# Simultaneous Coded Plane-Wave Imaging Using an Advanced Ultrasound Forward Model
## 3 Results
### Wire Phantom Simulation  
Here, the method is evaluated in simulation using the phantom described in Figure 3a. Figure 6 (left to right) shows the TRF estimations \(\hat{\gamma}_{3\text{PW-DAS}}\) obtained using DAS on successive PW acquisitions. \(\hat{\gamma}_{3\text{PW-DAS-L1}}\) is obtained by \(\ell_{1}\) denoising of \(\hat{\gamma}_{3\text{PW-DAS}}\), as described in Section 2.8). The TRF estimation \(\hat{\gamma}_{\text{SPW-DAS}}\) is obtained by applying DAS to the \(\hat{\beta}^{(k)}\) compressed echoes obtained after Wiener filtering. The TRF estimation \(\hat{\gamma}_{\text{SPW-DAS-L1}}\) is an \(\ell_{1}\) denoised version of \(\hat{\gamma}_{\text{SPW-DAS}}\), with the same denoising process described in Section 2.8. Finally, the estimated TRF with the proposed method \(\hat{\gamma}_{\text{SPW-FM-L1}}\) is shown using \(\lambda=0.01\).  
First, it can be seen that the point scatterers are correctly localized whatever the reconstruction method. However, image quality varies greatly between the techniques. It is noted that \(\hat{\gamma}_{\text{SPW-DAS}}\) is a poor-quality image with a very high amount of noise. In other words, since the excitation sequences \(e^{(k)}\) are not perfectly orthogonal (as described in Figure 5), the RF estimations \(\hat{\beta}^{(k)}\) are corrupted with crosstalk noise that affects the image negatively. Nevertheless, when applying \(\ell_{1}\) denoising, the TRF is relatively similar to the one of \(\hat{\gamma}_{\text{SPW-DAS-L1}}\).  
The TRF-estimated \(\hat{\gamma}_{\text{SPW-DAS}}\) contains artifacts that affect the final image negatively. From Table 1, it can be seen that the scatterers are reconstructed with a broader spatial resolution than for the proposed method. As the scatterers are punctual, thinner spatial resolution means a better reconstruction. When applying \(\ell_{1}\) denoising (\(\hat{\gamma}_{3\text{PW-DAS-L1}}\)), the DAS artifacts are suppressed and the spatial resolution decreases both axially and laterally. However, it can be seen that the TRF estimation is better with the proposed method asthe spatial resolution is point-like. This effect is clearly noted on the closely placed wires (e.g., at 31 mm, the rightmost points are not distinguishable), while the proposed method separates these targets.  
The performance of all the methods is evaluated using the PCID (Section 2.9) from 0.16\(\lambda\) to 2\(\lambda\) in both the axial and lateral directions. The results are presented in Figure 7. For the axial direction, scatterers spaced more than 0.16\(\lambda\) are separated by the proposed method, \(\hat{\gamma}_{\text{3 PW-DAS-L1}}\) and \(\hat{\gamma}_{\text{SPW-DAS-L1}}\), while \(\hat{\gamma}_{\text{3 PW-DAS}}\) allows us to separate scatterers spaced more than 0.81\(\lambda\) and \(\hat{\gamma}_{\text{SPW-DAS}}\) more than 0.89\(\lambda\). For the lateral direction, scatterers spaced more than 0.6\(\lambda\) are separated by the proposed method, \(\hat{\gamma}_{\text{3 PW-DAS-L1}}\) and \(\hat{\gamma}_{\text{SPW-DAS-L1}}\), while \(\hat{\gamma}_{\text{3 PW-DAS}}\) allows us to separate scatterers spaced more than 2\(\lambda\). Thus, the proposed method outperforms conventional DAS reconstruction in terms of spatial resolution in the simulated sparse medium. The performance is similar with \(\hat{\gamma}_{\text{3 PW-DAS-L1}}\) and \(\hat{\gamma}_{\text{SPW-DAS-L1}}\); however, \(\hat{\gamma}_{\text{SPW-DAS-L1}}\) contains a significant amount of noise even with few scatterers and will therefore be discarded for the rest of the study.