A Theory of Adaptive Quasi Linear Representations

Abstract

The analysis of the discrete multiscale edge representation is considered. A general signal description, called an inherently bounded Adaptive Quasi Linear Representation (AQLR), motivated by two important examples: the wavelet maxima representation and the wavelet zero-crossing 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, nonunique. Namely, for all maxima (zero-crossings) representation based on a dyadic wavelet transform, there exists a sequence having a nonunique representation. 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. The convergence of the algorithm is guaranteed for all inherently bounded AQLR. In the case, where the representation is based on a wavelet transform, this method yields an efficient, parallel algorithm, especially promising in an analog-hardware implementation.