Translation Invariance
and
the Wavelet Transform

John Weiss
Applied Mathematics Group
49 Grand View Road
Arlington, MA 02174

Abstract:

A translation invariant wavelet transform algorithm is defined. The algorithm is an extension of the best basis approach and can be used to define translation invariant best bases and wavelet transforms. The computational cost is a factor of m greater than the usual algorithms, where m is the multiplier of the wavelet system. Some applications to transient detection are presented.

A general form of an invariant wavelet transform is presented. This transform is shown to be invariant under a large group of symmetries described, most naturally, by the g-circulant transformations. The symmetries include translation and time-reversal of a periodic data vector. In our construction the expansion coefficients of g-circulant transformations of a data vector are shown to be simply related by periodic shifts of their expansion coefficients. Therefore, under g-circulant transformations the numerical values and ordering are invariant.

Keywords Wavelet Transform, Translation Invariance, Best Basis, Transient Detection.

 

• Contents
• Introduction
• The Translation Invariant Wavelet Transform
• The g-Invariant Wavelet Transform
• The g-Invariant Best-Basis Algorithm
• Transient Detection and Classification
• Conclusions
• Acknowledgement
• Appendix A: Higher Multiplier Constructions
• Appendix B: Congruences and g-Circulants
• References
• List of Figures
• About this document ...