Skip to main content
SpringerLink
Log in

Optimal noise suppression: A geometric nature of pseudoframes for subspaces

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

Pseudoframes for subspaces (PFFS) is a notion of frame-like expansions for a subspace \(\chi\) in a separable Hilbert space [11]. The spanning nature of the sequences \(\{x_n\}\) and \(\{x^*_n\}\) in a PFFS (relative to the subspace \(\chi\)) is generally very different from that of a frame. Incidentally, a PFFS constitutes generally a nonorthogonal projections onto \(\chi\). The directions of the projection determine the geometric meanings and its applications of a PFFS. PFFS also provides a means for the construction of nonorthogonal projections that arises in various linear reconstruction problems. This article is aimed at elaborations on such geometrical properties, demonstration of natural needs of nonorthogonal projections in applications and how PFFS can be applied, particularly for optimal noise suppressions. In this specific application, we show that PFFS is not only natural and sufficient but also necessary for generating an optimal solution among the class of all linear and series-based methods.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Benedetto, J.J.: Frame decompositions, sampling, and uncertainty principle inequalities. In: Benedetto, J.J., Frazier, M.W. (eds.) Wavelets: Mathematics and Applications, Chapter 7. CRC, Boca Raton, FL (1994)

    Google Scholar 

  2. Benedetto, J.J., Li, S.: Subband coding and noise reduction in multiresolution analysis frames. In: Proc. of SPIE Conf. on Mathematical Imaging, San Diego (July, 1994)

  3. Christensen, O., Edgar, Y.: Oblique dual frames and shift-invariant spaces. Appl. Comput. Harmon. Anal. 17(1), 48–68 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Daubechies, I.: Ten Lectures on Wavelets, SIAM, Philadelphia, Pennsylvania (1992)

  5. Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  6. Feichtinger, H.G., Grochenig, K.: Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view. In: Chui, C.K. (ed.) Wavelets: A Tutorial in Theory and Applications, vol. 2, pp. 359–398. Academic, Boston (1992)

    Google Scholar 

  7. Heil, C., Walnut, D.: Continuous and discrete wavelet transforms. SIAM Rev. 31, 628–666 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Li, S.: A generalized multiresolution structure and associated multirate systems. In: Proc. IEEE-SP Internl. Symp. TF-TS, Philadelphia, pp. 40–43 (Oct, 1994)

  9. Li, S.: A theory of generalized multiresolution structure and affine pseudoframes. J. Funct. Anal. Appl. 7(1), 23–40 (2001)

    Article  MATH  Google Scholar 

  10. Li, S.: New biorthogonal wavelets and filterbanks via the theory of pseudoframe for subspaces. In: Presented at SampTA'03, Strobl, Austria (May, 2003)

  11. Li, S., Ogawa, H.: Pseudoframes for subspaces with applications. J. Fourier Anal. Appl. 10(4), 409–431 (Oct, 2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ogawa, H.: A unified approach to generalized sampling theorem. In: Proc. of IEEE-IECEJ-ASJ Int. Conf. on Acoustics, Speech, and Signal Processing, pp. 1657–1660 (April, 1986)

  13. Ogawa, H.: Projection filter regularization of ill-conditioned problem. In: SPIE Vol. 808 Inverse Problems in Optics, pp. 189–196 (1987)

  14. Ogawa, H.: A generalized sampling theorem. Elec. Comm. Jap. 72(3), 97–105 (March, 1989)

    Article  MathSciNet  Google Scholar 

  15. Ogawa, H., Berrached, N.-E.: EPBOBs (extended pseudo biorthogonal bases) for signal recovery. IEICE Trans. Inf. Syst. E83-D(2), 223–232 (Feb, 2000)

    Google Scholar 

  16. Ogawa, H., Nakamura, N.: Projection filter restoration of degraded images. In: Proced. IEEE Seventh Intern. Conf. Patt. Recog., pp. 601–603 (July–Aug, 1984)

  17. Daubechies, I., Balan, R., Vaishampayan, V.: The analysis and design of windowed Fourier frame based multiple description encoding schemes. IEEE Trans. Inf. Theory 46(7), 2491–2537 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Balan, R.: Equivalence relations and distances between Hilbert frames. Proc. AMS 127(8), 2353–2366 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Heil, C., Balan, R., Casazza, P., Landau, Z.: Deficits and excesses of frames. Adv. Comput. Math. 18, 93–116 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Heil, C., Balan, R., Casazza, P., Landau, Z.: Excesses of Gabor frames. Appl. Comput. Harmon. Anal. 14, 87–106 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Shatten, R.: Norm Ideals of Completely Continuous Operators, 2nd Print. Springer, Berlin Heidelberg New York (1970)

    Google Scholar 

  22. Stark, H. (ed.) Image Recovery: Theory and Applications. Academic, New York (1987)

    MATH  Google Scholar 

  23. Unser, M., Aldroubi, A.: A general sampling theory for non-ideal acquisition devices. IEEE Trans. Signal Process. 42(11), 2915–2925 (Nov, 1994)

    Article  Google Scholar 

  24. Yamashita, Y., Ogawa, H.: Image restoration by averaged projection filter. IEICE Trans. J73-D-II(2), pp. 150–157 (1991) (Japanese); English translation in Syst. Comput. Jpn. 23(1), 79–88 (1992)

    Google Scholar 

  25. Yamashita, Y., Ogawa, H.: Properties of averaged projection filter. IEICE Trans. J73-D-II(2), 141–149 (1991) (Japanese); English translation in Syst. Comput. Jpn. 23(1), 69–78 (1992)

    Google Scholar 

  26. Young, R.: An Introduction to Nonharmonic Fourier Series. Academic, New York (1980)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, San Francisco State University, San Francisco, CA, 94132, USA

    Shidong Li

  2. Department of Computer Science, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8552, Japan

    Hidemitsu Ogawa

Authors
  1. Shidong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hidemitsu Ogawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shidong Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, S., Ogawa, H. Optimal noise suppression: A geometric nature of pseudoframes for subspaces. Adv Comput Math 28, 141–155 (2008). https://doi.org/10.1007/s10444-006-9014-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-006-9014-3

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation