Proceedings PaperWavelet packet representations from local maxima in their expansion coefficients
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A new method of adaptive waveform analysis, called shape adaptive wavelet analysis, is presented where the basis vectors (wavelet packets) are initially considered as functions of continuous variables, in particular position/time, frequency, and scale. A discrete subset of these wavelet packets is chosen from local maxima of the modulus of their expansion coefficients, in general resulting in a non-orthogonal basis. Redundancy removal, when required, is implemented using an algorithm loosely based on Gram-Schmidt orthogonalization, where basis vectors are chosen to maximize the modulus of the projection of the residual perpendicular to the space spanned by the basis vectors chosen previously, thus maximizing `information content.' In cases where this representation is not sufficiently accurate an additional process of feature substraction is available enabling representations of arbitrary accuracy. It is shown, however, that there are sometimes advantages in retaining redundancy in the wavelet packet characterization of the signal, in particular where noise is present. The method presented in this paper is compared with the wavelet packet transform of Coifman et al. and with the matching pursuit algorithm of Mallat et al., and is shown to provide an improved characterization under circumstances which are of relevance in surveillance, detection, and classification in noisy or cluttered backgrounds. The method is shown to be well suited to characterizations in higher-dimensional wavelet packet spaces, where additional shape parameters, for example chirp angle, characterize the wavelet packets in addition to position, frequency, and scale.