Uniqueness and Stability of Discrete Zero-Crossings and Maxima Wavelet Representations

A general signal description, called an inherently bounded Adaptive Quasi Linear Representation (AQLR), motivated by two important examples, namely the wavelet maxima representation, and the wavelet zero-crossings representation, is introduced. It is shown, that the dyadic wavelet maxima (zero-crossings) representation is, in general, nonunique. Using the idea of the inherently bounded AQLR, a global BIBO stability is shown. For a special case, where perturbations are limited to the continuous part of the representation, a Lipschitz condition is satisfied.