Far-field compressive ultrasound beamforming

We present a compressive beamforming method for coherent plane-wave compounding (CPWC) ultrasound imaging based on a far-field decomposition of the received radiofrequency (RF) data into virtual plane waves. This decomposition recasts the imaging operation entirely in the spatial frequency domain ($k$-space), allowing direct and flexible control over $k$-space sampling distributions based on the principle of coarrays. We present vernier-type sampling strategies designed to optimize the tradeoff between image contrast and resolution with minimum redundancy, including strategies that favor dense low-frequency sampling for high contrast, shifted schemes that extend the frequency support for improved resolution, and confocal or hybrid compounding schemes that approximate the spatial-frequency transfer function of conventional DAS beamforming. Our method, called KK beamforming, is validated with a calibration phantom and in-vivo human tissue data, demonstrating compression factors of an order of magnitude while maintaining image qualities comparable to conventional DAS. We further demonstrate that KK beamforming yields improvements in computational speed owing to its reduced memory footprint and more efficient cache utilization of the compressed data and associated look-up tables.