Properties of the multiscale maxima and zero-crossings representations

The analysis of a discrete multiscale edge representation is considered. 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. This paper mainly addresses the questions of uniqueness and stability. It is shown, that the dyadic wavelet maxima (zero-crossings) representation is, in general, nonunique. Nevertheless, using the idea of the inherently bounded AQLR, two stability results are proven. For a general perturbation, 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