Information-theoretic wavelet noise removal for inverse elastic wave scattering theory

Abstract

A discussion of noise removal in ultrasound (elastic wave) scattering for nondestructive evaluation is given. The methods used in this paper include a useful suboptimal Wiener filter, information theory and orthonormal wavelets. The multiresolution analysis (MRA), due to Mallat, is the key wavelet feature used here. Whereas Fourier transforms have a translational symmetry, wavelets have a dilation or affine symmetry which consists of the semi-direct product of a translation with a change of scale of the variable. The MRA describes the scale change features of orthonormal wavelet families. First, an empirical method of noise removal from scattered elastic waves using wavelets is shown to markedly improve the ${l}^{1}$ and ${l}^{2}$ error norms. This suggests that the wavelet scale can act as dial to ``tune out'' noise. Maximization of the Kullback-Liebler information is also shown to provide a scale-dependent noise removal technique that supports (but does not prove) the intuition that certain small energy coefficients that are retained contain large information content. The wavelet MRA thereby locates ``islands of information'' in the phase space of the signal. It is conjectured that this method holds more generally.