Deformation models and correlation analysis in elastography

Cross-correlation functions are derived with the purpose of determining how strain inhomogeneities affect the displacement estimates used in ultrasound-based elastography. Variations in the strain profile occur in most imaging situations and are caused by fluctuations in the stress field or elastic modulus of the sample. An analytical framework for developing signal processing strategies in elastography is described, and the limitations of correlation-based methods for measuring displacements in tissuelike media caused by static compression are emphasized. This paper includes (1) an accurate approximation for an inverse coordinate transformation that relates pre- and postcompression reflectivity profiles of the media, (2) a derivation of the echo-signal cross-correlation function in media with deterministic or stochastic strain profiles; (3) mathematical and graphical descriptions of the consequences that nonuniformities in the strain profile impose upon the uncertainty of displacement estimation; and (4) a demonstration of the advantages of echo signal conditioning and ultrasonic-pulse shaping to reduce the nonstationary effects that attenuate the cross-correlation peak and reduce the signal-to-noise ratio for displacement estimation.