Model-based estimation of ultrasonic echoes part I: Analysis and algorithms

The patterns of ultrasonic backscattered echoes represent valuable information pertaining to the geometric shape, size, and orientation of the reflectors as well as the microstructure of the propagation path. Accurate estimation of the ultrasonic echo pattern is essential in determining the object/propagation path properties. In this study, we model ultrasonic backscattered echoes in terms of superimposed Gaussian echoes corrupted by noise. Each Gaussian echo in the model is a nonlinear function of a set of parameters: echo bandwidth, arrival time, center frequency, amplitude, and phase. These parameters are sensitive to the echo shape and can be linked to the physical properties of reflectors and frequency characteristics of the propagation path. We address the estimation of these parameters using the maximum likelihood estimation (MLE) principle, assuming that all of the parameters describing the shape of the echo are unknown but deterministic. In cases for which noise is characterized as white Gaussian, the MLE problem simplifies to a least squares (LS) estimation problem. The iterative LS optimization algorithms when applied to superimposed echoes suffer from the problem of convergence and exponential growth in computation as the number of echoes increases. In this investigation, we have developed expectation maximization (EM)-based algorithms to estimate ultrasonic signals in terms of Gaussian echoes. The EM algorithms translate the complicated superimposed echoes estimation into isolated echo estimations, providing computational versatility. The algorithm outperforms the LS methods in terms of independence to the initial guess and convergence to the optimal solution, and it resolves closely spaced overlapping echoes.