FoCUS: Fourier-Based Coded Ultrasound

Signal-to-noise ratio of diverging waves in multiscattering media: Effects of signal duration and divergence angle

In this paper, SNR maximization in coded diverging waves is studied, and experimental verification of the results is presented. Complementary Golay sequences and binary phase shift keying modulation are used to code the transmitted signal. The SNR in speckle and pin targets is maximized with respect to chip signal length. The maximum SNR is obtained in diverging wave transmission when the chip signal is as short a duration as the array permits. We determined the optimum diverging wave profile to confine the transmitted ultrasound energy in the imaging sector. The optimized profile also contributes to the SNR maximization. The SNR performances of the optimized coded diverging wave and conventional single-focused phased array imaging are compared on a single frame basis. The SNR of the optimized coded diverging wave is higher than that of the conventional single-focused phased array imaging at all depths and regions.

Central Splitting of A2 Discrete Fourier–Weyl Transforms

Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2 constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A2 is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.

Fourier-Domain Beamforming and Structure-Based Reconstruction for Plane-Wave Imaging

Ultrafast imaging based on coherent plane-wave compounding is one of the most important recent developments in medical ultrasound. It significantly improves the image quality and allows for much faster image acquisition. This technique, however, requires large computational load motivating methods for sampling and processing rate reduction. In this work, we extend the recently proposed frequency-domain beamforming (FDBF) framework to plane-wave imaging. Beamforming in frequency yields the same image quality while using fewer samples. It achieves at least fourfold sampling and processing rate reduction by avoiding oversampling required by standard processing. To further reduce the rate, we exploit the structure of the beamformed signal and use compressed sensing methods to recover the beamformed signal from its partial frequency data obtained at a sub-Nyquist rate. Our approach obtains tenfold rate reduction compared with standard time-domain processing. We verify performance in terms of spatial resolution and contrast based on the scans of a tissue mimicking the phantom obtained by a commercial Aixplorer system. In addition, in vivo carotid and thyroid scans processed using standard beamforming and FDBF are presented for qualitative evaluation and visual comparison. Finally, we demonstrate the use of FDBF for shear-wave elastography by generating velocity maps from the beamformed data processed at sub-Nyquist rates.