# Simultaneous Coded Plane-Wave Imaging Using an Advanced Ultrasound Forward Model
## 3 Results
### Wire Phantom Experiment  
In this subsection, we apply the proposed method experimentally to the sparse medium described in Figure 3b. Figure 8 shows the estimated TRF using DAS (\(\hat{\gamma}_{\text{3 PW-DAS}}\)), using \(\ell_{1}\) denoising on \(\hat{\gamma}_{\text{3 PW-DAS}}\) (\(\hat{\gamma}_{\text{3 PW-DAS-L1}}\)), and using the proposed method with \(\lambda=0.01\). In order to avoid confusion, the location of the scatterers is indicated by the red arrows. First, it can be seen that the proposed method converges to a stable solution where the five scatterers are correctly localized. In fact, the estimated TRF contains 63 nonzero coefficients located around the physical position of the scatterers, which is more than the five expected but still considerably lower than the 836 and 36,405 coefficients from \(\hat{\gamma}_{\text{3 PW-DAS-L1}}\) and \(\hat{\gamma}_{\text{3 PW-DAS}}\). Due to the acquisition noise and the imperfect orthog \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Metric** & \(\hat{\gamma}_{\text{3 PW-DAS}}\) & \(\hat{\gamma}_{\text{3 PW-DAS-L1}}\) & \(\hat{\gamma}_{\text{SPW-DAS}}\) & \(\hat{\gamma}_{\text{SPW-DAS-L1}}\) & \(\hat{\gamma}_{\text{SPW-Fm-n}}\) \\ \hline _Ax. res_ WHM & 192.5 μm & 87.4 μm & 244 μm & 98.8 μm & 25 μm \\ _Lat. res_ WHM & 455 μm & 177.8 μm & 551.25 μm & 245 μm & 61.25 μm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Image quality metrics evaluated on the simulated wire phantom. Figure 6: Estimation of the TRF on simulated wire phantom. (**a**) Estimated using successive PW acquisitions with DAS algorithm (\(\hat{\gamma}_{\text{3 PW-DAS}}\)). (**b**) Estimated using \(\ell_{1}\) denoising on \(\hat{\gamma}_{\text{3 PW-DAS}}\) with \(\lambda=0.5\) (\(\hat{\gamma}_{\text{3 PW-DAS-L1}}\)). (**c**) Estimated using DAS and coherent compounding of the compressed echoes \(\hat{\delta}^{(k)}\) after simultaneous PW excitation (\(\hat{\gamma}_{\text{SPW-DAS}}\)). (**d**) Estimated using \(\ell_{1}\) denoising on \(\hat{\gamma}_{\text{SPW-DAS}}\) with \(\lambda=0.5\) (\(\hat{\gamma}_{\text{SPW-DAS-L1}}\)). (**e**) Estimated using the proposed method with simultaneous PW excitations (\(\hat{\gamma}_{\text{SPW-DAS}}\)). an ideal solution of five nonzero coefficients cannot be reached experimentally. However, these results confirm that the proposed method can be applied experimentally, even with the complexity of the simultaneously emitted waveforms. The spatial resolution FWHM from the central scatterer is compared between \(\hat{\gamma}_{\text{3 PW-DAS}}\), \(\hat{\gamma}_{\text{3 PW-DAS-L1}}\), and \(\hat{\gamma}_{\text{3 PW-FM-L1}}\). In the axial direction, the measured spatial resolution FWHM is 25, 250, and 175 \(\upmu\)m for the proposed method \(\hat{\gamma}_{\text{SPW-FM-L1}}\), \(\hat{\gamma}_{\text{3 PW-DAS}}\), and \(\hat{\gamma}_{\text{3 PW-DAS-L1}}\). Thus, the axial resolution is improved 10-fold using \(\hat{\gamma}_{\text{SPW-FM-L1}}\) rather than \(\hat{\gamma}_{\text{3 PW-DAS}}\). In the lateral direction, the same trend is observed. The measured spatial resolution FWHM is 61, 367, and 250 \(\upmu\)m for the proposed method \(\hat{\gamma}_{\text{SPW-FM-L1}}\), \(\hat{\gamma}_{\text{3 PW-DAS}}\), and \(\hat{\gamma}_{\text{3 PW-DAS-L1}}\). These measurements clearly show that the proposed method outperforms conventional techniques on sparse mediums in terms of spatial resolution, while increasing the frame rate by 55.57%. Figure 8: Estimation of the wires TRF. (**a**) Estimated using the delay and sum algorithm (\(\hat{\gamma}_{\text{3 PW-DAS}}\)). (**b**) Estimated using \(\ell_{1}\) denoising on \(\hat{\gamma}_{\text{3 PW-DAS}}\). (**c**) Estimated using the proposed approach (\(\hat{\gamma}_{\text{SPW-FM-L1}}\)). The physical location of the scatterers is indicated by the red arrows. It can be seen that for parts a and b, an additional step of envelope detection is applied before \(\log-\)compression. Figure 7: PCID measured on all reconstructed images. (**a**) Axial direction. (**b**) Lateral direction.