Completeness and stability of partial dyadic wavelet domain signal representations

  • Authors:

  • Hehong Zou;Ahmed H. Tewfik;Wenyun Xu

  • Affiliations:

  • Department of Electrical Engineering, University of Minnesota, Minneapolis, MN;Department of Electrical Engineering, University of Minnesota, Minneapolis, MN;Department of Electrical Engineering, University of Minnesota, Minneapolis, MN

  • Venue:
  • ICASSP'93 Proceedings of the 1993 IEEE international conference on Acoustics, speech, and signal processing: digital speech processing - Volume III
  • Year:
  • 1993

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Abstract

We give a necessary and sufficient condition for the completeness of any partial dyadic wavelet transform domain representation of discrete finite data length signal (including dyadic wavelet transform extrema and zero-crossings representations). We show that completeness depends only on the locations of the retained samples of the dyadic wavelet transform. Our completeness test is more convenient and easier to verify than previously derived tests. Furthermore, we explain why some conclusions reported in the literature hold for most signals except for some extreme cases. We also show how to ensure the completeness of the representation by adding additional information in those cases where the partial dyadic wavelet transform domain representation is incomplete. The numerical stability of such a representation is also discussed. The stability issue is important in the sense that a numerically unstable representation is useless from a practical point of view. Finally, we describe a fast FFT based reconstruction algorithm from such a signal representation.