We study the problem of choosing the optical wavelet basis with compact support for signal representation and provide a general algorithm for computing the optimal wavelet basis. We first briefly review the multiresolution property of wavelet decomposition and the conditions for generating a basis of compactly supported discrete wavelets in terms of properties of quadrature mirror filter (QMF) banks. We then parametrize the mother wavelet and scaling function through a set of real coefficients. We further introduce the concept of information measure as a distance measure between the signal and its projection onto the subspace spanned by the wavelet basis in which the signal is to be reconstructed. The optimal basis for a given signal is obtained through minimizing this information measure. We have obtained explicitly the sensitivity of dilations and shifts of the mother wavelet with respect to the coefficient set. A systematic approach is developed here to derive the information gradient with respect to the parameter set for a given square integrable signal and the optimal wavelet basis. A gradient-based optimization algorithm is developed in this paper for computing the optimal wavelet basis.
John S. Baras,
"Optimal wavelet basis selection for signal representation", Proc. SPIE 2242, Wavelet Applications, (15 March 1994); doi: 10.1117/12.170025; https://doi.org/10.1117/12.170025