# Simultaneous Coded Plane-Wave Imaging Using an Advanced Ultrasound Forward Model
## 4 Conclusions  
Here, we presented a general forward model that links the recorded RF data to the excitation signals, the geometric/acousto-electrical properties of the ultrasound probe, and the TRF. The general model was adapted to the concept of coherent PW compounding. To increase the frame acquisition rate, simultaneous coded emission of PWs was proposed, and the forward model was adapted accordingly. Three PWs were emitted simultaneously inside the medium to obtain an overdetermined forward model. The proposed excitation signals were a mix of three linear chirps. The maximal value of the cross-correlation/autocorrelation ratio obtained (between the excitation signals) did not exceed 14 dB. In addition, the proposed forward model accounts for the crosstalk noise, which was compensated for during the inversion of the forward model. Simulation results on wire phantom showed that the ability to separate scatterers is improved axially and laterally with the proposed method. For the estimated TRF \(\hat{\gamma}_{\text{SPW-FM-L1}}\), scatterers axially spaced by more than 0.16 \(\lambda\) were separated, while DAS achieved a similar performance at 0.89 \(\lambda\). In the lateral direction, the scatterers separated by more than 0.6 \(\lambda\) were separated using the proposed method, while for the same task, DAS-reconstructed TRF required more than 2 \(\lambda\). Furthermore, the PCID was globally lower than \(-60\) dB for the proposed method, while the best value for DAS was \(-31.6\) dB. We also observed that the PSF inside the proposed TRF estimation was quasi-constant as a function of the scatterer depth and lateral position (unlike the DAS). Such improvements were obtained because the proposed model is not based on the classic DAS assumptions (a pulse as the excitation signal, an ultrasound probe of infinite bandwidth, far-field scatterings). In addition, we implemented the proposed method experimentally on an ultrasonic system and imaged a sparse medium composed of five wires in water. The spatial resolution was better than the one obtained with conventional DAS and DAS with \(\ell_{1}\) denoising (\(\hat{\gamma}_{\text{3 PW-DAS}}\) and \(\hat{\gamma}_{\text{3 PW-DAS-L1}}\)). This confirms that the model is accurate, even on real experimental data. However, since the model is based on free-field propagation within nonattenuating media, it may not be accurate for media with heterogeneous speed of sound and attenuation. In the proposed method, three planes waves were emitted simultaneously. More plane waves could be emitted simultaneously to increase the image quality and the frame rate. However, due to the crosstalk noise (Section 2.4), which can cause degradation in the image quality, it would require a higher number of orthogonal sequences. The prospectivesof this work are the following. First, the performance of the method will be evaluated in sparse mediums containing microbubbles such as in ultrasound localization microscopy [57], and in phantoms mimicking blood vessels, where the gain in frame rate may be truly beneficial. Safety measures will be quantified with a view toward in vivo application of the technique. The feasibility of using a larger number of simultaneous plane waves will be studied. Finally, regularization techniques using deep learning (see, e.g., [58]) will be studied for TRF reconstruction. Conceptualization, D.B. and D.F. ; methodology, D.B., H.L., D.F. and B.N. ; software, F.N. and D.B. ; validation, F.N., D.B., H.L. and B.N. ; formal analysis, D.B. ; investigation, F.N. and D.B. ; data curation, F.N. and D.B. ; writing--original draft preparation, all; writing--review and editing, all; visualization, F.N. and D.B. ; supervision, E.C., H.L., D.F. and B.N. ; project administration, E.C., H.L. and B.N. ; funding acquisition, E.C., H.L., D.F. and B.N. All authors have read and agreed to the published version of the manuscript.