A Study on Discrete Multiscale Edge 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 addresses the questions of uniqueness, stability, and reconstruction. It is shown, that the dyadic wavelet maxima (zero-crossings) representation is, in general, non unique. Nevertheless, these representations are always stable. 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. A reconstruction algorithm, based on the minimization of an appropriate cost function, is proposed.