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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 2 — Jan. 21, 2008
  • pp: 975–992

A unified iterative denoising algorithm based on natural image statistical models: derivation and examples

Shan Tan and Licheng Jiao  »View Author Affiliations

Optics Express, Vol. 16, Issue 2, pp. 975-992 (2008)

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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

OCIS Codes
(100.0100) Image processing : Image processing
(110.7410) Imaging systems : Wavelets

ToC Category:
Image Processing

Original Manuscript: October 10, 2007
Revised Manuscript: November 21, 2007
Manuscript Accepted: November 23, 2007
Published: January 11, 2008

Shan Tan and Licheng Jiao, "A unified iterative denoising algorithm based on natural image statistical models: derivation and examples," Opt. Express 16, 975-992 (2008)

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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).

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