Back-propagation based beamformer design for Transverse Oscillations in Echocardiography

Transverse oscillation (TO) imaging is a technique that produces ultrasound radiofrequency images featuring oscillations in both spatial dimensions. This is natural for axial direction but necessitates specific beamforming for transverse direction. Conventional TO beamformer design using Fraunhofer approximation has been properly developed in linear geometry. Using cosine function modulated by Gaussian as a lateral profile, the apodization function defined as inverse Fourier transform of the profile thus corresponds to the convolution between a Gaussian and two Dirac. This approach works well in linear geometry but is limited for sector scan as in echocardiography. We propose to use back-propagation based on reciprocity theorem, which appropriately takes into account the fact that the expected oscillation profiles are in polar coordinate. To validate with simulations using Filled II, the comparison of PSFs obtained using FA and back-propagation with theoretical profiles is quantified using root mean square error. The result clearly shows that PSFs obtained using back-propagation are more accurate than using FA. But it has declining advantages than FA at larger depth because the transformation from Cartesian to polar coordinate becomes closer for FA. The next step is to validate that the obtained PSF and associated TO allow more accurate motion estimates.