On the optimal choice of a wavelet for signal representation

Two techniques for finding the discrete orthogonal wavelet of support less than or equal to some given integer that leads to the best approximation to a given finite support signal up to a desired scale are presented. The techniques are based on optimizing certain cost functions. The first technique consists of minimizing an upper bound that is derived on the L/sub 2/ norm of error in approximating the signal up to the desired scale. It is shown that a solution to the problem of minimizing that bound does exist and it is explained how the constrained minimization over the parameters that define discrete finite support orthogonal wavelets can be turned into an unconstrained one. The second technique is based on maximizing an approximation to the norm of the projection of the signal on the space spanned by translates and dilates of the analyzing discrete orthogonal wavelet up to the desired scale. Both techniques can be implemented much faster than the optimization of the L/sub 2/ norm of either the approximation to the given signal up to the desired scale or that of the error in that approximation.<>