Far-field compressive ultrasound beamforming

Abstract

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.

Figures (7)

• Figure 1: Top row: Conventional CPWC DAS beamforming. For every transmit angle $\theta_i$, the received RF data is a function of $u_l$ and $t$. Time delays to sample coordinate $\mathbf{r}$ are shown in left panel. Bottom row. KK beamforming. RF data is first compressed (Eq. \ref{['compression']}) and then beamformed (Eq. \ref{['KK']}) using time delays to sample coordinate $\mathbf{r}$ shown in left panel.
• Figure 2: Examples of KK frequency support. a) Transmit angles $\theta_i$ and (sorted) receive angles $\theta_o$ for different shift parameters $j$, where angular range is $\pm 12^{\circ}$ for both and $N=M=15$. b) Resultant distribution of difference angles $\Delta \theta = \theta_o - \theta_i$, (proportional to $\Delta k_x$ in the paraxial limit). c) Stacked histograms of $\Delta \theta$.
• Figure 3: Example of KK frequency support. a) Transmit angles $\theta_i$ and (sorted) receive angles $\theta_o$, where angular range is $\pm 12^{\circ}$ for both and $N=M=15$. b) Resultant distribution of difference angles $\Delta \theta = \theta_o - \theta_i$, (proportional to $\Delta k_x$ in the paraxial limit). c) Histogram of $\Delta \theta$ closely resembles confocal-like frequency transfer of conventional CPWC DAS.
• Figure 4: Resolution point-target images. Top row: full field-of-view (FOV) images from DAS and KK beamforming with various receive sampling schemes. $M$ indicates number of array elements in the case of DAS or the number of synthesized receive angles in the case of KK beamforming. FOV sizes are indicated in Table \ref{['tableMethods']}. The red box indicates the zoomed region shown in middle rows. Middle rows: zoomed views of the closest-spaced wire targets. Bottom panel: normalized intensity profiles along red dashed line, comparing DAS with KK $j=0$, 3, 6. The $j=6$ scheme achieves improved separability of the leftmost 0.25 mm-spaced targets owing to its broader spatial frequency support, at the cost of increased susceptibility to aliasing.
• Figure 5: Anechoic inclusion with associated gCNR values (inclusion defined by blue circle; background defined by red rectangle). All panels obtained with $N=15$ transmit angles; $M$ indicates the number of receive elements or angles. Row 1 shows conventional DAS image ($M=192$) and KK images obtained with $27\times$ compressed RF data ($M=7$) using Eq. \ref{['uniform']}. Row 2 shows KK images obtained from confocal beamforming (Eq. \ref{['confocal']} -- left panel) and from compounding $j=0,3,6$ panels directly above, either coherently (Eq. \ref{['coherent']}) or incoherently (Eq. \ref{['incoherent']}). For all panels in Row 2, $M=21$ overall . Rows 3 and 4 are the same as Rows 1 and 2, except with threefold more receive angles. The resulting gCNR values indicated in top/right of each panel are comparable to DAS for a compression factor of $3.4 \times$ ($M=57$), though somewhat deteriorate with increasing compression (decreasing $M$).