Introduction

We have found a translation invariant wavelet transform algorithm. 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. In our construction the expansion coefficients of periodic shifts are shown to be simply related by the periodic shifts of their expansion coefficients. Therefore, under translation the numerical values and ordering are invariant. Some applications to transient detection are presented.

In this paper we present a general form of an invariant wavelet transform. This transform is shown to be invariant under a large group of symmetries described, most naturally, by the g-circulant transformation. 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.