## 5 Discussions and Conclusions  
In this study, DMAS beamforming has been extended to use 2D spatial coherence in multi-angle PW imaging. The 2D spatial coherence is derived from the echo matrix of every image pixel in both the dimensions of PW transmit angle and receiving channel. Specifically, the proposed 2D-DMAS method extracts the 2D signal coherence by magnitude-scaling every entry in the echo matrix by \(p\)th root operation while maintaining its phase. After 2D summation of the magnitude-scaled echo matrix, \(p\)th power operation is used to restore the dimensionality of the final image output. Here, the aforementioned \(p\) value determines the extent of 2D signal coherence included in the beamforming output, and thus provides a flexibly tunable manipulation of image quality. Moreover, due to the baseband nature of the proposed 2D-DMAS beamforming, its implementation does not demand for oversampling as the original DMAS in the RF domain. Simulated and experimental B-mode images of wire reflectors and speckle phantom from PICMUS data indicate that, compared to other one-dimensional DMAS methods such as Rx-DMAS and Tx-DMAS, the proposed 2D-DMAS beamforming is capable of the most effective improvement not only in image resolution but also in ghost-artifact suppression in multi-angle PW imaging. For image contrast, both 2D-DMAS and Rx-DMAS lead to higher image CRs and GCNRs than the Tx-DMAS counterpart. B-mode images of in vivo carotid artery further indicate that, in clinical scenarios, 2D-DMAS beamforming provides the most apparent suppression of image artifacts within the vessel even in the presence of acoustic reverberations. Nonetheless, in terms of image CNR, all DMAS beamforming methods remain inferior to the conventional CPWC image (i.e., 2D-DAS) due to the noticeably elevated speckle variation, especially for 2D-DMAS. This observation is as expected since the speckle variation has been known to increase with the extent of spatial coherence considered in B-mode imaging. Since 2D-DMAS emphasizes the signal coherence in the beamforming output more than the other one-dimensional DMAS counterparts, it suffers from more noticeable increases of speckle variation. Nonetheless, it is noticeable that 2D-DMAS also provides higher or at least comparable GCNRs to the 2D-DAS. To improve the image CNR in DMAS beamforming, either frequency compounding [30] or receive aperture compounding [31] can be a feasible solution to smoothing the granular speckle appearance of B-mode image. Though 2D-DMAS effectively improves the image contrast and resolution in B-mode imaging, its application in motion or blood flow estimation may be limited due to the low signal coherence of moving objects in the dimension of the PW transmit angle. The 2D-DMAS beamforming in this study is implemented as described in Equation (3) by firstly accumulating the received channel data from distinct PW transmit angles to construct the echo matrix before applying 2D-DMAS processing. Consequently, it demands for huge memory allocation for the echo matrix, especially when the number of PW transmit angles is high. This is also why the number of PW transmit angles is currently limited to only seven in this study. Nonetheless, note that the formulation of 2D-DMAS can be rewritten in an alternative form as follows:  
\[Y_{\text{2D-DMAS}}=\left(\frac{1}{M}\!\sum_{k=1}^{M}\!\left(\!\frac{1}{N}\! \sum_{n=1}^{N}\sqrt[]{a_{nk}}e^{j\phi_{nk}}\!\right)\right)^{p} \tag{9}\]  
In this alternative form of 2D-DMAS implementation, the received channel data from each PW transmit angle is firstly magnitude-scaled by \(p\)th root operation and then combined to fulfill the inner summation in Equation (7). In other words, for each PW transmit angle, the received channel data has been summed with magnitude-scaling to a single value for the corresponding image pixel. This produces one magnitude-scaled low-quality image for each PW transmit angle.