## 4 Results  
### Phantom Simulations  
Simulated B-mode images of wire reflectors in Figure 1a-d show that the 2D-DMAS, Rx-DMAS, and Tx-DMAS beamforming methods can effectively suppress side-lobe and ghost artifact compared to conventional CPWC imaging (i.e., 2D-DAS). The \(p\) value of DMAS beamforming in Figure 1 is set to be 2.0 as an example. Visual comparison reveals that 2D-DMAS provides the lowest side-lobe and ghost-artifact levels among all three DMAS methods. B-mode images of anechoic cysts in Figure 1e-h also show that these DMAS beamforming methods lead to noticeably lower clutter artifacts within the cyst, especially in the depth ranging from 15 mm to 35 mm. This is because the presence of ghost artifacts in 2D-DAS beamforming is relatively obvious in the shallow depth while spatial-coherence-based beamforming like DMAS is capable of alleviating the ghost artifact in the case of short transmit pulse [29]. On the other hand, speckle variation in the background region is also noticeably higher due to the emphasis of signal coherence in these DMAS beamforming methods. This is particularly evident for 2D-DMAS and Rx-DMAS. Nonetheless, it should be noted that 2D-DMAS beamforming appears to provide the lowest clutter within the cysts compared to the Rx-DMAS and Tx-DMAS counterparts.  
Quantitative analyses of each beamforming method with \(p\) values ranging from 1.5 to 3.0 are shown in Figure 2. Though Figure 2a,b shows that the image resolution generally improves with \(p\) value for all DMAS methods, it should be noted that the change of AW with \(p\) value is minor for Rx-DMAS and Tx-DMAS. On the contrary, only the AW in 2D-DMAS significantly decreases from 0.38 mm to 0.33 mm when the \(p\) value increases from 1.5 to 3.0. For LW, on the other hand, all 2D-DMAS, Rx-DMAS, and Tx-DMAS demonstrate a lower LW with the increase of \(p\) value. Among them, note that 2D-DMAS consistently provides the lowest LW when the same \(p\) value is considered. Figure 2c also shows that the resultant ghost-artifact level decreases with \(p\) value for all DMAS methods but that 2D-DMAS consistently outperforms the other two. For \(p=2.0\) as an example, the corresponding ghost-artifact levels are \(-90.8\), \(-81.5\), and \(-73.5\) dB, respectively, for 2D-DMAS, Rx-DMAS, and Tx-DMAS. For image contrast, Figure 2d shows that the CR markedly improves with \(p\) value, especially for 2D-DMAS and Rx-DMAS. However, it should be indicated in Figure 2e that the conventional 2D-DAS still provides the highest CNR while 2D-DMAS has the lowest due to the elevated granular pattern in the speckle background. Note that the difference in CNR between 2D-DMAS and the other DMAS counterparts becomes relatively minor with decreasing \(p\) value. For example, the CNRs are respectively 0.59, 0.69, and 0.89 for 2D-DMAS, Rx-DMAS, and Tx-DMAS when \(p=3.0\). This corresponds to a CNR difference of 0.3 between 2D-DMAS and Tx-DMAS. However, when \(p\) decreases to 2.0, the CNR becomes respectively 0.89, 0.96, and 1.02 for 2D-DMAS, Rx-DMAS, and Tx-DMAS. In other words, the CNR difference between 2D-DMAS and Tx-DMAS now decreases to 0.13. When the GCNR is considered, however, it should be noted that 2D-DMAS and Rx-DMAS outperforms 2D-DAS. In other words, 2D-DMAS and Rx-DMAS indeed provide improvements in image contrast even with the elevated speckle variation.  
Figure 1: Simulated B-mode images of wire reflectors and anechoic cysts for two-dimensional delay-and-sum (2D-DAS), Rx-DMAS, Tx-DMAS, and two-dimensional delay-multiply-and-sum (2D-DMAS) beamforming: The \(p\) value in DMAS beamforming is fixed to 2.0.