## 2 Theory
### 2D-DMAS Processing for Multi-Angle PW Imaging  
Multi-angle PW imaging can be understood as 2D signal processing of the received echo matrix of which the dimensions are the receiving channel and the PW transmit angle. Each PW transmission with specified a transmit angle corresponds to one transmit event, and thus, the echoes after time delay and phase rotation can be represented as a three-dimensional signal with the dimensions of image width, image depth, and receiving channel. Therefore, for each pixel (\(x\), \(z\)) in the image where \(x\) and \(z\) are respectively the lateral and axial positions, a 2D echo matrix comprising the received channel signal from every transmit event of different PW transmit angles can be constructed. The echo matrix can be processed to produce the final B-mode output by any combination of DAS and DMAS beamforming in the two dimensions of PW transmit angle and receiving channel. CPWC imaging depends on the 2D coherent summation of the echo matrix to achieve both receive and transmit focusing and is referred to as 2D-DAS in this study. Given the baseband channel data in multi-angle PW imaging after time compensation and phase rotation as \(s_{nk}=a_{nk}e^{i\phi_{nk}}\) where \(n\) is the index of receiving channel (\(n=1\)\(2\),..., \(N\)) and \(k\) is the index of PW transmit angle (\(k\) = 1, 2,..., \(M\)), the 2D-DAS beamforming (i.e., CPWC) can be formulated as follows:  
\[\text{y}_{\text{2D-DAS}}=\frac{1}{MN}\sum\limits_{k=1}^{M}\sum\limits_{n=1}^{N}s _{nk} \tag{2}\]  
On the contrary, the proposed 2D-DMAS method extracts the 2D signal coherence by directly applying the DMAS process on the echo matrix. In other words, each entry in the echo matrix is magnitude-scaled by \(p\)th rooting while maintaining its phase. Then, the signal dimensionality is restored by \(p\)th power after 2D summation, as shown in the following:  
\[\text{y}_{\text{2D-DMAS}}=\left(\frac{1}{MN}\sum\limits_{k=1}^{M}\sum\limits_{n =1}^{N}\sqrt[]{a_{nk}}e^{j\phi_{nk}}\right)^{p} \tag{3}\]  
Other DMAS beamforming methods proposed in the literatures for multi-angle PW imaging are considered for comparison, and they are referred to as Rx-DMAS [25] and Tx-DMAS [26]. Rx-DMAS and Tx-DMAS also consider the spatial coherence of the echo matrix but in only one dimension. In other words, the DMAS operation is performed in the dimension of either the PW transmit angle or the receiving channel. For Rx-DMAS, the echo matrix is firstly summed in the dimension of the PW transmit angle for synthetic transmit focusing. Then, the DMAS processing is applied on the channel data to estimate the spatial coherence in the dimension of the receiving channel. For Tx-DMAS, on the contrary, the echo matrix is firstly summed in the dimension of the receiving channel to achieve dynamic receive focusing before the DMAS processing is applied in the dimension of the PW transmit angle to emphasize the signal coherence from distinct PW transmissions. Specifically, Tx-DMAS beamforming can be understood as the \(p\)th-root magnitude-scaling of low-quality images from several PW transmit angles, coherently summing these images to produce a high-quality image but with \(p\)th-power restoring of signal dimensionality. Using the same representation of baseband echo matrix in Equation (2), Rx-DMAS and Tx-DMAS can be formulated in Equations (4) and (5), respectively:  
\[\text{y}_{\text{Rx-DMAS}}=\left(\frac{1}{N}\sum\limits_{n=1}^{N}\sqrt[]{c_{n} }e^{j\gamma_{n}}\right)^{p}\text{with}\quad c_{n}e^{j\gamma_{n}}=\frac{1}{M} \sum\limits_{k=1}^{M}s_{nk} \tag{4}\]  
\[\text{y}_{\text{Tx-DMAS}}=\left(\frac{1}{M}\sum\limits_{k=1}^{M}\sqrt[]{b_{k }}e^{j\beta_{k}}\right)^{p}\text{with}\quad b_{k}e^{j\beta_{k}}=\frac{1}{N} \sum\limits_{n=1}^{N}s_{nk} \tag{5}\]