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

摘要

We construct an algorithm for implementing the discrete wavelet transform bymeans of matrices in SO_2(R) for orthonormal compactly supported wavelets andmatrices in SL_m(R), m > = 2, for compactly supported biorthogonal wavelets. Weshow that in 1 dimension the total operation count using this algorithm can bereduced to about 50% of the conventional convolution and downsampling by2-operation for both orthonormal and biorthogonal filters. In the special caseof biorthogonal symmetric odd-odd filters, we show an implementation yielding atotal operation count of about 38% of the conventional method. In 2 dimensionswe show an implementation of this algorithm yielding a reduction in the totaloperation count of about 70% when the filters are orthonormal, a reduction ofabout 62% for general biorthogonal filters, and a reduction of about 70% if thefilters are symmetric odd-odd length filters. We further extend these resultsto 3 dimensions. We also show how the SO_2(R)-method for implementing thediscrete wavelet transform may be exploited to compute short FIR filters, andwe construct edge mappings where we try to improve upon the degree ofpreservation of regularity in the conventional methods. We also consider atwo-class waveform discrimination problem. A statistical space-frequencyanalysis is performed on a training data set using the LDB-algorithm of N.Saitoand R.Coifman. The success of the algorithm on this particular problem isevaluated on a disjoint test data set.