Fast Computational Algorithms for the Discrete Wavelet Transform and Applications of Localized Orthonormal Bases in Signal Classification

In the first part of this paper we construct an algorithm for implementing the discrete wavelet transform by means of matrices in SO2(R) for orthonormal compactly supported wavelets and matrices in SLm(R),m ≥ 2, for compactly supported biorthogonal wavelets. We show that in 1 dimension the total operation count using this algorithm can be reduced to about 50% of the conventional convolution and downsampling by 2-operation for both orthonormal and biorthogonal filters. In the special case of biorthogonal symmetric odd-odd filters, we show an implementation yielding a total operation count of about 38% of the conventional method. In 2 dimensions we show an implementation of this algorithm yielding a reduction in the total operation count of about 70% when the filters are orthonormal, a reduction of about 62% for general biorthogonal filters, and a reduction of about 70% if the filters are symmetric odd-odd length filters. We further extend these results to 3 dimensions. In the second part of the paper we show how the SO2(R)method for implementing the discrete wavelet transform may be exploited to compute short FIR filters, and we construct edge mappings where we try to improve upon the preservation of regularity due to conventional methods. In the third part of the paper we consider the problem of discriminating two classes of radar signals generated from some number of point sources distributed randomly in a bounded plane domain. A statistical space-frequency analysis is performed on a set of training signals using the LDB-algorithm of N.Saito and R.Coifman. In this analysis we consider several dictionaries of orthonormal bases. The resulting most discriminating basis functions are used to construct classifiers. The success of different dictionaries is measured by computing the misclassification rates of the classifiers on a set of test signals. Fast Computational Algorithms for the Discrete Wavelet Transform and Applications of Localized Orthonormal Bases in Signal Classification