Advertisement

Multi-scale Geometric Analysis and Its Application of De-noising

  • Wu Guoning
  • Cao Siyuan
  • Duan Qingquan
Chapter
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 126)

Abstract

The essence of multi-scale geometric analysis is to achieve optimal approximation of signal interested. This paper firstly introduces the backgrounds and recent developments of the subject, unveil the impetus behind this root causes. Finally we compare the differences of wavelet transform, contourlet transform and curvelet transform in suppression of random noise, and through practical experiments we confirm that multi-scale geometric analysis are better (sparser) than wavelet in approximating multidimensional signal of interested.

Keywords

Random Noise Noise Image Image Representation Short Time Fourier Transform Laplacian Pyramid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Stein, E.M.: Fourier analysis: an introduction, pp. 5–171. Princeton University Press (2006)Google Scholar
  2. 2.
    Xinda, Z., Zheng, B.: Non-stationary signal analysis, pp. 17–178. National defense industry Press (1998)Google Scholar
  3. 3.
    Gabor, D.: Theory of communication. Journal of Institute for Electrical Engineering 93, 429–457 (1946)Google Scholar
  4. 4.
    Mallat, S.: A Wavelet Tour of Signal Processing, 2nd edn., pp. 67–216. Academic Press (1999)Google Scholar
  5. 5.
    Mallat, S.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans., PAMI 11(7), 674–693 (1989)zbMATHCrossRefGoogle Scholar
  6. 6.
    Sarkar, T.K., Su, C.: A tutorial on wavelets from an electrical engineering perspective, Part 2: the continuous case. IEEE Antennas & Propagation Magazine 40(6), 36–48 (1988)CrossRefGoogle Scholar
  7. 7.
    Daubechies, I.: Orthonormal bases of compactly supported wavelets. Comm. Pure and Applied Math. 41, 909–996 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Aldroubi, A., Laine, A.F., Unser, M.A.: Wavelet applications in signal and image proc-essing VI. In: Proc. SPIE, vol. 3458, pp. 24–37 (1998)Google Scholar
  9. 9.
    Donoho, D.L., Vetterli, M., DeVore, R.A., Daubechies, I.: Data compression and harmonic analysis. IEEE Trans. Inform. 44(6), 2435–2476 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Jiao, L., Tan, S.: Development and Prospect of Image Multiscale Geometric Analysis. Chinese Journal of Electronics 31, 1975–1981 (2003)Google Scholar
  11. 11.
    Pennec, E.L., Mallat, S.: Sparse geometric image representation with bandelets. IEEE Trans. on Image Processing 14(4), 423–438 (2005)CrossRefGoogle Scholar
  12. 12.
    Hubel, D.H., Wiesel, T.N.: Receptive fields, binocular interaction and functional ar-chitecture in the cat’s visual cortex. Journal of Physiology 160, 106–154 (1962)Google Scholar
  13. 13.
    Do, M.N., Vetterli, M.: The contourlet transform: an efficient directional multiresolution image representation. IEEE Trans. on Image Processing 14(12), 2091–2106 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pennec, E.L., Mallat, S.: Image compression with Geometrical wavelets. IEEE ICIP, Canada (2000)Google Scholar
  15. 15.
    Candès, E.J.: Ridgelets: Theory and Applications. Stanford University, California (1998)Google Scholar
  16. 16.
    Hong, B., Liu, F., Jiao, L.: Linear feature detection based on ridgelet transform. Science in China (E) 33(1), 65–73 (2003)Google Scholar
  17. 17.
    DeVore, R.A.: Nonlinear approximation. Acta Numer. 7, 51–150 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Candès, E.J., Donoho, D.L.: Ridgelets: a key to higher-dimensional intermittency. Phil. Trans. R Soc. Lond. (1999)Google Scholar
  19. 19.
    Donoho, D.L., Flesia, A.G.: Can recent innovations in harmonic analysis ex-plain key findings in natural image statistics. Computation in Neural Systems 12, 371–393 (2001)zbMATHGoogle Scholar
  20. 20.
    Candès, E.J., Donoho, D.L.: Curvelets: a surprisingly effective nonadaptive representation for objects with edges, pp. 105–120. Vandebilt University Press, Nashville (2000)Google Scholar
  21. 21.
    Do, M.N.: Directional multiresolution image representation. Swiss federal institute of technology, Lausanne (2001)Google Scholar
  22. 22.
    Do, M.N., Vetterli, M.: Contourlets: a directional multiresolution image representation. In: International Conference on Image Process., Rochester (2002)Google Scholar
  23. 23.
    Candès, E.J., Donoho, D.L.: Curvelets, multiresolution representation and scaling laws. In: Wavelet Applications in Signal and Image Processing (2001)Google Scholar
  24. 24.
    Candès, E.J., Donoho, D.L.: Fast discrete curvelet transform. Cali-fornia Institute of Technology, California (2005)Google Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.College of ScienceChina University of PetroleumBeijingChina
  2. 2.College of Geophysics and Information EngineeringChina University of PetroleumBeijingChina
  3. 3.College of Mechanical and Transportation EngineeringChina University of PetroleumBeijingChina

Personalised recommendations