Advertisement

International Journal of Computer Vision

, Volume 75, Issue 2, pp 209–230 | Cite as

Multivariate Statistical Models for Image Denoising in the Wavelet Domain

  • Shan Tan
  • Licheng Jiao
Article

Abstract

We model wavelet coefficients of natural images in a neighborhood using the multivariate Elliptically Contoured Distribution Family (ECDF) and discuss its application to the image denoising problem. A desirable property of the ECDF is that a multivariate Elliptically Contoured Distribution (ECD) can be deduced directly from its lower dimension marginal distribution. Using the property, we extend a bivariate model that has been used to successfully model the 2-D joint probability distribution of a two dimension random vector—a wavelet coefficient and its parent—to multivariate cases. Though our method only provides a simple and rough characterization of the full probability distribution of wavelet coefficients in a neighborhood, we find that the resulting denoising algorithm based on the extended multivariate models is computably tractable and produces state-of-the-art restoration results. In addition, we discuss the equivalence relation between our denoising algorithm and several other state-of-the-art denoising algorithms. Our work provides a unified mathematic interpretation of a type of statistical denoising algorithms. We also analyze the limitations and advantages of algorithms of this type.

Keywords

natural image statistics multivariate model elliptically contoured distribution family image denoising 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, T.W. and Fang, K.T. 1987. On the theory of multivariate elliptically contoured distributions. Sankhya, 49 (Series A):305–315.Google Scholar
  2. Anderson, T.W. and Fang, K.T. 1992. Theory and applications elliptically contoured and related distributions. In The Development of Statistics: Recent Contributions from China, Longman, London, pp. 41–62.Google Scholar
  3. Andrews, D. and Mallows, C. 1974. Scale mixtures of normal distributions. J. R. Statist. Soc, 36:99.zbMATHMathSciNetGoogle Scholar
  4. Awate, S.P. and Whitaker, R.T. 2005. Higher-order image statistics for unsupervised, information-theoretic, adaptive, image filtering. In Proc. IEEE Int. Conf. Computer Vision and Pattern Recognition. Available: http://www.cs.utah.edu/research/techreports/
  5. Brehm, H. and Stammler, W. 1987. Description and generation of spherically invariant speech-model signals. Signal Processing, 9:119–141.CrossRefGoogle Scholar
  6. Candès, E.J. 1999. Harmonic analysis of neural netwoks. Appl. Comput. Harmon. Anal., 6:197–218.zbMATHCrossRefMathSciNetGoogle Scholar
  7. Candès, E.J. and Donoho, D.L. 1999. Curvelets—A surprisingly effective nonadaptive representation for objects with edges. In Curve and Surface Fitting: Saint-Malo 1999, A. Cohen, C. Rabut, and L.L. Schumaker (eds.). Van-derbilt Univ. Press, Nashville, TN.Google Scholar
  8. Chang, S.G., Yu, B., and Vetterli, M. 1998. Spatially adaptive wavelet thresh-olding with context modeling for image denoising. In Proc. 5th IEEE Int. Conf. Image Processing, Chicago.Google Scholar
  9. Crouse, M.S., Nowak, R.D., and Baraniuk, R.C. 1998. Wavelet-based statistical signal processing using hidden Markov models. IEEE Trans. Signal Processing, 46:886–902.CrossRefMathSciNetGoogle Scholar
  10. Do, M.N. and Vetterli, M. 2003. The finite ridgelet transform for image representation. IEEE Trans. Image Processing, 12:16–28.CrossRefMathSciNetGoogle Scholar
  11. Do, M.N. and Vetterli, M. 2005. The contourlet transform: An efficient directional multiresolution image representation. IEEE Trans. Image Processing, 14:2091–2106.CrossRefMathSciNetGoogle Scholar
  12. Donoho, D.L. 2000. Orthonormal ridgelet and linear singularities. SIAM J. Math Anal., 31:1062–1099.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Fang, K.T. and Zhang, Y.T. 1990. Generalized Multivariate Analysis. Science Press and Springer-Verlag, Beijing and Berlin.zbMATHGoogle Scholar
  14. Field, D. 1987. Relations between the statistics of natural images and the response properties of cortical cells. J. Opt. Soc. Amer. A, 4(12):2379–2394.CrossRefGoogle Scholar
  15. Figueiredo, M. and Nowak., R. 2001. Wavelet-based image estimation: An empirical Bayes approach using Jeffrey’s noninformative prior. IEEE Trans. Image Processing, 10:1322–1331.zbMATHCrossRefMathSciNetGoogle Scholar
  16. Gehler, P.V. and Welling, M. 2005. Product of Edgeperts. Advances in Neural Information Processing System, 18, 8, MIT Press, Cambridge, MA.Google Scholar
  17. Huang, J. 2000. Statistics of natural images and models. Ph.D. Thesis, Division of Appled Mathematics, Brown University, RI.Google Scholar
  18. Lee, A.B., Pedersen, K.S., and Mumford, D. 2003. The nonlinear statistics of high-contrast patches in natural Images. International Journal of Computer Vision, 54:83–103.zbMATHCrossRefGoogle Scholar
  19. Liu, J. and Moulin, P. 2001. Information-theoretic analysis of interscale and intrascale dependencies between image wavelet coefficients. IEEE Trans. Image Processing, 10:1647–1658.Google Scholar
  20. Mallat, S.G. 1989. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Pattern Anal. Machine Intell., 11:674–693.zbMATHCrossRefGoogle Scholar
  21. Mihcak, M.K., Kozintsev, I., Ramchandran, K., and Moulin, P. 1999. Low complexity image denoising based on statistical modeling of wavelet coefficients. IEEE Signal Processing Lett., 6:300–303.CrossRefGoogle Scholar
  22. Moulin, P. and Liu, J. 1999. Analysis of multiresolution image denoising schemes using a generalized Gaussian and complexity priors. IEEE Trans. Inform. Theory, 45:909–919.zbMATHCrossRefMathSciNetGoogle Scholar
  23. Mumford, D. 2005. Empirical statistics and stochastic models for visual signals. In New Directions in Statistical Signal Processing: From Systems to Brain, S. Haykin, J.C. Principe, T.J. Sejnowski, J. McWhirter (eds.). MIT Press, Cambridge, MA.Google Scholar
  24. Pižurica, A., Philips, W., Lemahieu, I., and Acheroy, M. 2002. A joint inter-and intrascale statistical model for Bayesian wavelet based image de-noising. IEEE Trans. Image Processing, 11:545–557.CrossRefGoogle Scholar
  25. Po, D.D.-Y. and Do, M.N. 2006. Directional multiscale modeling of images using the contourlet transform. IEEE Trans. Image Processing, 15:1610–1620.CrossRefMathSciNetGoogle Scholar
  26. Portilla, J., Strela, V., Wainwright, M.J., and Simoncelli, E.P. 2003. Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Trans. Image Processing, 12:1338–1351.CrossRefMathSciNetGoogle Scholar
  27. Rangaswamy, M., Weiner, D., and Ozturk, A. 1993. Non-Gaussian random vector identification using spherically invariant random processes. IEEE Trans. Aerosp. Electron. Syst., 29:111–123.CrossRefGoogle Scholar
  28. Schoenberg, I.J. 1938. Metric spaces and completely monotone functions. Ann. Math., 39:811–841.CrossRefMathSciNetGoogle Scholar
  29. Sendur, L. and Selesnick, I.W. 2002a. Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency. IEEE Trans. Signal Processing, 50:2744–2756.CrossRefGoogle Scholar
  30. Sendur, L. and Selesnick, I.W. 2002b. Bivariate shrinkage with local variance estimation. IEEE Signal Processing Lett., 9:438–441.CrossRefGoogle Scholar
  31. Shapiro, J. 1993. Embedded image coding using zerotrees of wavelet coefficients. IEEE Trans. Sig. Proc., 41(12):3445–3462.zbMATHCrossRefGoogle Scholar
  32. Simoncelli. E.P. 1997. Statistical models for images: Compression, restoration and synthesis. In Proc. 31st Asilomar Conf. on Signals, Systems and Computers. Available: http://www.cns.nyu.edu/∼eero/publications.html
  33. Simoncelli, E.P. and Adelson, E.H. 1996. Noise removal via Bayesian wavelet coring. In Proc. 3rd Int. Conf. on Image Processing, Lausanne, Switzerland, Vol. I, pp. 379–382.Google Scholar
  34. Srivastava, A., Lee, A.B., Simoncelli, E.P., and Zhu, S.-C. 2003. On Advances in Statistical Modeling of Natural Images. Journal of Mathematical Imaging and Vision, 18:17–33.zbMATHCrossRefMathSciNetGoogle Scholar
  35. Starck, J.L., Candès, E.J., and Donoho, D.L. 2001. Very high quality image restoration. In Proc. SPIE, Vol. 4478, pp. 9–19.Google Scholar
  36. Starck, J.L., Candès, E.J., and Donoho, D.L. 2002. The curvelet transform for image denoising. IEEE Trans. Image Processing, 11:670–684.CrossRefGoogle Scholar
  37. Tan, S. and Jiao, L.C. 2006a. Ridgelet bi-frame. Appl. Comput. Harmon. Anal., 20:391–402.zbMATHCrossRefMathSciNetGoogle Scholar
  38. Tan, S. and Jiao, L.C. 2006b. Image denoising using the ridgelet bi-frame. J. Opt. Soc. Amer. A, 23:2449–2461.CrossRefMathSciNetGoogle Scholar
  39. Torralba, A. and Oliva, A. 2003. Statistics of natural image categories. Network: Computation in Neural System, 14:391–412.CrossRefGoogle Scholar
  40. Voloshynovskiy, S., Koval, O., and Pun, T. 2005. Image denoising based on the edge-process model. Signal Processing, 85:1950–1969.CrossRefGoogle Scholar
  41. Wainwright, M.J. and Simoncelli, E.P. 2000. Scale mixtures of Gaussians and the statistics of natural images. In Advances in Neural Information Processing Systems, S.A. Solla, T.K. Leen, and K.-R. Muller (eds.), pp. 855–861.Google Scholar
  42. Wegmann, B. and Zetzsche, C. 1990. Statistical dependence between orientation filter outputs used in a human vision based image code. In Proceedings of Visual Communication and Image Processing, Society of Photo-Optical Instrumentation Engineers, Vol. 1360, pp. 909–922.Google Scholar
  43. Yao, K. 1973. A representation theorem and its applications to sphericallyinvariant random processes. IEEE Trans. Inform. Theory, IT-19:600–608.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • Shan Tan
    • 1
  • Licheng Jiao
    • 1
  1. 1.Institute of Intelligent Information ProcessingXidian UniversityXi’anChina

Personalised recommendations