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Multivariate Statistical Models for Image Denoising in the Wavelet Domain

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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.

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References

  • Anderson, T.W. and Fang, K.T. 1987. On the theory of multivariate elliptically contoured distributions. Sankhya, 49 (Series A):305–315.

    Google Scholar 

  • 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 

  • Andrews, D. and Mallows, C. 1974. Scale mixtures of normal distributions. J. R. Statist. Soc, 36:99.

    MATH  MathSciNet  Google Scholar 

  • 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/

  • Brehm, H. and Stammler, W. 1987. Description and generation of spherically invariant speech-model signals. Signal Processing, 9:119–141.

    Article  Google Scholar 

  • Candès, E.J. 1999. Harmonic analysis of neural netwoks. Appl. Comput. Harmon. Anal., 6:197–218.

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • 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.

  • 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.

    Article  MathSciNet  Google Scholar 

  • Do, M.N. and Vetterli, M. 2003. The finite ridgelet transform for image representation. IEEE Trans. Image Processing, 12:16–28.

    Article  MathSciNet  Google Scholar 

  • Do, M.N. and Vetterli, M. 2005. The contourlet transform: An efficient directional multiresolution image representation. IEEE Trans. Image Processing, 14:2091–2106.

    Article  MathSciNet  Google Scholar 

  • Donoho, D.L. 2000. Orthonormal ridgelet and linear singularities. SIAM J. Math Anal., 31:1062–1099.

    Article  MATH  MathSciNet  Google Scholar 

  • Fang, K.T. and Zhang, Y.T. 1990. Generalized Multivariate Analysis. Science Press and Springer-Verlag, Beijing and Berlin.

    MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • Gehler, P.V. and Welling, M. 2005. Product of Edgeperts. Advances in Neural Information Processing System, 18, 8, MIT Press, Cambridge, MA.

  • Huang, J. 2000. Statistics of natural images and models. Ph.D. Thesis, Division of Appled Mathematics, Brown University, RI.

  • 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.

    Article  MATH  Google Scholar 

  • 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 

  • Mallat, S.G. 1989. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Pattern Anal. Machine Intell., 11:674–693.

    Article  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Schoenberg, I.J. 1938. Metric spaces and completely monotone functions. Ann. Math., 39:811–841.

    Article  MathSciNet  Google Scholar 

  • Sendur, L. and Selesnick, I.W. 2002a. Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency. IEEE Trans. Signal Processing, 50:2744–2756.

    Article  Google Scholar 

  • Sendur, L. and Selesnick, I.W. 2002b. Bivariate shrinkage with local variance estimation. IEEE Signal Processing Lett., 9:438–441.

    Article  Google Scholar 

  • Shapiro, J. 1993. Embedded image coding using zerotrees of wavelet coefficients. IEEE Trans. Sig. Proc., 41(12):3445–3462.

    Article  MATH  Google Scholar 

  • 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

  • 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.

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • Starck, J.L., Candès, E.J., and Donoho, D.L. 2001. Very high quality image restoration. In Proc. SPIE, Vol. 4478, pp. 9–19.

  • Starck, J.L., Candès, E.J., and Donoho, D.L. 2002. The curvelet transform for image denoising. IEEE Trans. Image Processing, 11:670–684.

    Article  Google Scholar 

  • Tan, S. and Jiao, L.C. 2006a. Ridgelet bi-frame. Appl. Comput. Harmon. Anal., 20:391–402.

    Article  MATH  MathSciNet  Google Scholar 

  • Tan, S. and Jiao, L.C. 2006b. Image denoising using the ridgelet bi-frame. J. Opt. Soc. Amer. A, 23:2449–2461.

    Article  MathSciNet  Google Scholar 

  • Torralba, A. and Oliva, A. 2003. Statistics of natural image categories. Network: Computation in Neural System, 14:391–412.

    Article  Google Scholar 

  • Voloshynovskiy, S., Koval, O., and Pun, T. 2005. Image denoising based on the edge-process model. Signal Processing, 85:1950–1969.

    Article  Google Scholar 

  • 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.

  • 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.

  • Yao, K. 1973. A representation theorem and its applications to sphericallyinvariant random processes. IEEE Trans. Inform. Theory, IT-19:600–608.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Intelligent Information Processing, Xidian University, Xi’an, China

    Shan Tan & Licheng Jiao

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  1. Shan Tan
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  2. Licheng Jiao
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Tan, S., Jiao, L. Multivariate Statistical Models for Image Denoising in the Wavelet Domain. Int J Comput Vis 75, 209–230 (2007). https://doi.org/10.1007/s11263-006-0019-7

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  • DOI: https://doi.org/10.1007/s11263-006-0019-7

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