Skip to main content

Nonlinear Multiscale Transforms

  • Conference paper
Book cover Multiscale and Multiresolution Methods

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 20))

Abstract

We present in this paper several approaches for performing nonlinear multiscale transforms to an image. Both redundant and non-redundant transformations are discussed. The median based multiscale transforms and some of their applications are detailed. Finally we show how a multiscale vision model can be used to decompose an image, and how several transformations can be combined in order to benefit of the advantages of each of them.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
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. E. H. Adelson, E. Simoncelli, and R. Hingorani. Optimal image addition using the wavelet transform. SPIE Visual Communication and Image Processing II, 845:50–58, 1987.

    Article  Google Scholar 

  2. M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies. Image coding using wavelet transform. IEEE Transactions on Image Processing, 1(2):205–220, 1992.

    Article  Google Scholar 

  3. A. Bijaoui and F. Rué. A multiscale vision model adapted to astronomical images. Signal Processing, 46:229–243, 1995.

    Article  Google Scholar 

  4. F. Bonnarel, P. Fernique, F. Genova, J. G. Bartlett, O. Bienaymé, D. Egret, J. Florsch, H. Ziaeepour, and M. Louys. ALADIN: A reference tool for identification of astronomical sources. Astronomical Data Analysis Software and Systems VIII, ASP Conference Series, Vol. 172. Ed. David M. Mehringer, Raymond L. Plante, and Douglas A. Roberts. ISBN: 1-886733-94-5 (1999), p. 229., 8:229–232, 1999.

    Google Scholar 

  5. E. J. Breen, R. J. Jones, and H. Talbot. Mathematical morphology: A useful set of tools for image analysis. Statistics and Computing journal, 10:105–120, 2000.

    Article  Google Scholar 

  6. C. S. Burrus, R. A. Gopinath, and H. Guo. Introduction to Wavelets and Wavelet Transforms. Prentice Hall, 1998.

    Google Scholar 

  7. P. J. Burt and A. E. Adelson. The Laplacian pyramid as a compact image code. IEEE Tranactions on Communications, 31:532–540, 1983.

    Article  Google Scholar 

  8. R. Calderbank, I. Daubechies, W. Sweldens, and B.-L. Yeo. Wavelet transforms that map integers to integers. Appl. Comput. Harmon. Anal., 5(3):332–369, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  9. E. J. Candès. Harmonic analysis of neural netwoks. Applied and Computational Harmonic Analysis, 6:197–218, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  10. E. J. Candès and D. Donoho. Curvelets. Technical report, Statistics, Stanford University, 1999.

    Google Scholar 

  11. E. J. Candès and D. Donoho. Ridgelets: the key to high dimensional intermittency? Phil. trans; R. Soc. Lond. A, 357:2495–2509, 1999.

    Article  MATH  Google Scholar 

  12. S. S. Chen, D. L. Donoho, and M. A. Saunder. Atomic decomposition by basis pursuit. SIAM J. Math. Anal., 20(1):33–61, 1998.

    Google Scholar 

  13. C. H. Chui. Wavelet Analysis and Its Applications. Academic Press, 1992.

    Google Scholar 

  14. R. Claypoole, G. M. Davis, W. Sweldens, and R. Baraniuk. Nonlinear wavelet transforms for image coding via lifting. IEEE Transactions on Image Processing, 2000. submitted.

    Google Scholar 

  15. R. L. Claypoole, R. G. Baraniuk, and R. D. Nowak. Lifting construction of nonlinear wavelet transforms. In IEEE International Conference IEEE-SP Time-Frequency and Time-Scale Analysis, pages 49–52, 1998.

    Google Scholar 

  16. A. Cohen, I. Daubechies, and J. C. Feauveau. Biorthogonal bases of compactly supported wavelets. Communications in Pure and Applied Mathematics, 45:485–560, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  17. I. Daubechies. Time-frequency localization operators: A geometric phase space approach. IEEE Transactions on Information Theory, 34(1):605–612, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  18. I. Daubechies. Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1992.

    Book  MATH  Google Scholar 

  19. I. Daubechies, I. Guskov, P. Schröder, and W. Sweldens. Wavelets on irregular point sets. Phil. Trans. R. Soc. Lond. A, To be published.

    Google Scholar 

  20. I. Daubechies and W. Sweldens. Factoring wavelet transforms into lifting steps. J. Fourier Anal. Appl., 4(3):245–267, 1998.

    Article  MathSciNet  Google Scholar 

  21. D. L. Donoho. Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data. Proceedings of Symposia in Applied Mathematics, 47, 1993.

    Google Scholar 

  22. D. L. Donoho. Nonlinear pyramid transforms based on median-interpolation. SIAM J. Math. Anal., 60(4):1137–1156, 2000.

    Article  Google Scholar 

  23. O. Egger and W. Li. Very low bit rate image coding using morphological operators and adaptive decompositions. In IEEE International Conference ICIP-94 on Image Processing, volume 2, pages 326–330, 1994.

    Google Scholar 

  24. 24. D. A. F. Florêncio and R. W. Schafer. Perfect reconstructing nonlinear filter banks. In IEEE International Conference ICASSP-96 on Acoustics, Speech, and Signal Processing, volume 3, pages 1814–1817, 1996.

    Google Scholar 

  25. J. Goutsias and H. J. A. M. Heijmans. Nonlinear multiresolution signal decomposition schemes. part 1: Morphological pyramids. IEEE Transactions on Image Processing, 9(11):1862–1876, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  26. F. J. Hampson and J.-C. Pesquet. A nonlinear subband decomposition with perfect reconstruction. In IEEE International Conference ICASSP-96 on Acoustics, Speech, and Signal Processing, volume 3, pages 1523–1526, 1996.

    Google Scholar 

  27. H. J. A. M. Heijmans and J. Goutsias. Multiresolution signal decomposition schemes. part 2: Morphological wavelets. IEEE Transactions on Image Processing, 9(11):1897–1913, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  28. L. Huang and A. Bijaoui. Astronomical image data compression by morphological skeleton transformation. Experimental Astronomy, 1:311–327, 1991.

    Article  Google Scholar 

  29. X. Huo. Sparse Image Representation via Combined Transforms. PhD thesis, Stanford Univesity, August 1999.

    Google Scholar 

  30. M. Louys, J.-L. Starck, S. Mei, F. Bonnarel, and F. Murtagh. Astronomical image compression. Astronomy and Astrophysics, Suppl. Ser., 136:579–590, 1999.

    Article  Google Scholar 

  31. S. Mallat. A Wavelet Tour of Signal Processing. Academic Press, 1998.

    Google Scholar 

  32. S. Mallat and F. Falzon. Analysis of low bit rate image transform coding. IEEE Transactions on Signal Processing, 46(4):1027–42, 1998.

    Article  Google Scholar 

  33. S. G. Mallat. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7):674–693, 1989.

    Article  MATH  Google Scholar 

  34. S. G. Mallat and Z. Zhang. Atomic decomposition by basis pursuit. IEEE Transactions on Signal Processing, 41(12):3397–3415, 1993.

    Article  MATH  Google Scholar 

  35. G. Matheron. Elements pour une théorie des milieux poreux. Masson, Paris, 1967.

    Google Scholar 

  36. G. Matheron. Random Sets and Integral Geometry. Willey, New York, 1975.

    MATH  Google Scholar 

  37. F. Murtagh, J. L. Starck, and M. Louys. Very high quality image compression based on noise modeling. International Journal of Imaging Systems and Technology, 9:38–45, 1998.

    Article  Google Scholar 

  38. A. Said and W. A. Pearlman. A new, fast, and efficient image codec based on set partitioning in hierarchival trees. IEEE Trans. on Circ. and Syst. for Video Tech., 6(3):243–250, 1996.

    Article  Google Scholar 

  39. Peter Schröder and Wim Sweldens. Spherical wavelets: Efficiently representing functions on the sphere. Computer Graphics Proceedings (SIGGRAPH 95), pages 161–172, 1995.

    Google Scholar 

  40. J. Serra. Image Analysis and Mathematical Morphology. Academic Press, London, 1982.

    MATH  Google Scholar 

  41. J. M. Shapiro. Embedded image coding using zerotrees of wavelet coefficients. IEEE Transactions on Signal Processing, 41(12):3445–3462, 1993.

    Article  MATH  Google Scholar 

  42. J. L. Starck, A. Bijaoui, I. Vatchanov, and F. Murtagh. A combined approach for object detection and deconvolution. Astronomy and Astrophysics, Suppl. Ser., 147:139–149, 2000.

    Article  Google Scholar 

  43. J. L. Starck, E. Candès, and D. L. Donoho. The curvelet transform for image denoising. IEEE Transactions on Image Processing, 2001. submitted.

    Google Scholar 

  44. J. L. Starck and F. Murtagh. Multiscale entropy filtering. Signal Processing, 76(2):147–165, 1999.

    Article  MATH  Google Scholar 

  45. J. L. Starck, F. Murtagh, and A. Bijaoui. Image Processing and Data Analysis: The Multiscale Approach. Cambridge University Press, Cambridge (GB), 1998.

    Book  Google Scholar 

  46. J. L. Starck, F. Murtagh, and R. Gastaud. A new entropy measure based on the wavelet transform and noise modeling. Special Issue on Multirate Systems, Filter Banks, Wavelets, and Applications of IEEE Transactions on CAS II, 45(8), 1998.

    Google Scholar 

  47. J. L. Starck, F. Murtagh, B. Pirenne, and M. Albrecht. Astronomical image compression based on noise suppression. Publications of the Astronomical Society of the Pacific, 108:446–455, 1996.

    Article  Google Scholar 

  48. G. Strang and T. Nguyen. Wavelet and Filter Banks. Wellesley-Cambridge Press, Box 812060, Wellesley MA 02181, fax 617-253-4358, 1996.

    Google Scholar 

  49. W. Sweldens. The lifting scheme: A construction of second generation wavelets. SIAM J. Math. Anal., 29(2):511–546, 1997.

    Article  MathSciNet  Google Scholar 

  50. W. Sweldens and P. Schröder. Building your own wavelets at home. In Wavelets in Computer Graphics, pages 15–87. ACM SIGGRAPH Course notes, 1996.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Starck, JL. (2002). Nonlinear Multiscale Transforms. In: Barth, T.J., Chan, T., Haimes, R. (eds) Multiscale and Multiresolution Methods. Lecture Notes in Computational Science and Engineering, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56205-1_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-56205-1_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42420-8

  • Online ISBN: 978-3-642-56205-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics