Back-propagation beamformer design for motion estimation in echocardiography

Transverse oscillation (TO) techniques have shown their potential for improving the accuracy of local motion estimation in the transverse direction (i.e., the direction perpendicular to the beam axis). The conventional design of TOs in linear geometry, which is based on the Fraunhofer approximation, relates point spread function (PSF) and apodization function through a Fourier transform. Motivated by the adaptation of TOs in echocardiography, we propose a specific beamforming approach based on back-propagation (BP) to build TOs in sector-shaped geometry. Numerical simulations and experimental data give a comparison between proposed and conventional beamforming for TOs. The accuracy is first quantified by comparing the generated and theoretical PSF using the root mean square error (RMSE) and shows that BP-based beamforming approximates the desired TOs more closely than the conventional approach. Motion estimation is then evaluated. The axial and lateral displacements are within the range [0-0.6] mm and [0°-6.4°], respectively, which correspond to 0.8 times the axial (0.73 mm) and lateral (8°) wavelengths. The result shows that the proposed method yields a clear improvement for lateral displacements, by reducing the error by 28.6% compared with Fourier transform-based beamforming, while maintaining the same error for axial motion estimation. Experimental measurements are discussed to complete this study and confirm that BP-based beamforming leads to better controlled TO images than conventional Fourier-based beamforming.