Abstract

Lots of prior models for natural image wavelet coefficients have been proposed in the last two decades. Although most of them belong to the Scale Mixture of Gaussian (GSM) models, they are of obviously different analytical forms. As a result, Bayesian image denoising algorithms based on these prior models are also very different from each other. In this paper, we develop a novel image denoising algorithm by combining the Expectation Maximization (EM) scheme and the properties of the GSM models. The developed algorithm is of a simple iterative form and can converge quickly. It only uses the derivative information of a probability density function and is suitable for all GSM-type prior models that have an analytical probability density function. The developed algorithm can be viewed as a unified Bayesian image denoising framework. As examples, several classical and recently-proposed prior models for natural image wavelet coefficients are tested and some new results are obtained.

© 2008 Optical Society of America

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References

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  1. A. Srivastava, B. Lee, E. P. Simoncelli, and S.-C. Zhu, "On advances in statistical modeling of natural images," Journal of Mathematical Imaging and Vision 18, 17-33 (2003).
    [CrossRef]
  2. E. P. Simoncelli and E. H. Adelson, "Noise removal via Bayesian wavelet coring," in Proc. 3rd Int. Conf. on Image Processing (Lausanne, Switzerland, 1996), pp. 379-382.
    [CrossRef]
  3. S. G. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Pattern Anal. Machine Intell. 11, 674-693 (1989).
    [CrossRef]
  4. A. Achim, A. Bezerianos, and P. Tsakalides, "Novel Bayesian multiscale method for speckle removal in medical ultrasound images," IEEE Trans. Med. Imag. 20, 772-783 (2001).
    [CrossRef]
  5. A. Achim, P. Tsakalides, and A. Bezerianos, "SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling," IEEE Trans. Geosci. Remote Sensing 41, 1773-1784 (2001).
    [CrossRef]
  6. J. Huang, "Statistics of natural images and models," PhDthesis, Division of Appled Mathematics, Brown University (2000).
  7. U. Grenander and A. Srivastava, "Probability models for clutter in natural images," IEEE Pattern Anal. Machine Intell. 23, 424-429 (2001).
    [CrossRef]
  8. A. Srivastava, X. Liu, and U. Grenander, "Universal analytical forms for modeling image probability," IEEE Pattern Anal. Machine Intell. 28, 1200-1214 (2002).
    [CrossRef]
  9. J. M. Fadili and L. Boubchir, "Analytical form for a Bayesian wavelet estimator of images using the Bessel K Form densities," IEEE Trans. Image Processing 14, 231-240 (2005).
    [CrossRef]
  10. B. Vidakovic, "Nonlinear wavelet shrinkage with Bayes rules and Bayes factors," J. Amer. Stat. Assoc. 93, 173- 179 (1998).
    [CrossRef]
  11. H. A. Chipman, E. D. Kolaczyk, and R. E. McCulloch, "Adaptive Bayesian wavelet shrinkage," J. Amer. Stat. Assoc. 92, 1413-1421 (1997).
    [CrossRef]
  12. 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]
  13. L. Sendur and I. W. Selesnick, "Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency," IEEE Trans. Signal Processing 50, 2744-2756 (2002).
    [CrossRef]
  14. L. Sendur and I.W. Selesnick, "Bivariate shrinkage with local variance estimation," IEEE Signal Processing Lett. 9, 438-441 (2002).
    [CrossRef]
  15. T. Eltoft, K. Taesu, and Te-Won Lee, "On the multivariate Laplace distribution," IEEE Signal Processing Lett. 13, 300-303 (2006).
    [CrossRef]
  16. S. Tan and L. Jiao, "Multishrinkage: analytical form for a Bayesian wavelet estimator based on the multivariate Laplacian model," Opt. Lett. 32, 2583-2585 (2007).
    [CrossRef] [PubMed]
  17. S. Tan and L. Jiao, "Wavelet-based Bayesian image estimation: from marginal and bivariate prior models to multivariate prior models," IEEE Trans. Image Processing, submitted, (2006).
  18. A. Achim and E. E. Kuruoglu, "Image denoising using bivariate-stable distributions in the complex wavelet domain," IEEE Signal Processing Lett. 12, 17-20 (2005).
    [CrossRef]
  19. J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Processing 12, 1338-1351 (2003).
    [CrossRef]
  20. T. A. Øigard, A. Hanssen, R. E. Hansen and F. Godtliebsen, "EM-estimation and modeling of heavy-tailed processes with the multivariate normal inverse Gaussian distribution," Signal Process. 85, 1655-1673 (2005).
    [CrossRef]
  21. S. Tan and L. Jiao, "Multivariate statistical models for image denoising in the wavelet domain," International Journal of Computer Vision 75, 209-230 (2007).
    [CrossRef]
  22. P. V. Gehler and M. Welling, "Product of Edgeperts," in Advances in Neural Information Processing System, Y. Weiss, B. Scholkopf, and J. Platt, eds. (Cambridge, MA., 2005), pp. 419-426.
  23. M. J. Wainwright and E. P. Simoncelli, "Random cascades on wavelet trees and their use in analyzing and modeling natural images," Applied and Computational Harmonic Analysis 11, 89-123 (2001).
    [CrossRef]
  24. J. M. Bioucas-Dias, "Bayesian wavelet-based image deconvolution: a GEMalgorithmexploiting a class of heavytailed priors," IEEE Trans. Image Processing 15, 937-951 (2006).
    [CrossRef]
  25. R. W. Buccigrossi and E. P. Simoncelli, "Image compression via joint statistical characterization in the wavelet domain," IEEE Trans. Image Processing 8, 1688-1701 (1999).
    [CrossRef]
  26. M. Abramowitz and C. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (New York: Dover, 1972).

2007 (2)

S. Tan and L. Jiao, "Multivariate statistical models for image denoising in the wavelet domain," International Journal of Computer Vision 75, 209-230 (2007).
[CrossRef]

S. Tan and L. Jiao, "Multishrinkage: analytical form for a Bayesian wavelet estimator based on the multivariate Laplacian model," Opt. Lett. 32, 2583-2585 (2007).
[CrossRef] [PubMed]

2006 (3)

T. Eltoft, K. Taesu, and Te-Won Lee, "On the multivariate Laplace distribution," IEEE Signal Processing Lett. 13, 300-303 (2006).
[CrossRef]

J. M. Bioucas-Dias, "Bayesian wavelet-based image deconvolution: a GEMalgorithmexploiting a class of heavytailed priors," IEEE Trans. Image Processing 15, 937-951 (2006).
[CrossRef]

S. Tan and L. Jiao, "Wavelet-based Bayesian image estimation: from marginal and bivariate prior models to multivariate prior models," IEEE Trans. Image Processing, submitted, (2006).

2005 (3)

A. Achim and E. E. Kuruoglu, "Image denoising using bivariate-stable distributions in the complex wavelet domain," IEEE Signal Processing Lett. 12, 17-20 (2005).
[CrossRef]

J. M. Fadili and L. Boubchir, "Analytical form for a Bayesian wavelet estimator of images using the Bessel K Form densities," IEEE Trans. Image Processing 14, 231-240 (2005).
[CrossRef]

T. A. Øigard, A. Hanssen, R. E. Hansen and F. Godtliebsen, "EM-estimation and modeling of heavy-tailed processes with the multivariate normal inverse Gaussian distribution," Signal Process. 85, 1655-1673 (2005).
[CrossRef]

2003 (2)

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Processing 12, 1338-1351 (2003).
[CrossRef]

A. Srivastava, B. Lee, E. P. Simoncelli, and S.-C. Zhu, "On advances in statistical modeling of natural images," Journal of Mathematical Imaging and Vision 18, 17-33 (2003).
[CrossRef]

2002 (3)

A. Srivastava, X. Liu, and U. Grenander, "Universal analytical forms for modeling image probability," IEEE Pattern Anal. Machine Intell. 28, 1200-1214 (2002).
[CrossRef]

L. Sendur and I. W. Selesnick, "Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency," IEEE Trans. Signal Processing 50, 2744-2756 (2002).
[CrossRef]

L. Sendur and I.W. Selesnick, "Bivariate shrinkage with local variance estimation," IEEE Signal Processing Lett. 9, 438-441 (2002).
[CrossRef]

2001 (4)

M. J. Wainwright and E. P. Simoncelli, "Random cascades on wavelet trees and their use in analyzing and modeling natural images," Applied and Computational Harmonic Analysis 11, 89-123 (2001).
[CrossRef]

A. Achim, A. Bezerianos, and P. Tsakalides, "Novel Bayesian multiscale method for speckle removal in medical ultrasound images," IEEE Trans. Med. Imag. 20, 772-783 (2001).
[CrossRef]

A. Achim, P. Tsakalides, and A. Bezerianos, "SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling," IEEE Trans. Geosci. Remote Sensing 41, 1773-1784 (2001).
[CrossRef]

U. Grenander and A. Srivastava, "Probability models for clutter in natural images," IEEE Pattern Anal. Machine Intell. 23, 424-429 (2001).
[CrossRef]

1999 (1)

R. W. Buccigrossi and E. P. Simoncelli, "Image compression via joint statistical characterization in the wavelet domain," IEEE Trans. Image Processing 8, 1688-1701 (1999).
[CrossRef]

1998 (1)

B. Vidakovic, "Nonlinear wavelet shrinkage with Bayes rules and Bayes factors," J. Amer. Stat. Assoc. 93, 173- 179 (1998).
[CrossRef]

1997 (1)

H. A. Chipman, E. D. Kolaczyk, and R. E. McCulloch, "Adaptive Bayesian wavelet shrinkage," J. Amer. Stat. Assoc. 92, 1413-1421 (1997).
[CrossRef]

1989 (1)

S. G. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Pattern Anal. Machine Intell. 11, 674-693 (1989).
[CrossRef]

1987 (1)

Achim, A.

A. Achim and E. E. Kuruoglu, "Image denoising using bivariate-stable distributions in the complex wavelet domain," IEEE Signal Processing Lett. 12, 17-20 (2005).
[CrossRef]

A. Achim, A. Bezerianos, and P. Tsakalides, "Novel Bayesian multiscale method for speckle removal in medical ultrasound images," IEEE Trans. Med. Imag. 20, 772-783 (2001).
[CrossRef]

A. Achim, P. Tsakalides, and A. Bezerianos, "SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling," IEEE Trans. Geosci. Remote Sensing 41, 1773-1784 (2001).
[CrossRef]

Bezerianos, A.

A. Achim, P. Tsakalides, and A. Bezerianos, "SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling," IEEE Trans. Geosci. Remote Sensing 41, 1773-1784 (2001).
[CrossRef]

A. Achim, A. Bezerianos, and P. Tsakalides, "Novel Bayesian multiscale method for speckle removal in medical ultrasound images," IEEE Trans. Med. Imag. 20, 772-783 (2001).
[CrossRef]

Bioucas-Dias, J. M.

J. M. Bioucas-Dias, "Bayesian wavelet-based image deconvolution: a GEMalgorithmexploiting a class of heavytailed priors," IEEE Trans. Image Processing 15, 937-951 (2006).
[CrossRef]

Boubchir, L.

J. M. Fadili and L. Boubchir, "Analytical form for a Bayesian wavelet estimator of images using the Bessel K Form densities," IEEE Trans. Image Processing 14, 231-240 (2005).
[CrossRef]

Buccigrossi, R. W.

R. W. Buccigrossi and E. P. Simoncelli, "Image compression via joint statistical characterization in the wavelet domain," IEEE Trans. Image Processing 8, 1688-1701 (1999).
[CrossRef]

Chipman, H. A.

H. A. Chipman, E. D. Kolaczyk, and R. E. McCulloch, "Adaptive Bayesian wavelet shrinkage," J. Amer. Stat. Assoc. 92, 1413-1421 (1997).
[CrossRef]

Eltoft, T.

T. Eltoft, K. Taesu, and Te-Won Lee, "On the multivariate Laplace distribution," IEEE Signal Processing Lett. 13, 300-303 (2006).
[CrossRef]

Fadili, J. M.

J. M. Fadili and L. Boubchir, "Analytical form for a Bayesian wavelet estimator of images using the Bessel K Form densities," IEEE Trans. Image Processing 14, 231-240 (2005).
[CrossRef]

Field, D. J.

Grenander, U.

A. Srivastava, X. Liu, and U. Grenander, "Universal analytical forms for modeling image probability," IEEE Pattern Anal. Machine Intell. 28, 1200-1214 (2002).
[CrossRef]

U. Grenander and A. Srivastava, "Probability models for clutter in natural images," IEEE Pattern Anal. Machine Intell. 23, 424-429 (2001).
[CrossRef]

Jiao, L.

S. Tan and L. Jiao, "Multivariate statistical models for image denoising in the wavelet domain," International Journal of Computer Vision 75, 209-230 (2007).
[CrossRef]

S. Tan and L. Jiao, "Multishrinkage: analytical form for a Bayesian wavelet estimator based on the multivariate Laplacian model," Opt. Lett. 32, 2583-2585 (2007).
[CrossRef] [PubMed]

S. Tan and L. Jiao, "Wavelet-based Bayesian image estimation: from marginal and bivariate prior models to multivariate prior models," IEEE Trans. Image Processing, submitted, (2006).

Kolaczyk, E. D.

H. A. Chipman, E. D. Kolaczyk, and R. E. McCulloch, "Adaptive Bayesian wavelet shrinkage," J. Amer. Stat. Assoc. 92, 1413-1421 (1997).
[CrossRef]

Kuruoglu, E. E.

A. Achim and E. E. Kuruoglu, "Image denoising using bivariate-stable distributions in the complex wavelet domain," IEEE Signal Processing Lett. 12, 17-20 (2005).
[CrossRef]

Lee, B.

A. Srivastava, B. Lee, E. P. Simoncelli, and S.-C. Zhu, "On advances in statistical modeling of natural images," Journal of Mathematical Imaging and Vision 18, 17-33 (2003).
[CrossRef]

Liu, X.

A. Srivastava, X. Liu, and U. Grenander, "Universal analytical forms for modeling image probability," IEEE Pattern Anal. Machine Intell. 28, 1200-1214 (2002).
[CrossRef]

Mallat, S. G.

S. G. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Pattern Anal. Machine Intell. 11, 674-693 (1989).
[CrossRef]

McCulloch, R. E.

H. A. Chipman, E. D. Kolaczyk, and R. E. McCulloch, "Adaptive Bayesian wavelet shrinkage," J. Amer. Stat. Assoc. 92, 1413-1421 (1997).
[CrossRef]

Portilla, J.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Processing 12, 1338-1351 (2003).
[CrossRef]

Selesnick, I. W.

L. Sendur and I. W. Selesnick, "Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency," IEEE Trans. Signal Processing 50, 2744-2756 (2002).
[CrossRef]

Selesnick, I.W.

L. Sendur and I.W. Selesnick, "Bivariate shrinkage with local variance estimation," IEEE Signal Processing Lett. 9, 438-441 (2002).
[CrossRef]

Sendur, L.

L. Sendur and I.W. Selesnick, "Bivariate shrinkage with local variance estimation," IEEE Signal Processing Lett. 9, 438-441 (2002).
[CrossRef]

L. Sendur and I. W. Selesnick, "Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency," IEEE Trans. Signal Processing 50, 2744-2756 (2002).
[CrossRef]

Simoncelli, E. P.

A. Srivastava, B. Lee, E. P. Simoncelli, and S.-C. Zhu, "On advances in statistical modeling of natural images," Journal of Mathematical Imaging and Vision 18, 17-33 (2003).
[CrossRef]

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Processing 12, 1338-1351 (2003).
[CrossRef]

M. J. Wainwright and E. P. Simoncelli, "Random cascades on wavelet trees and their use in analyzing and modeling natural images," Applied and Computational Harmonic Analysis 11, 89-123 (2001).
[CrossRef]

R. W. Buccigrossi and E. P. Simoncelli, "Image compression via joint statistical characterization in the wavelet domain," IEEE Trans. Image Processing 8, 1688-1701 (1999).
[CrossRef]

Srivastava, A.

A. Srivastava, B. Lee, E. P. Simoncelli, and S.-C. Zhu, "On advances in statistical modeling of natural images," Journal of Mathematical Imaging and Vision 18, 17-33 (2003).
[CrossRef]

A. Srivastava, X. Liu, and U. Grenander, "Universal analytical forms for modeling image probability," IEEE Pattern Anal. Machine Intell. 28, 1200-1214 (2002).
[CrossRef]

U. Grenander and A. Srivastava, "Probability models for clutter in natural images," IEEE Pattern Anal. Machine Intell. 23, 424-429 (2001).
[CrossRef]

Strela, V.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Processing 12, 1338-1351 (2003).
[CrossRef]

Taesu, K.

T. Eltoft, K. Taesu, and Te-Won Lee, "On the multivariate Laplace distribution," IEEE Signal Processing Lett. 13, 300-303 (2006).
[CrossRef]

Tan, S.

S. Tan and L. Jiao, "Multishrinkage: analytical form for a Bayesian wavelet estimator based on the multivariate Laplacian model," Opt. Lett. 32, 2583-2585 (2007).
[CrossRef] [PubMed]

S. Tan and L. Jiao, "Multivariate statistical models for image denoising in the wavelet domain," International Journal of Computer Vision 75, 209-230 (2007).
[CrossRef]

S. Tan and L. Jiao, "Wavelet-based Bayesian image estimation: from marginal and bivariate prior models to multivariate prior models," IEEE Trans. Image Processing, submitted, (2006).

Tsakalides, P.

A. Achim, A. Bezerianos, and P. Tsakalides, "Novel Bayesian multiscale method for speckle removal in medical ultrasound images," IEEE Trans. Med. Imag. 20, 772-783 (2001).
[CrossRef]

A. Achim, P. Tsakalides, and A. Bezerianos, "SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling," IEEE Trans. Geosci. Remote Sensing 41, 1773-1784 (2001).
[CrossRef]

Vidakovic, B.

B. Vidakovic, "Nonlinear wavelet shrinkage with Bayes rules and Bayes factors," J. Amer. Stat. Assoc. 93, 173- 179 (1998).
[CrossRef]

Wainwright, M. J.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Processing 12, 1338-1351 (2003).
[CrossRef]

M. J. Wainwright and E. P. Simoncelli, "Random cascades on wavelet trees and their use in analyzing and modeling natural images," Applied and Computational Harmonic Analysis 11, 89-123 (2001).
[CrossRef]

Zhu, S.-C.

A. Srivastava, B. Lee, E. P. Simoncelli, and S.-C. Zhu, "On advances in statistical modeling of natural images," Journal of Mathematical Imaging and Vision 18, 17-33 (2003).
[CrossRef]

Applied and Computational Harmonic Analysis (1)

M. J. Wainwright and E. P. Simoncelli, "Random cascades on wavelet trees and their use in analyzing and modeling natural images," Applied and Computational Harmonic Analysis 11, 89-123 (2001).
[CrossRef]

IEEE Pattern Anal. Machine Intell. (3)

S. G. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Pattern Anal. Machine Intell. 11, 674-693 (1989).
[CrossRef]

U. Grenander and A. Srivastava, "Probability models for clutter in natural images," IEEE Pattern Anal. Machine Intell. 23, 424-429 (2001).
[CrossRef]

A. Srivastava, X. Liu, and U. Grenander, "Universal analytical forms for modeling image probability," IEEE Pattern Anal. Machine Intell. 28, 1200-1214 (2002).
[CrossRef]

IEEE Signal Processing Lett. (3)

A. Achim and E. E. Kuruoglu, "Image denoising using bivariate-stable distributions in the complex wavelet domain," IEEE Signal Processing Lett. 12, 17-20 (2005).
[CrossRef]

L. Sendur and I.W. Selesnick, "Bivariate shrinkage with local variance estimation," IEEE Signal Processing Lett. 9, 438-441 (2002).
[CrossRef]

T. Eltoft, K. Taesu, and Te-Won Lee, "On the multivariate Laplace distribution," IEEE Signal Processing Lett. 13, 300-303 (2006).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (1)

A. Achim, P. Tsakalides, and A. Bezerianos, "SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling," IEEE Trans. Geosci. Remote Sensing 41, 1773-1784 (2001).
[CrossRef]

IEEE Trans. Image Processing (5)

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Processing 12, 1338-1351 (2003).
[CrossRef]

J. M. Bioucas-Dias, "Bayesian wavelet-based image deconvolution: a GEMalgorithmexploiting a class of heavytailed priors," IEEE Trans. Image Processing 15, 937-951 (2006).
[CrossRef]

R. W. Buccigrossi and E. P. Simoncelli, "Image compression via joint statistical characterization in the wavelet domain," IEEE Trans. Image Processing 8, 1688-1701 (1999).
[CrossRef]

J. M. Fadili and L. Boubchir, "Analytical form for a Bayesian wavelet estimator of images using the Bessel K Form densities," IEEE Trans. Image Processing 14, 231-240 (2005).
[CrossRef]

S. Tan and L. Jiao, "Wavelet-based Bayesian image estimation: from marginal and bivariate prior models to multivariate prior models," IEEE Trans. Image Processing, submitted, (2006).

IEEE Trans. Med. Imag. (1)

A. Achim, A. Bezerianos, and P. Tsakalides, "Novel Bayesian multiscale method for speckle removal in medical ultrasound images," IEEE Trans. Med. Imag. 20, 772-783 (2001).
[CrossRef]

IEEE Trans. Signal Processing (1)

L. Sendur and I. W. Selesnick, "Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency," IEEE Trans. Signal Processing 50, 2744-2756 (2002).
[CrossRef]

International Journal of Computer Vision (1)

S. Tan and L. Jiao, "Multivariate statistical models for image denoising in the wavelet domain," International Journal of Computer Vision 75, 209-230 (2007).
[CrossRef]

J. Amer. Stat. Assoc. (2)

B. Vidakovic, "Nonlinear wavelet shrinkage with Bayes rules and Bayes factors," J. Amer. Stat. Assoc. 93, 173- 179 (1998).
[CrossRef]

H. A. Chipman, E. D. Kolaczyk, and R. E. McCulloch, "Adaptive Bayesian wavelet shrinkage," J. Amer. Stat. Assoc. 92, 1413-1421 (1997).
[CrossRef]

J. Opt. Soc. Am. A (1)

Journal of Mathematical Imaging and Vision (1)

A. Srivastava, B. Lee, E. P. Simoncelli, and S.-C. Zhu, "On advances in statistical modeling of natural images," Journal of Mathematical Imaging and Vision 18, 17-33 (2003).
[CrossRef]

Opt. Lett. (1)

Signal Process. (1)

T. A. Øigard, A. Hanssen, R. E. Hansen and F. Godtliebsen, "EM-estimation and modeling of heavy-tailed processes with the multivariate normal inverse Gaussian distribution," Signal Process. 85, 1655-1673 (2005).
[CrossRef]

Other (4)

J. Huang, "Statistics of natural images and models," PhDthesis, Division of Appled Mathematics, Brown University (2000).

M. Abramowitz and C. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (New York: Dover, 1972).

E. P. Simoncelli and E. H. Adelson, "Noise removal via Bayesian wavelet coring," in Proc. 3rd Int. Conf. on Image Processing (Lausanne, Switzerland, 1996), pp. 379-382.
[CrossRef]

P. V. Gehler and M. Welling, "Product of Edgeperts," in Advances in Neural Information Processing System, Y. Weiss, B. Scholkopf, and J. Platt, eds. (Cambridge, MA., 2005), pp. 419-426.

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Figures (4)

Fig. 1.
Fig. 1.

PSNR of the solution versus the number of iterations for the Pepper image: For the asymptotic univariate Bessel K Form model (a) at noise level 20 and (b) at noise level 30; For the multivariate exponential model (3×3 neighborhood structure) (c) at noise level 20 and (d) at noise level 30; For the multivariate Gaussian model (3×3 neighborhood structure) (e) at noise level 20 and (f) at noise level 30

Fig. 2.
Fig. 2.

Restoration images for visual comparison using different univariate prior models on the Pepper image (with additive Gaussian white noise of standard deviation 20, PSNR=22.08dB) (a) Hard threshold, PSNR=26.84dB; (b) MMSE estimator based on the GL model proposed by Simoncelli et. al (GL_MMSE) in [2], PSNR=28.24dB; (c) Unified denoising algorithm based on the Generalized Laplacian model (GL_UA), PSNR=28.31dB; (d) Unified denoising algorithm based on the Bessel K Form model (Bessel K Form_UA), PSNR=28.12dB; (e) Unified denoising algorithm based on the Asymptotic Bessel K Form model (Asy. Bessel K Form_UA), PSNR=28.07dB; (f) Unified denoising algorithm based on the Laplacian model (Laplacian_UA), PSNR=28.28dB

Fig. 3.
Fig. 3.

Restoration images for visual comparison using different multivariate prior models on the Pepper image (with additive Gaussian white noise of standard deviation 20, PSNR=22.08dB) (a) 3×3 Wiener filtering in the wavelet domain, PSNR=27.84dB; (b) Unified denoising algorithm based on the Gaussian model (Gaussian_UA: 3×3), PSNR=27.85dB; (c) GSM-based MMSE algorithm developed by Portilla et. al in [19] (GSM_MMSE: 3×3+1), PSNR=29.45dB; (d) Unified denoising algorithm based on the Laplacian model (Laplacian_UA: 3×3+1), PSNR=29.43dB; (e) Unified denoising algorithm based on the Elliptically Contoured Distribution model (MECDF_UA: 3×3+1), PSNR=29.19dB; (f) Unified denoising algorithm based on the exponential model (MEM_UA: 3×3+1), PSNR=29.31dB

Fig. 4.
Fig. 4.

Crop of restoration images for visual comparison using different multivariate prior models on the Boat image (with additive Gaussian white noise of standard deviation 20, PSNR=22.10dB) (a) GSM-based MMSE algorithm developed by Portilla et. al in [19] (GSM_MMSE: 3×3+1), PSNR=29.62dB; (b) Unified denoising algorithm based on the Laplacian model (Laplacian_UA: 3×3+1), PSNR=29.76dB; (c) Unified denoising algorithm based on the Elliptically Contoured Distribution model (MECDF_UA: 3×3+ 1), PSNR=29.58dB; (d) Unified denoising algorithm based on the exponential model (MEM_UA: 3×3+1), PSNR=29.67dB

Tables (2)

Tables Icon

Table 1. Comparison of denoising performance using different prior models in terms of PSNR for Barbara and Lena image

Tables Icon

Table 2. Comparison of denoising performance using different prior models in terms of PSNR for Boat and Pepper image

Equations (60)

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Y i = X i + ε ,
Q ( X , X t ) = E Z { log P ( X , Y , Z ) | Y , X t } .
Q ( X , X t ) = Z [ ( log P ( Y | X ) + log P ( X | Z ) + log P ( Z ) ) ] P ( Z | Y , X t ) d Z
= log P ( Y | X ) + Z log P ( X | Z ) P ( Z | Y , X t ) d Z + C ( Z , Y , X t ) ,
Q ( X , X t ) = log P ( Y X ) 1 2 i = 1 M X i T ρ 1 X i z z i 1 P ( Z X t ) d Z + C ( Z , Y , X t )
= log P ( Y X ) 1 2 i = 1 M X i T ρ 1 X i z i z i 1 P ( z i X i t ) d z i + C ( Z , Y , X t ) .
X i t + 1 = arg max X i Q ( X , X t ) = arg max X i [ log P ( Y | X ) 1 2 i = 1 N X i T ρ 1 X i Ω ( X i t ) ]
  = arg max X i [ 1 2 i = 1 M ( Y i X i ) T ρ ε 1 ( Y i X i ) 1 2 i = 1 M X i T ρ 1 X i Ω ( X i t ) ]
= arg zero d d X i [ i = 1 M ( Y i X i ) T ρ ε 1 ( Y i X i ) + i = 1 M X i T ρ 1 X i Ω ( X i t ) ]
= arg zero [ 2 ρ ε 1 ( Y i X i ) + 2 Ω ( X i t ) ρ 1 X i ] .
Ω( X i t ) ρ 1 X i t+1 = ρ ε 1 ( Y i X t t+1 ),
X i t + 1 = ( Ω ( X i t ) ρ 1 + ρ ε 1 ) 1 ρ ε 1 Y i .
X i t + 1 = ( 2 f ( r ) d dr f ( r ) ρ 1 + ρ ε 1 ) 1 ρ ε 1 Y i
= ( 2 d dr log f ( r ) ρ 1 + ρ ε 1 ) 1 ρ ε 1 Y i .
X i = ( 2 d dr log f ( r ) ρ 1 + ρ ε 1 ) 1 ρ ε 1 Y i .
X i = ( ρ 1 + ρ ε 1 ) 1 ρ ε 1 Y i ,
P ( X i ) = 3 2 π σ 2 exp ( 3 σ x 1 2 + x 2 2 )
                = f ( r ) = 3 2 π σ 2 exp ( 3 r 1 2 ) ,
x 1 ̂ = ( y 1 2 + y 2 2 3 σ n 2 σ ) + y 1 2 + y 2 2 y 1 ,
S T ρ 1 S = ( S 1 ρ S T ) 1 = ( Q Λ Q T ) 1 = Q Λ 1 Q T .
X i t + 1 = S Q ( I d 2 d d r log f ( r ) Λ 1 ) 1 ( SQ ) 1 Y i
= SQ [ λ i λ i 2 d dr log f ( r ) ] 1 d ( SQ ) 1 Y i ,
r = ( X i t ) T ρ 1 X i t = ( X i t ) T S T S T ρ 1 S S 1 X i t = ( X i t ) T S T Q Λ 1 Q T S 1 X i t
= ( Q T S 1 X i t ) T Λ 1 Q T S 1 X i t = i = 1 d 1 λ i V i 2 ,
P ( x ) = ( 2 π ) 1 2 σ 1 exp ( r 2 ) = C · f G ( r ) ,
d dr log f G ( r ) = 1 2 .
P ( x ) = 1 Ψ ( s , p ) exp ( x s p ) ,
σ 2 = σ n 2 + s 2 Γ ( 3 p ) Γ ( 1 p ) , m 4 = 3 σ n 4 + 6 σ n 2 s 2 Γ ( 3 p ) Γ ( 1 p ) + s 4 Γ ( 5 p ) Γ ( 1 p ) .
P ( x ) = C · exp ( x 2 p ) = C · exp [ ( x 2 s 2 ) p 2 ] = C · exp [ ( x 2 σ 2 ) p 2 ( σ 2 s 2 ) p 2 ]
= C · exp [ ( σ s ) p r p 2 ] = C · exp ( α r β ) = C · f GL ( r ) ,
d dr log f GL ( r ) = α β r β 1 .
P ( x ) = 1 π Γ ( p ) ( s 2 ) p 2 1 4 x 2 p 1 2 K p 1 2 ( 2 s x ) ,
σ ~ 2 = ps + σ n 2 , κ ~ = 3 p + 3 ,
P ( x ) = C · r p 2 1 4 K p 1 2 ( 2 pr ) = C · f BK ( r ) ,
d dz [ K v ( z ) ] = d dz [ z v K v ( z ) z v ] = K v 1 ( z ) v z 1 K v ( z ) .
d dr log f BK ( r ) = ( p 2 1 4 ) 1 r + d dr K p 1 2 ( 2 p r ) K p 1 2 ( 2 p r ) = p 2 p r K p 3 2 ( 2 p r ) K p 1 2 ( 2 p r ) .
P ( x ) = 1 Γ ( p ) ( 2 s ) p 2 x p 1 exp ( 2 s x ) .
P ( x ) = 1 Γ ( p ) ( 2 s ) p 2 ( x 2 ) p 1 2 exp ( 2 s ( x 2 ) 1 2 )
= 1 Γ ( p ) ( 2 s ) p 2 σ p 1 ( x 2 σ 2 ) p 1 2 exp ( 2 σ 2 s ( x 2 σ 2 ) 1 2 )
= C · r p 1 2 exp ( r 1 2 2 σ 2 s ) .
P ( x ) = C · r p 1 2 exp ( 2 pr ) = C · f AABK ( r ) ,
d dr log f AABK ( r ) = p 1 2 r p 2 r .
P ( x ) = 1 p exp ( x s ) ,
P ( x ) = 1 p exp ( 2 r ) = C · f BE ( r ) ,
d dr log f BE ( r ) = 1 2 r .
P ( X ) = ( 2 π ) N 2 1 2 exp [ 1 2 X T ρ 1 X ] = C · f G ( r ) ,
d dr log f G ( r ) = 1 2 .
P ( X ) = ( 2 π ) d 2 2 λ K d 2 1 ( 2 λ q ( X ) ) ( λ 2 q ( X ) ) d 2 1 ,
P ( X ) = ( 2 π ) d 2 2 1 2 + d 4 λ d 2 K d 2 1 ( 2 r ) r d 4 1 2 = C · K d 2 1 ( 2 r ) r d 4 1 2 = C · f ML ( r ) ,
d dr log f ML ( r ) = d dr log K d 2 1 ( 2 r ) 1 r ( d 4 1 2 ) = K d 2 2 ( 2 r ) 2 r K d 2 1 ( 2 r ) ( d 2 1 ) 1 r .
P ( X ) = 3 2 π ρ 1 2 exp ( 3 r 1 2 ) = C · f ECDF _ 2 ( r ) ,
d dr log f ECDF _ 2 ( r ) = 3 2 r 1 2 .
P ( X ) = 3 3 4 π 2 ρ 1 2 r 1 2 exp ( 3 r 1 2 ) = C · f ECDF _ 4 ( r ) ,
d dr log f ECDF _ 4 ( r ) = 1 2 r 3 2 r 1 2 .
P ( X ) = 9 3 32 π 5 ρ 1 2 ( 5 r 7 2 + 5 3 r 3 + 6 r 5 2 + 3 r 2 ) exp ( 3 r 1 2 ) = f ECDF _ 10 ( r ) ,
d dr log f ECDF _ 10 ( r ) = 3 r 2 6 3 r 3 2 15 r 7 3 r 1 2 15 15 3 r 1 2 17.5 r 1 3 r 3 2 + 6 r + 5 3 r 1 2 + 5 .
P ( X ) = C · exp ( a 1 a 2 r a 3 ) = C · f MEM _ d ( r ) = C · exp ( a 2 r a 3 ) ,
d dr log f MEM _ d ( r ) = a 2 a 3 r a 3 1 .
dP ( X i t z i ) dX i _ cent t = ( 2 π ) d 2 z i ρ 1 2 exp [ 1 2 ( X i t ) T ( z i ρ ) 1 X i t ] ( 1 2 d dX i _ cent t ( X i t ) T ( z i ρ ) 1 X i t )
= P ( X i t z i ) ( 1 2 z i 1 ) d ( X i t ) T ρ 1 X i t dX i _ cent t = 1 2 z i 1 P ( X i t z i ) d ( X i t ) T ρ 1 X i t dX i _ cent t ,

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