural images," Vision Res. 37, 3385-3398 (1997).
    [CrossRef]
  16. W. H. Hsiao and R. P. Millane, "Effects of occlusion edges and scaling on the power spectra of natural images," J. Opt. Soc. Am. A 22, 1789-1797 (2005).
    [CrossRef]
  17. D. J. Tolhurst, Y. Tadmor, and T. Chao, "Amplitude spectra of natural images," Ophthalmic Physiol. Opt. 12, 229-232 (1992).
    [CrossRef] [PubMed]
  18. I. S. Gradshtein, I. M. Ryzhik, and A. Jeffrey, Table of Integrals, Series, and Products (Academic, 1980).

2005 (3)

J. H. Chang, "Image probability distribution based on generalized Gamma function," IEEE Signal Process. Lett. 12, 325-328 (2005).
[CrossRef]

J. W. Shin, J. H. Chang, and N. Kim, "Statistical modeling of speech signals based on generalized Gamma distribution," IEEE Signal Process. Lett. 12, 258-261 (2005).
[CrossRef]

W. H. Hsiao and R. P. Millane, "Effects of occlusion edges and scaling on the power spectra of natural images," J. Opt. Soc. Am. A 22, 1789-1797 (2005).
[CrossRef]

2003 (1)

S. Gazor and W. Zhang, "Speech probability distri-bution," IEEE Signal Process. Lett. 10, 204-207 (2003).
[CrossRef]

1999 (2)

J. Ye, K. Webb, C. Bouman, and R. Millane, "Optical diffusion tomography by iterative-coordinate-descent optimization in a Bayesian framework," J. Opt. Soc. Am. A 16, 2400-2412 (1999).
[CrossRef]

S. Baskaran and R. Millane, "Bayesian image reconstruction from partial image and aliased spectral intensity data," IEEE Trans. Image Process. 8, 1420-1434 (1999).
[CrossRef]

1998 (1)

A. Mohammad-Djafari, "From deterministic to probabilistic approaches to solve inverse problems," in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE 3459, 2-11 (1998).
[CrossRef]

1997 (1)

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

1995 (1)

R. L. Joshi and T. R. Fischer, "Comparison of generalized Gaussian and Laplacian modeling in DCT image-coding," IEEE Signal Process. Lett. 2, 81-82 (1995).
[CrossRef]

1993 (1)

F. Muller, "Distribution shape of 2-dimensional DCT coefficients of natural images," Electron. Lett. 29, 1935-1936 (1993).
[CrossRef]

1992 (1)

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

1990 (1)

1984 (1)

J. K. Jao, "Amplitude distribution of composite terrain radar clutter and the k-distribution," IEEE Trans. Antennas Propag. AP-32, 1049-1062 (1984).

1964 (1)

D. L. Richards, "Statistical properties of speech," Proc. Inst. Electr. Eng. 111, 941-949 (1964).
[CrossRef]

Baskaran, S.

S. Baskaran and R. Millane, "Bayesian image reconstruction from partial image and aliased spectral intensity data," IEEE Trans. Image Process. 8, 1420-1434 (1999).
[CrossRef]

Bouman, C.

Chang, J. H.

J. H. Chang, "Image probability distribution based on generalized Gamma function," IEEE Signal Process. Lett. 12, 325-328 (2005).
[CrossRef]

J. W. Shin, J. H. Chang, and N. Kim, "Statistical modeling of speech signals based on generalized Gamma distribution," IEEE Signal Process. Lett. 12, 258-261 (2005).
[CrossRef]

Chao, T.

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

Conte, E.

E. Conte and M. Longo, "Characterization of radar clutter as a spherically invariant random process," IEE Proc. F, Radar Signal Process. 134, 191-196 (1987).

Fischer, T. R.

R. L. Joshi and T. R. Fischer, "Comparison of generalized Gaussian and Laplacian modeling in DCT image-coding," IEEE Signal Process. Lett. 2, 81-82 (1995).
[CrossRef]

Gazor, S.

S. Gazor and W. Zhang, "Speech probability distri-bution," IEEE Signal Process. Lett. 10, 204-207 (2003).
[CrossRef]

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, and A. Jeffrey, Table of Integrals, Series, and Products (Academic, 1980).

Hsiao, W. H.

W. H. Hsiao and R. P. Millane, "Effects of occlusion edges and scaling on the power spectra of natural images," J. Opt. Soc. Am. A 22, 1789-1797 (2005).
[CrossRef]

W. H. Hsiao and R. P. Millane, "Effects of Fourier amplitude and phase errors on image reconstruction," in Image Reconstruction from Incomplete Data III, P. J. Bones, M. A. Fiddy, and R. P. Millane, eds., Proc. SPIE 5562, 27-37 (2004).

Jao, J. K.

J. K. Jao, "Amplitude distribution of composite terrain radar clutter and the k-distribution," IEEE Trans. Antennas Propag. AP-32, 1049-1062 (1984).

Jeffrey, A.

I. S. Gradshtein, I. M. Ryzhik, and A. Jeffrey, Table of Integrals, Series, and Products (Academic, 1980).

Joshi, R. L.

R. L. Joshi and T. R. Fischer, "Comparison of generalized Gaussian and Laplacian modeling in DCT image-coding," IEEE Signal Process. Lett. 2, 81-82 (1995).
[CrossRef]

Kim, N.

J. W. Shin, J. H. Chang, and N. Kim, "Statistical modeling of speech signals based on generalized Gamma distribution," IEEE Signal Process. Lett. 12, 258-261 (2005).
[CrossRef]

Longo, M.

E. Conte and M. Longo, "Characterization of radar clutter as a spherically invariant random process," IEE Proc. F, Radar Signal Process. 134, 191-196 (1987).

Millane, R.

J. Ye, K. Webb, C. Bouman, and R. Millane, "Optical diffusion tomography by iterative-coordinate-descent optimization in a Bayesian framework," J. Opt. Soc. Am. A 16, 2400-2412 (1999).
[CrossRef]

S. Baskaran and R. Millane, "Bayesian image reconstruction from partial image and aliased spectral intensity data," IEEE Trans. Image Process. 8, 1420-1434 (1999).
[CrossRef]

Millane, R. P.

W. H. Hsiao and R. P. Millane, "Effects of occlusion edges and scaling on the power spectra of natural images," J. Opt. Soc. Am. A 22, 1789-1797 (2005).
[CrossRef]

R. P. Millane, "Phase retrieval in crystallography and optics," J. Opt. Soc. Am. A 7, 394-411 (1990).
[CrossRef]

W. H. Hsiao and R. P. Millane, "Effects of Fourier amplitude and phase errors on image reconstruction," in Image Reconstruction from Incomplete Data III, P. J. Bones, M. A. Fiddy, and R. P. Millane, eds., Proc. SPIE 5562, 27-37 (2004).

Mohammad-Djafari, A.

A. Mohammad-Djafari, "From deterministic to probabilistic approaches to solve inverse problems," in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE 3459, 2-11 (1998).
[CrossRef]

Muller, F.

F. Muller, "Distribution shape of 2-dimensional DCT coefficients of natural images," Electron. Lett. 29, 1935-1936 (1993).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, 1978).

Richards, D. L.

D. L. Richards, "Statistical properties of speech," Proc. Inst. Electr. Eng. 111, 941-949 (1964).
[CrossRef]

Ruderman, D. L.

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

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, and A. Jeffrey, Table of Integrals, Series, and Products (Academic, 1980).

Shin, J. W.

J. W. Shin, J. H. Chang, and N. Kim, "Statistical modeling of speech signals based on generalized Gamma distribution," IEEE Signal Process. Lett. 12, 258-261 (2005).
[CrossRef]

Tadmor, Y.

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

Tolhurst, D. J.

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

Webb, K.

Ye, J.

Zhang, W.

S. Gazor and W. Zhang, "Speech probability distri-bution," IEEE Signal Process. Lett. 10, 204-207 (2003).
[CrossRef]

Electron. Lett. (1)

F. Muller, "Distribution shape of 2-dimensional DCT coefficients of natural images," Electron. Lett. 29, 1935-1936 (1993).
[CrossRef]

IEEE Signal Process. Lett. (4)

J. H. Chang, "Image probability distribution based on generalized Gamma function," IEEE Signal Process. Lett. 12, 325-328 (2005).
[CrossRef]

J. W. Shin, J. H. Chang, and N. Kim, "Statistical modeling of speech signals based on generalized Gamma distribution," IEEE Signal Process. Lett. 12, 258-261 (2005).
[CrossRef]

R. L. Joshi and T. R. Fischer, "Comparison of generalized Gaussian and Laplacian modeling in DCT image-coding," IEEE Signal Process. Lett. 2, 81-82 (1995).
[CrossRef]

S. Gazor and W. Zhang, "Speech probability distri-bution," IEEE Signal Process. Lett. 10, 204-207 (2003).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. K. Jao, "Amplitude distribution of composite terrain radar clutter and the k-distribution," IEEE Trans. Antennas Propag. AP-32, 1049-1062 (1984).

IEEE Trans. Image Process. (1)

S. Baskaran and R. Millane, "Bayesian image reconstruction from partial image and aliased spectral intensity data," IEEE Trans. Image Process. 8, 1420-1434 (1999).
[CrossRef]

J. Opt. Soc. Am. A (3)

Ophthalmic Physiol. Opt. (1)

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

Proc. Inst. Electr. Eng. (1)

D. L. Richards, "Statistical properties of speech," Proc. Inst. Electr. Eng. 111, 941-949 (1964).
[CrossRef]

Proc. SPIE (1)

A. Mohammad-Djafari, "From deterministic to probabilistic approaches to solve inverse problems," in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE 3459, 2-11 (1998).
[CrossRef]

Vision Res. (1)

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

Other (4)

I. S. Gradshtein, I. M. Ryzhik, and A. Jeffrey, Table of Integrals, Series, and Products (Academic, 1980).

W. H. Hsiao and R. P. Millane, "Effects of Fourier amplitude and phase errors on image reconstruction," in Image Reconstruction from Incomplete Data III, P. J. Bones, M. A. Fiddy, and R. P. Millane, eds., Proc. SPIE 5562, 27-37 (2004).

W. K. Pratt, Digital Image Processing (Wiley, 1978).

E. Conte and M. Longo, "Characterization of radar clutter as a spherically invariant random process," IEE Proc. F, Radar Signal Process. 134, 191-196 (1987).

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Figures (1)

Fig. 1
Fig. 1

(a) A typical image and (b) its amplitude histogram (dots) and the density function Eq. (19) (solid curve).

Equations (21)

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F ( u , v ) = f ( x , y ) exp ( i 2 π ( u x + v y ) ) d x d y ,
F ( u , v ) = F ( u , v ) exp ( i ϕ ( u , v ) ) .
f ( x , y ) = n = 1 N f n ( x x n , y y n ) ,
F ( u , v ) = n = 1 N F n ( u , v ) exp ( i 2 π ( u x n + v y n ) ) ,
F ( u , v ) = n = 1 N F n ( ρ ) exp ( i 2 π ( u x n + v y n ) ) .
σ 2 ( ρ ) = 1 2 N n = 1 N F n ( ρ ) 2 .
P ( ρ ; A ) = A σ 2 ( ρ ) exp ( A 2 2 σ 2 ( ρ ) ) .
f ( x , y ) = n = 1 N b n δ ( x x n , y y n ) ,
F ( u , v ) = n = 1 N b n exp ( i 2 π ( u x n + v y n ) ) ,
P ( A ) = A σ 2 exp ( A 2 2 σ 2 ) ,
P ( ρ ; I ) = 1 2 σ 2 ( ρ ) exp ( I 2 σ 2 ( ρ ) ) .
I ( ρ ) ϕ = 2 σ 2 ( ρ ) .
I ( ρ ) ϕ = k ρ γ ,
σ 2 ( ρ ) = k 2 ρ γ .
P ( ρ ; I ) = ρ γ k exp ( ρ γ I k ) ,
P ( ρ ; A ) = 2 A ρ γ k exp ( ρ γ A 2 k ) .
P ( I ) = ρ 1 ρ 2 P ( ρ ; I ) 2 π ρ d ρ ρ 1 ρ 2 2 π ρ d ρ = 2 k ( ρ 2 2 ρ 1 2 ) ρ 1 ρ 2 ρ γ + 1 exp ( I ρ γ k ) d ρ .
P ( I ) = 2 k 2 γ γ ( ρ 2 2 ρ 1 2 ) I 1 + 2 γ I ρ 1 γ k I ρ 2 γ k t 2 γ exp ( t ) d t = 2 k 2 γ γ ( ρ 2 2 ρ 1 2 ) I 1 + 2 γ [ Γ ( 1 + 2 γ , ρ 1 γ I k ) Γ ( 1 + 2 γ , ρ 2 γ I k ) ] ,
P ( A ) = 2 ( c 1 ) A 1 2 c ( b c 1 a c 1 ) [ Γ ( c , a A 2 ) Γ ( c , b A 2 ) ] ,
( a , b , c ) = arg max ( a , b , c > 1 ) [ i log P a b c ( A i ) ] ,
e = { i [ α P ( A i ) h ( A i ) ] 2 i [ h ( A i ) ] 2 } 1 2

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