Advertisement

Efficient Time and Frequency Methods for Sampling Filter Functions

  • Fadel M. Adib
  • Hazem M. Hajj
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6134)

Abstract

In this paper, we seek to determine the adequate number of samples for an analog filter function f(t). The proposed approaches provide discrete filters that can be used for multiresolution analysis. We introduce two methods that provide sampling results for localization: one of them defines an approximate Nyquist rate, and the other samples in a manner that ensures time-frequency consistency between the generated samples and the analog filter function. The key contribution of the paper is that it establishes robust mathematical and programmable foundations for a previously established empirical method. Analytically, we show that the time-frequency method is based on minimizing aliasing while maximizing decimation. The method can be programmed by introducing a mean square error (MSE) threshold across scales. Afterwards, we provide the outcomes of experiments that demonstrate success of localization with the proposed time-frequency method.

Keywords

Multiresolution analysis multiscale analysis sampling filters time-frequency 

References

  1. 1.
    Qiu, P.: Image Processing and Jump Regression Analysis. Wiley, Hoboken (2005)zbMATHCrossRefGoogle Scholar
  2. 2.
    Foucher, S.: Multiscale filtering of SAR images using scale and space consistency. In: International Geoscience and Remote Sensing Symposium, July 2007, pp. 3878–3882 (2007)Google Scholar
  3. 3.
    Scharcanski, J., Jung, C., Clarke, R.T.: Adaptive Image Denoising Using Scale and Space Consistency. IEEE Transactions on Image Processing 11(9), 1091–1101 (2002)CrossRefGoogle Scholar
  4. 4.
    Marin-Jimenez, M.J., de la Blanca, N.P.: Empirical Study of Multi-scale Filter Banks for Object Categorization Pattern Recognition. In: 18th International Conference on Pattern Recognition, vol. 1, pp. 578–581 (2006)Google Scholar
  5. 5.
    Foucher, S., Benie, G., Boucher, J.-M.: Multiscale MAP Filtering of SAR Images. IEEE Transactions on Image Processing 10(1), 49–60 (2001)zbMATHCrossRefGoogle Scholar
  6. 6.
    Polak, I.: Segmentation and Restoration via Nonlinear Multiscale Filtering. IEEE Signal Processing Magazine, 26–35 (September 2002)Google Scholar
  7. 7.
    Witkin, A.P.: Scale-space Filtering. In: Proceedings of the Eighth International Joint Conference on Artificial Intelligence, Karlsruhe, FRG, pp. 1019–1022 (1983)Google Scholar
  8. 8.
    Ziou, D., Tabbone, S.: A Multiscale Edge Detector. Pattern Recognition 26(9), 1305–1314 (1993)CrossRefGoogle Scholar
  9. 9.
    Xu, Y., Weaver, J.B., Healy Jr., D., Lu, J.: Wavelet Transform Domain Filters: A Spatially Selective Noise Filtration Technique. IEEE Trans. on Image Processing 3(6), 747–758 (1994)CrossRefGoogle Scholar
  10. 10.
    Mallat, S., Hwang, W.L.: Singularity Detection and Processing with Wavelets. Proceedings of the IEEE Trans. on Inf. Theory, part 2 38, 617–643 (March 1992)Google Scholar
  11. 11.
    Mallat, S.G.: A wavelet Tour of Signal Processing, 2nd edn. Academic, New York (1998)zbMATHGoogle Scholar
  12. 12.
    Daubechies, I.: Orthonormal basis of compactly supported wavelets, Coom. Pure Appl. Math. 41, 909–996 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)zbMATHGoogle Scholar
  14. 14.
    Guan, K., Singer, A.C.: A Level-Crossing Sampling Scheme for Non-Bandlimited Signals. Presented at 2006 IEEE Sarnoff Symposium, Princeton, NJ, USA (2006)Google Scholar
  15. 15.
    Al-Alaoui, M.A.: Novel Approach to Analog-to-Digital Transforms. IEEE Transactions on Circuits and Systems 54(2), 338–350 (2007)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Liu, Y.-L., et al.: doi:10.1016/j.sigpro.2009.09.030 Google Scholar
  17. 17.
    Chen, Q.H., Wang, Y.B., Wang, Y.: A sampling theorem for non- bandlimited signals using generalized Sinc functions. Comput. Math. Appl. 56(6), 1650–1661 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hajj, H., Nguyen, T., Chin, R.: On Multi-Scale Feature Detection using Filter Banks. In: Proceedings of the Twenty Ninth Annual Asilomar Conference, vol. 29 (1996)Google Scholar
  19. 19.
    Boche, H., Mönich, U.: Limits of signal processing performance under thresholding. Signal Processing 89(8), 1634–1646 (2009)zbMATHCrossRefGoogle Scholar
  20. 20.
    Bose, T.: Digital Signal and Image Processing. Wiley, Hoboken (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Fadel M. Adib
    • 1
  • Hazem M. Hajj
    • 1
  1. 1.Department of Electrical and Computer EngineeringAmerican University of BeirutBeirutLebanon

Personalised recommendations