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.

💡 Research Summary

The paper introduces “KK beamforming,” a compressive ultrasound imaging technique that recasts both transmit and receive operations as far‑field plane waves, enabling the entire beamforming process to be performed in spatial frequency (k‑space). The authors first transform raw RF channel data into virtual receive plane‑wave data by applying a temporal shear followed by horizontal summation, reducing an L × T data matrix to M × T and achieving compression ratios on the order of ten. Beamforming then proceeds by summing over all transmit angles θᵢ and virtual receive angles θₒ with simple linear delays τ_in = sᵢ·r/c and τ_out = sₒ·r/c, yielding a fully spatially invariant formulation.

A central contribution is the design of θₒ sampling strategies based on coarray theory. “Vernier‑type” schemes densely fill the gaps between transmit angles, producing a small kₓ step size (δkₓ ≈ 2 δkᵢ/M) that maximizes image contrast. To increase resolution, the receive angle set is shifted by an integer j, expanding the k‑space support up to roughly twice that of the transmit set while trading off sampling density. Intermediate j values give a bimodal kₓ distribution that balances contrast and resolution. Multiple j‑specific images can be coherently or incoherently compounded, further improving quality at the cost of a modest increase in effective compression factor (L/(J·M)).

The authors also propose a “confocal” KK configuration that mimics conventional DAS transfer functions: when transmit and receive apertures are identical, their spectra are rectangular, and the combined transfer function becomes triangular, preserving DAS‑like resolution and contrast while still benefiting from compression.

Experimental validation on a calibrated phantom and in‑vivo human tissue demonstrates that KK beamforming retains PSNR, CNR, and visual resolution comparable to standard DAS, despite a ten‑fold reduction in data volume. Memory footprint and cache usage are dramatically lowered, and the use of pre‑computed lookup tables yields 2–3× faster processing.

Overall, KK beamforming leverages k‑space coarray concepts to achieve simultaneous data compression, computational speed‑up, and high‑quality ultrasound imaging, offering a flexible framework where users can select sampling patterns tailored to specific contrast‑resolution trade‑offs.