Astron. Astrophys. 347, 630-633 (1999)

The relation of the Allan- and -variance to the continuous wavelet transform

M. Zielinsky and J. Stutzki

I. Physikalisches Institut, Universität zu Köln, Zülpicherstrasse 77, D-50937 Köln, Germany (zielinsky, stutzki@ph1.uni-koeln.de)

Received 23 December 1998 / Accepted 14 April 1999

Abstract

This paper is understood as a supplement to the paper by Stutzki et al. 1998, where we have shown the usefulness of the Allan-variance and its higher dimensional generalization, the -variance, for the characterization of molecular cloud structures. In this study we present the connection between the Allan- and -variance and a more popular structure analysis tool: the wavelet transform. We show that the Allan- and -variances are the variances of wavelet transform coefficients.

Key words: ISM: clouds ISM: structure

Send offprint requests to: zielinsky@ph1.uni-koeln.de

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Contents

• 1. Introduction
• 2. Allan-variance and wavelets
  • 2.1. Allan-variance
  • 2.2. Wavelets
  • 2.3. The Allan-variance as variance of wavelet transform coefficients
• 3. -variance and -wavelet
  • 3.1. Two-dimensional continuous wavelet transform
  • 3.2. -variance in two dimensions
  • 3.3. -variance and two-dimensional continuous wavelet transform
• 4. Summary and outlook
• Acknowledgements
• Appendix A: evaluation of the factors
  • A.1. Modified Haar-wavelet
  • A.2. -wavelet
    • A.2.1. -wavelet in one dimension
    • A.2.2. -wavelet in two dimensions
• References

© European Southern Observatory (ESO) 1999

Online publication: June 30, 1999
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