## 1 Introduction  
In medical ultrasound imaging, delay-and-sum (DAS) beamforming is the standard technique to produce image outputs. However, DAS beamforming intrinsically suffers from limited image resolution and insufficient rejection of off-axis clutter interference. The loss of image quality is particularly evident in the case of plane-wave (PW) imaging due to the lack of transmit focusing [1]. In PW imaging, an unfocused wave is transmitted to illuminate a wide field-of-view, and then the backscattered echoes in the receiving array are coherently summed after time compensation of geometric paths to generate low-quality images at frame rates on the order of kHz. To improve the image quality of PW imaging, coherent plane wave compounding (CPWC) uses multi-angle PW transmissions to achieve synthetic transmit focusing [2; 3]. Unlike incoherent compounding, which combines the envelope image without phase information to reduce speckle variation, CPWC depends on the summation of image data with phase information to achieve synthetic focusing. In other words, low-quality images are firstly obtained with different PW transmit angles and then combined coherently to form the final high-quality image. Therefore, each image pixel in CPWC imaging is actually constructed by the two-dimensional (2D) DAS beamforming of the corresponding echo matrix of which the dimensions are the PW transmit angle and the receiving channel. Though CPWC imaging provides a trade-off between the frame rate and the image quality, its image quality heavily depends on the number of PW transmit angles. With only a few PW transmit angles, the suppression of side-lobe clutter and axial-lobe artifacts is generally unsatisfactory, and thus leads to ghost artifact in the image [4]. In order to improve image quality, many adaptive imaging methods have been proposed to process the echo matrix in multi-angle PW imaging. One of the widely used methods is minimum variance (MV) beamforming. Joint transmitting-receiving (JTR) beamforming calculates both the transmit and the receiving MV weights and then applies them to the echo matrix after sub-matrix averaging [5]. In double MV [6], adaptive weights of received channel data are firstly applied to each PW image, and then the second weights in the PW dimension are estimated from the resultant image data with different transmit angles. Spatial-coherence-based MV (DCT-MV and DCR-MV) beamforming is also feasible in which the covariance matrix is estimated through different combinations of the echo matrix [7]. Other methods mainly rely on adaptive weighting to suppress the image artifacts in multi-angle PW imaging. The coherence factor (CF) can represent the focusing quality, which is defined as the ratio between the coherent power and the total power in the channel data [8]. The CF weighting can also be combined with MV beamforming as suggested in References [9; 10]. In multi-angle PW imaging, it can be calculated from the output of JTR-MV beamforming in both the transmit and receive dimensions [11]. Other definitions of focusing factor include generalized coherence factor (GCF) [12], short-lag spatial coherence (SLSC) [13; 14], normalized autocorrelation factor (NAF) [15], and cross-coherence factor [4]. These factors are calculated using received echoes from multi-angle PW transmissions and then applied to the final CPWC image output to suppress low-quality pixels. Singular value decomposition (SVD) filter is also proposed for side-lobe suppression of CPWC by transforming the multi-angle PW image data into a 2D spatio-angular matrix before applying the SVD filter to extract the high-coherence component [16]. Recently, a novel nonlinear beamforming has been proposed for ultrasound imaging by multiplying radiofrequency (RF) echoes between every possible channel pair after time compensation. It is referred to as the delay-multiply-and-sum (DMAS) beamforming [17; 18]. The purpose of channel-domain multiplication is to introduce spatial coherence in the channel data into the beamforming process.