Skip to main content
SpringerLink
Log in

Signal representation, uncertainty principles and localization measures

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

Abstract

The following collection of articles addresses one of the most basic problems in signal and image processing, namly the search for function systems (basis, frames, dictionaries) which allow efficient representations of certain classes of signals/images. Such representations are essential for decomposition and synthesis of signals, hence they are at the core of almost any application (coding, compression, pattern matching, feature extraction, classification, etc.) in this field. Accordingly, this is one of the best-studied topics in data analysis and a multitude of different concepts also addressing discretization/algorithmic issues has been investigated in this context. The starting point for reviving activities in this field was a recently rediscovered inconsistency in the concept of constructing optimally localized basis functions by minimizing uncertainty principles. In this short introductory note, we shortly sketch the basic dilemma, which was the starting point for this research approximately three years ago. However, the subsequent investigations presented in this collection of papers cover a much wider range of more general localization measures, discretization concepts as well as discussing algorithmic efficiency and stability.

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. Dahlke, S., Kutyniok, G., Maass, P., Sagiv, C., Stark, H.-G., Teschke, G.: The uncertainty principle associated with the continuous shearlet transform. Int. J. Wave. Multires. Inf. Process. 6(2), 157–181 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Dahlke, S., Lorenz, D., Maass, P., Sagiv, C., Teschke, G.: The canonical states associated with quotients of the affine weyl – heisenberg group. J. Appl. Func. Anal. 3(2), 215–232 (2008)

    MATH  MathSciNet  Google Scholar 

  3. Dahlke, S., Maass, P.: The affine uncertainty principle in one and two dimensions. Comput. Math. Appl. 30, 293–305 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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

  5. Feichtinger, H.G., Strohmer, T.: Gabor Analysis and Applications. Birkhäuser, Cambridge (1998)

    MATH  Google Scholar 

  6. Folland, G.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989)

    MATH  Google Scholar 

  7. Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Cambridge (2001)

    Book  MATH  Google Scholar 

  8. Stark, R.L.H.-G., Lieb, F., Sochen, N.: Adjoint translation, adjoint localization and uncertainty. In: ACOM, (not yet assigned) (2013)

  9. Hans, A.G., Feichtinger, G., Onchis, D.: Approximate dual gabor atoms via the adjoint lattice method. In: ACOM, (not yet assigned) (2013)

  10. Kobarg, J.H., Maass, P., Oetjen, J., Tropp, O., Hirsch, E., Sagiv, C., Golbabaee, M., Vandergheynst, P.: Numerical experiments with maldi imaging data. In: ACOM, (not yet assigned) (2013)

  11. Louis, A., Maass, P., Rieder, A.: Wavelets: Theory and Applications. Wiley, New York (1997)

    MATH  Google Scholar 

  12. Maass, P., Sagiv, C., Sochen, N., Stark, H.-G.: Do uncertainty minimizers attain minimal uncertainty?. J. Fourier Anal. Appl. 16(3), 448–469 (2010)

    Article  MATH  Google Scholar 

  13. Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, New York (2008)

    Google Scholar 

  14. Onchis, D.: Optimized frames and multi-dimensional challenges in time-frequency analysis. In: ACOM, (not yet assigned) (2013)

  15. Ricaud, B., Stempfel, G., Lachambre, H., Torresani, B., Wiesmayr, C., Onchis, D.: An optimally concentrated gabor transform for localized time-frequency components. In: ACOM, (not yet assigned) (2013)

  16. Ricaud, B., Torresani, B.: A survey of uncertainty principles and some signal processing applications. In: ACOM, (not yet assigned) (2013)

  17. Stark, H.-G.: Wavelet and Signal Processing. Springer, New York (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich 3, Universität Bremen, 28334, Bremen, Germany

    Peter Maass & Chen Sagiv

  2. SagivTech, Hasadna st. 8, P.O.Box 2622, Ra’anana, 4365104, Israel

    Chen Sagiv

  3. Fachhochschule Aschaffenburg, Mathematik/Informatik/Technomathematik, 63743, Aschaffenburg, Germany

    Hans-Georg Stark

  4. I2M, Centre de Mathématique et Informatique, Aix-Marseille Université, 13453, Marseille, France

    Bruno Torresani

Authors
  1. Peter Maass
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chen Sagiv
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hans-Georg Stark
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bruno Torresani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Maass.

Additional information

Communicated by: Peter Maass, Hans G. Feichtinger, Bruno Torresani, Darian M. Onchis, Benjamin Ricaud and David Shuman

All authors have been supported by EU-grant ICT-FET UnLocX

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maass, P., Sagiv, C., Stark, HG. et al. Signal representation, uncertainty principles and localization measures. Adv Comput Math 40, 597–607 (2014). https://doi.org/10.1007/s10444-014-9341-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-014-9341-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation