The SVD Beamformer: Physical Principles and Application to Ultrafast Adaptive Ultrasound

A shift of paradigm is currently underway in biomedical ultrasound thanks to plane and diverging waves for ultrafast imaging. One remaining challenge consists in the correction of phase and amplitude aberrations induced during propagation through complex layers. Unlike conventional line-per-line imaging, ultrafast ultrasound provides for each transmission, backscattering information from the whole imaged area. Here, we take benefit from this feature and propose an efficient approach to perform fast aberration correction based on the Singular Value Decomposition of an ultrafast compound matrix built from backscattered data for several plane wave transmissions. First, we explain the physical signification of SVD and associated singular vectors within the ultrafast matrix formalism. We theoretically demonstrate that the spatial and angular variables separation, rendered by SVD on ultrafast data, provides an elegant and straightforward way to optimize angular coherence of backscattered data. In heterogeneous media with an aberrating phase screen approximation, we demonstrate that the first spatial and angular singular vectors retrieve on one side the non-aberrated image, and on the other, the phase and amplitude of the aberration law. In vitro results prove the efficiency of the image correction, but also the accuracy of the aberrator determination. Based on spatial and angular coherence, we introduce a complete methodology for adaptive beamforming of ultrafast data, performed on successive isoplanatism patches undergoing SVD beamforming. The simplicity of this method paves the way to real-time adaptive ultrafast ultrasound imaging and provides a theoretical framework for future quantitative ultrasound applications.