Error analysis in acoustic elastography. I. Displacement estimation.

Correlation between acoustic echo signals obtained before and after application of an external compressional force provides information about the internal deformation of an elastic medium. In this paper, the variance for displacement estimated from an echo data segment and the covariance between two windowed segments that may overlap are derived. The signal and noise spectra are Gaussian and independent. The dependence of the displacement variance on input signal-to-noise ratio (SNRi), time-bandwidth product W, fractional bandwidth Y-1, and the rate of displacement variation with depth a is investigated. The relationship between a and the other experimental parameters is crucial for understanding how signal decorrelation affects displacement error. The expression for displacement variance reduces to the Cramer-Rao lower bound result when a = 0 and W > > 1 for both bandpass and base-band signals. When a not equal to 0 displacement variance increases, and there is an optimal window length at W = square root of 20/a square root of 1 + Y2 for which the displacement variance is minimum. Narrow-band signals produce larger errors than broadband signals for long observation windows when a not equal to 0 and just the opposite when a = 0. Errors are greatest for displacements estimated from the envelope of narrow-band signals. Finally, a general expression for the minimum displacement variance for arbitrary signal and noise spectra is derived as a function of the experimental parameters. These results form a framework for analyzing strain estimates in elastography, the subject of a companion paper.