Skip to main content
SpringerLink
Log in

Wavelet Statistical Models and Besov Spaces

  • Chapter
Book cover Nonlinear Estimation and Classification

Part of the book series: Lecture Notes in Statistics ((LNS,volume 171))

Abstract

Natural image models provide a foundation for framing numerous problems encountered in image processing, from compressing images to detecting tumors in medical scans. A good model must capture the key properties of the images of interest. A typical photograph of a natural scene consists of piecewise smooth or textured regions separated by step edge discontinuities along contours. Modeling both the smoothness and edge structure is essential for maximum processing performance.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Donoho and I. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Amer. Stat. Assoc, vol. 90, pp. 1200–1224, Dec. 1995.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. A. DeVore, B. Jawerth, and B. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans, on Information Theory, vol. 38, no. 2, pp. 719–746, March 1992.

    Article  Google Scholar 

  3. P. Moulin and J. Liu, “Analysis of multiresolution image denoising schemes using generalized-Gaussian priors,” in Proc. IEEE-SP Int. Symp. Time-Freq. and Time-Scale Anal, Pittsburgh, PA, Oct. 6-9 1998, pp. 633-636.

    Google Scholar 

  4. M. S. Crouse, R. D. Nowak, and R. G. Baraniuk, “Wavelet-based statistical signal processing using hidden Markov models,” IEEE Trans. Signal Proc, vol. 46, no. 4, pp. 886–902, April 1998.

    Article  MathSciNet  Google Scholar 

  5. F. Abramovich, T. Sapatinas, and B. W. Silverman, “Wavelet thresholding via a Bayesian approach,” J. Roy Stat. Soc. Ser. B, vol. 60, pp. 725–749, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  6. E. P. Simoncelli and E. H. Adelson, “Noise removal via Bayesian wavelet coring,” in Proc. IEEE Int. Conf. on Image Proc. — ICIP 1996, Lausanne, Switzerland, Sept. 1996.

    Google Scholar 

  7. H. Choi and R. G. Baraniuk, “Image segmentation using wavelet-domain classification,” in Proc. SPIE Conf. Math. Modeling, Bayesian Estimation, and Inverse Problems, Denver, CO, July 1999, vol. 3816, pp. 306-320.

    Google Scholar 

  8. S. Jaffard, “Beyond Besov spaces,” Preprint, 2001.

    Google Scholar 

  9. I. Daubechies, Ten Lectures on Wavelets, SIAM, New York, 1992.

    Book  MATH  Google Scholar 

  10. E. A. Abbot and A. Lightman, Flatland: A Romance of many dimensions, Penguin Classics, New York: NY, 1998.

    Google Scholar 

  11. A. B. Lee, K. S. Pederson, and D. Mumford, “The complex statistics of high-contrast patches in natural images,” in Proc. 2nd International IEEE Workshop on Statistical and Computational Theories of Vision, Vancouver, July 2001.

    Google Scholar 

  12. M. T. Orchard, Personal Communication, 2000.

    Google Scholar 

  13. S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998.

    MATH  Google Scholar 

  14. M. Vetterli and J. Kovačević, Wavlets and Subband Coding, Prentice Hall, Englewood Cliffs, NJ, 1995.

    Google Scholar 

  15. S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, no. 7, pp. 674–693, July 1989.

    Article  MATH  Google Scholar 

  16. D. L. Ruderman and W. Bialek, “Statistics of natural images: Scaling in the woods,” Phys. Rev. Lett, vol. 73, no. 6, pp. 814–817, 1994.

    Article  Google Scholar 

  17. Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1992.

    MATH  Google Scholar 

  18. L. C. Evans and R. F. Gariepy, Measure Theory and Find Properties of Functions, CRC Press, Boca Raton, FL, 1992.

    Google Scholar 

  19. S. Osher, L. I. Rudin and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, pp. 259–268, 1992.

    Article  MATH  Google Scholar 

  20. C. S. Güntürk, “Harmonic analysis of two problems in signal quantization and compression,” Ph.D Thesis, Princeton University, 2000.

    Google Scholar 

  21. S. LoPresto, K. Ramchandran, and M. T. Orchard, “Image coding based on mixture modeling of wavelet coefficients and a fast estimation-quantization framework,” in Data Compression Conference, Snowbird, Utah, 1997, pp. 221-230.

    Google Scholar 

  22. S. LoPresto, K. Ramchandran, and M. T. Orchard, “Wavelet image coding via rate-distortion optimized adaptive classification,” in Proc. of NJIT Symposium on Wavelet, Subband and Block Transforms in Communications, New Jersey Institute of Technology, 1997.

    Google Scholar 

  23. M. K. Mihcak, I. Kozintsev, and K. Ramchandran, “Spatially adaptice statistical modeling of wavelet image coefficients and its application to denoising,” in Proc. IEEE Int. Conf. on Acoust., Speech, Signal Proc. — ICASSP’ 99, Phoenix, AZ, March 1999.

    Google Scholar 

  24. H. Choi and R. Baraniuk, “Wavelet-domain statistical models and Besov spaces,” in Proc. of SPIE Conf. Wavelet Applications in Signal Proc. VII, Denver, July 1999, vol. 3813, pp. 489-501.

    Google Scholar 

  25. J. Romberg, H. Choi, and R. G. Baraniuk, “Bayesian wavelet domain image modeling using hidden Markov models,” IEEE Trans, on Image Proc, vol. 10, pp. 1056–1068, July 2001.

    Article  Google Scholar 

  26. A. Chambolle, R. A. DeVore, N. Lee, and B. J. Lucier, “Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage,” IEEE Trans, on Image Proc, vol. 7, pp. 319–355, March 1998.

    Article  MATH  Google Scholar 

  27. H. Choi and R. Baraniuk, “Interpolation and denoising of nonuniformly sampled data using wavelet domain processing,” in Proc IEEE Int. Conf. on Acoust., Speech, Signal Proc. — ICASSP’ 99, Phoenix, AZ, March 1999.

    Google Scholar 

  28. T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, Inc., New York, 1991.

    Book  MATH  Google Scholar 

  29. J. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Signal Proc, vol. 41, no. 12, pp. 3445–3462, Dec. 1993.

    Article  MATH  Google Scholar 

  30. A. Cohen, I. Daubechies, O. G. Guleryuz, and M. T. Orchard, “On the importance of combining wavelet-based non-linear approximation with coding strategies,” IEEE Trans, on Information Theory, Submitted.

    Google Scholar 

  31. N. Saito, “Simultaneous noise supression and signal compression using a library of orthonormal bases and the MDL criterion,” in Wavelets in Geophysics. 1994, pp. 299–324, New York: Academic Press, Editors: E. Foufoula-Georgiou and P. Kumar.

    Google Scholar 

  32. E. Candès and D. Donoho, “Ridgelets: The key to high-dimensional intermittency?,” Phil. Trans. R. Soc. Lond. A., vol. 357, pp. 2495–2509, 1999.

    Article  MATH  Google Scholar 

  33. E. Candès and D. Donoho, “Curvelets: A surprisingly effective non-adaptive representation of objects with edges,” in Curves and Surface Fitting: Saint-Malo 1999. 2000, Vanderbilt University Press, Nashville, TN, Editors A. Cohen, C. Rabut and L. L. Schumaker.

    Google Scholar 

  34. E. L. Pennec and S. Mallat, “Image compression with geometrical wavelets,” in IEEE Int. Conf. on Image Proc. — ICIP’ 01, Thessaloniki, Greece, Oct. 7-10 2001.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Electrical and Computer Engineering, Rice University, USA

    Hyeokho Choi (Research Professor/Lecturer) & Richard G. Baraniuk (Professor)

Authors
  1. Hyeokho Choi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Richard G. Baraniuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Imperial College, 180 Queen’s Gate, London, SW7 2BZ, UK

    David D. Denison  & Christopher C. Holmes  & 

  2. Room 2C283 Bell Laboratories, Lucent Technologies, 600 Mountain Avenue, Murray Hill, NJ, 07974-0636, USA

    Mark H. Hansen

  3. Statistical Department, Texas A&M University, College Station, TX, 77843-3143, USA

    Bani Mallick

  4. Department of Statistics, University of California, Berkeley, Berkeley, CA, 94720-3860, USA

    Bin Yu

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Science+Business Media New York

About this chapter

Cite this chapter

Choi, H., Baraniuk, R.G. (2003). Wavelet Statistical Models and Besov Spaces. In: Denison, D.D., Hansen, M.H., Holmes, C.C., Mallick, B., Yu, B. (eds) Nonlinear Estimation and Classification. Lecture Notes in Statistics, vol 171. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21579-2_2

Download citation

  • DOI: https://doi.org/10.1007/978-0-387-21579-2_2

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-95471-4

  • Online ISBN: 978-0-387-21579-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation