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) representations are, in general nonunique. Nevertheless, using the idea of the inherently bounded AQLR, two stability results are proven. For a special case, where perturbations are limited to the continuous part of the representation, a Lipschitz condition is satisfied.