Wavelet Regularity of Iterated Filter Banks with Rational Sampling Changes

The regularity property was first introduced by wavelet theory for octave-band “dyadic” filter banks. In this paper, we provide a detailed theoretical analysis of the regularity property in the more flexible case of filters banks with rational sampling changes. Such filter banks provide a finer analysis on fractions of an octave, and regularity is equally important as in the dyadic case. Sharp regularity estimates for any filter bank are given. The major difficulty of the rational case, as compared to the dyadic case, is that one obtains “wavelets” that are not shifted versions of each other at a given scale. We show, however, that under regularity conditions, shift invariance can be almost obtained. This is a desirable property for e.g. coding applications and for efficient filter bank implementation of a continuous wavelet transform.