<title>Why adaptive wavelet transform?</title>

Abstract

The freedom of choosing an appropriate kernel of a linear transform, which is given to us by the recent mathematical foundation of the wavelet transform, is exploited fully and is generally called the adaptive wavelet transform. However, there are several levels of adaptivity: (1) Optimum Coefficients: adjustable transform coefficients chosen with respect to a fixed mother kernel for better invariant signal representation; (2) Super-Mother: grouping different scales of daughter wavelets of same or different eother wavelets at different shift locations into a new family called a superposition mother kernel for better speech signal classification; (3) Variational Calculus to determine ab initio a constraint optimization mother for a specific task. The tradeoff between the mathematical rigor of the complete orthonormality and the speed of order (N) with the adaptive flexibility is finally up to the users' decisions to get their jobs done with the desirable properties. Then, to illustrate (1), a new invariant optoelectronic architecture of a wedge-shape filter in the WT domain is given for a scale-invariant signal classification by neural networks.