## 3 Results and Discussion
### Performances Evaluation of \(p\)-DAS on Image Quality
#### 3.2.1 Single Plane Wave Imaging  
First, the results obtained for single plane wave imaging (1 PW) are considered. The images obtained on the four phantoms with 1 PW are shown in Figure 2. The measurements are summed in Figure 4. As a first observation, the resolution is enhanced with \(p\)-DAS, due to its better rejection of ill-rephased wave fronts: in Figure 4, as \(p\) increases, improved sensitivity in the lateral direction is highlighted, both in simulation (Figure 4b; DAS, 0.73 mm; 2-DAS, 0.53 mm; 3-DAS, 0.46 mm) and experiment (Figure 4f; DAS, 0.81 mm; 2-DAS, 0.58 mm; 3-DAS, 0.48 mm). The mean axial FWHM over the 20 targets relies mainly on the excitation waveform, and it is slightly decreased for \(p=2\) and \(p=3\) (Figure 4a,e). For the simulated phantom (Figure 2a,c), the 20 axial FWHM are quasi constant for all of the scatterers with DAS (0.39-0.41 mm), but they decrease for the interfering scatterers when using 2-DAS (0.31-0.41 mm). The target that is most affected by interference located at (\(x=0\) mm, \(z=20\) mm) has an axial FWHM of 0.40 mm with  
Figure 4: Performances obtained with DAS, FDMAS, 2-DAS, and 3-DAS beamformers, using a single plane wave in simulations (**a–d**) and experiments (**e–h**). The results obtained are given for axial resolution (**a,e**), lateral resolution (**b,f**), contrast ratio (**c,g**), and contrast-to-noise ratio (**d,h**). DAS, whereas this is 0.31 mm with 2-DAS. However, the FWHM of the isolated scatterer placed at (\(x=0\) mm, \(z=10\) mm) is 0.40 mm for DAS and 2-DAS. Indeed, the rejection of interference for \(p\)-DAS is nonlinear with respect to amplitude, as seen for Equation (12). Then, the same interference noise is not rejected in the same way whether it interferes with the top of the pulse, or with the low-amplitude edges of the pulse. Finally, \(p\)-DAS tends to shrink the axial FWHM in the presence of interference. In addition, the results demonstrate that the CR increases with the \(p\)-value for both simulation (Figure 4c; DAS, 16.4 dB; 2-DAS, 24.9 dB; 3-DAS, 31.0 dB) and experiment (Figure 4g; DAS, 12.7 dB; 2-DAS, 19.9 dB; 3-DAS, 24.2 dB). These observations confirm the conclusions of Section 3.1: the higher the \(p\)-value is, the more rejected the side-lobes are. As a result, the artifacts from the bright speckle into the dark cysts are attenuated for \(p=2\) (Figure 2g,o), and even more so for \(p=3\) (Figure 2h,p), with respect to those for DAS (Figure 2e,m). Conversely, the CNR decreases as the \(p\)-value increases, both for simulation (Figure 4d; DAS, 5.9 dB; 2-DAS, 2.1 dB; 3-DAS, \(-1.6\) dB) and experiment (Figure 4h; DAS, 6.0 dB; 2-DAS, 2.5 dB; 3-DAS, \(-0.2\) dB). This means that the variance of the pixel intensities inside the cysts and inside the speckle is stronger with \(p=2\) and even more with \(p=3\). Looking at the simulated images, the speckle obtained with DAS (Figure 2e,m) is relatively homogeneous with gray pixels, whereas for \(p=2\) (Figure 2g,o) and \(p=3\) (Figure 2h,p) the speckle is more heterogeneous, with a background that is darker. Indeed, the coherent bright spots in the speckle are not strongly impacted by \(p\)-DAS, whereas the incoherent dark pixels of the speckle are heavily rejected. As a result, the CNR drops for high \(p\)-values because it varies in the opposite way to the increased variance. The same trend is obtained for the experiment (Figure 2m,o,p). Moreover, roughly coherent wave fronts back-scattered from point targets appear as relatively brighter spots when using \(p=2\) or \(p=3\) (Figure 2k,l), rather than conventional DAS (Figure 2i). Note that the energy of such high coherent targets tends to darken the speckle in their near lateral neighborhood. This phenomenon was identified by [PERSON] et al. in [18] as an inner characteristic of beamformers based on coherence. For this reason, the \(p\)-value can be adjusted to enhance the resolution and the CR (useful for lesion detectability), while preserving the speckle structure and CNR (used for texture analysis, and so, tissue characterization) [19].