Dynamics of errors in 3D motion estimation and implications for strain-tenser imaging in acoustic elastography

For the purpose of quantifying the noise in acoustic elastography, a displacement covariance matrix is derived analytically for the cross-correlation based 3D motion estimator. Static deformation induced in tissue from an external mechanical source is represented by a second-order strain tensor. A generalized 3D model is introduced for the ultrasonic echo signals. The components of the covariance matrix are related to the variances of the displacement errors and the errors made in estimating the elements of the strain tensor. The results are combined to investigate the dependences of these errors on the experimental and signal-processing parameters as well as to determine the effects of one strain component on the estimation of the other. The expressions are evaluated for special cases of axial strain estimation in the presence of axial, axial-shear and lateral-shear type deformations in 2D. The signals are shown to decorrelate with any of these deformations, with strengths depending on the reorganization and interaction of tissue scatterers with the ultrasonic point spread function following the deformation. Conditions that favour the improvements in motion estimation performance are discussed, and advantages gained by signal companding and pulse compression are illustrated.