Existence and Construction of Optimal Wavelet Basis for Signal Representation

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We study the problem of choosing the optimal 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 decomposition entropy as an information measure to describe the distance between the given 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 in this paper to derive the information gradient with respect to the parameter set from a given square integrable signal and a discrete basis of wavelets. The existence of the optimal basis for the wavelets has been proven in this paper. a gradient based optimization algorithm is developed for computing the optimal wavelet basis.