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 the present work, the authors provide a detailed theoretical analysis of the regularity property in the more flexible case of filter banks with rational sampling changes. Such filter banks provide a finer analysis of fractions of an octave, and regularity is as important as in the dyadic case. Sharp regularity estimates for any filter bank are given. The major difficulty of the rational case, as compared with the dyadic case, is that one obtains wavelets that are not shifted versions of each other at a given scale. It is shown, however, that, under regularity conditions, shift invariance can almost be obtained. This is a desirable property for, e.g. coding applications and for efficient filter bank implementation of a continuous wavelet transform.<>