Generalizations and 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 thesis 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, 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. Two reconstruction algorithms, based on the minimization of an appropriate cost function, are proposed. The first is based on the integration of the gradient of the cost function; the second is a standard steepest descent algorithm. Both algorithms are shown to converge. The last part of this dissertation describes possible modifications in the basic multiscale maxima representations. The main idea is to preserve the structure of the inherently bounded AQLR, while allowing a trade-off between reconstruction quality and amount oof information required for representation. In partucular, it is shown how quantization can be considered as an integral part of the representation.