Ultrasonic Signal Decomposition via Matching Pursuit with an Adaptive and Interpolated Dictionary

Matching pursuit is an iterative method whereby a signal is decomposed into a linear combination of functions that are selected from a redundant dictionary. In the original paper by Mallat and Zhang [1], a dictionary of Gabor functions is proposed. Each Gabor function is the product of a Gaussian function with a complex sinusoid, and is specified by time, frequency and scale. Since these functions are qualitatively and quantitatively very similar to ultrasonic echoes, it is appropriate to use the matching pursuit method to decompose ultrasonic signals to locate and identify discrete echoes embedded in complex signals. In this paper, a modified implementation of the matching pursuit algorithm is described, where the algorithm is specifically designed for an efficient decomposition of ultrasonic signals. The size of the wavelet dictionary is adaptively determined by the spectrum of the ultrasonic signal and is further controlled by additional physically meaningful restrictions. In each iterative step, the pursuit of the matching function begins with a coarse grid in the parameter space of the dictionary, and the highest energy matching function is found by interpolation of this coarse grid over the parameters. The algorithm is applied to a variety of measured ultrasonic signals. Signals consisting of multiple echoes are successfully decomposed, and the individual wavelets are well-matched to the original echoes.