Scale filtering in the wavelet domain

Given a basic wavelet /spl psi/(t), we establish two correspondences (called C1 and C2) between frequency filters, defined in the frequency domain through multiplication by a transfer function W(f), and scale filters, defined in the wavelet domain through multiplication by a scale transfer function w(/spl sigma/). W(f) is obtained by performing a scaling convolution of w(/spl sigma/) with /spl psi//spl circ/(f)* (for C1) or its spectral energy density |/spl psi//spl circ/(f)|/sup 2/ (for C2). These relations can be solved for w(/spl sigma/) in terms of W(f) by applying the Mellin transform, subject to certain admissibility conditions being satisfied jointly by /spl psi//spl circ/(f) and W(f). This generalizes the usual wavelet reconstruction formula, making it possible to filter the wavelet transform by scale before transforming back to the time domain. In particular, the identity operator (W(f)/spl equiv/1) is C2-admissible if and only if the wavelet /spl psi/ is admissible in the conventional sense. If this is the case, then C2-equivalence reduces to the usual reconstruction formula.