An Adaptive Quasi-Linear Representation-A Generalization of the Multiscale Edge Representation

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 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, nonunique. Nevertheless, these representations are always stable. 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, of the wavelet transform, this method yields an efficient, parallel algorithm, especially promising in an analog-hardware implementation