# Simultaneous Coded Plane-Wave Imaging Using an Advanced Ultrasound Forward Model
## 2 Materials and Methods  
### Model Using Two Transducers  
Let us define \(u\) and \(v\) as punctual transducers such that \(u\) is the emitter placed in \(r_{u}=(x_{u},z_{u})\) and \(v\) the receiver placed in \(r_{v}=(x_{v},z_{v})\). The pressure variation generated by \(u\) inside a medium \(\Omega\) is tied to its electrical excitation signal \(a(t)\) through its acousto-electrical impulse response (IR) \(h_{u}(t)\)[38]. Let us assume that the pressure wave generated by \(u\) propagates under free-field conditions until it reaches a scatterer placed at \(r=(x,y)\). The pressure wave is partially reflected by the scatterer [39]; consequently, the punctual receiver \(v\) receives an electrical radiofrequency (RF) signal \(y_{uv}(t)\) that depends on its acousto-electrical IR \(h_{p}(t)\) and on the incident wave pressure [40]. The RF signal, \(y_{uv}(t)\), can be written as [41]:  
\[y_{uv}(t)=a(t)*h_{u}(t)*g_{u}(r,t)*h_{v}(t)*g_{v}(r,t)\gamma(r), \tag{1}\]  
where \(*\) is the continuous time convolution product, \(g_{u}(r,t)\) is the spatial impulse response between the medium point \(r\) and the emitter position \(r_{u}\), \(g_{v}(r,t)\) is the spatial impulse response between the medium point \(r\) and the receiver position \(r_{v}\), and \(\gamma(r)\) is the TRF value in \(r\). [PERSON]'s functions \(g_{u}(r,t)\) and \(g_{v}(r,t)\) model the free-field propagation of a pressure wave inside a homogeneous medium [39]. The pulse echo spatial impulse response \(g_{uv}(r,t)\) between the source \(u\), the medium point \(r\), and the receiver \(v\) can be expressed as:  
\[ \begin{split}& g_{uv}(r,t)=g_{u}(r,t)*g_{v}(r,t)=\\ &\frac{1}{4\pi^{2}d_{u}(r)d_{v}(r)}\delta\Big{(}t-\frac{d_{u}(r) +d_{v}(r)}{c}\Big{)}.\end{split}  \tag{2}\]  
where \(d_{u}(r)=\|r-r_{u}\|_{2}\) is the distance between \(u\) and the scatterer, \(d_{v}(r)=\|r-r_{v}\|_{2}\) is the distance between the scatterer and \(v\), and \(c\) is the sound speed in the medium. The term \(g_{uv}(t)\) is obtained by integrating the contributions of \(g_{uv}(r,t)\) on the medium \(\Omega\) such as:  
\[g_{uv}(t)=\int_{r\in\Omega}g_{uv}(r,t)\gamma(r)dr. \tag{3}\]  
Let us now consider a more realistic case where \(j\) and \(i\) are planar transducers of surface \(S_{j}\) and \(S_{i}\) such that \(j\) is an emitter and \(i\) a receiver. The transducers are of height \(h\) (off-axis) and width \(w\) (along the \(x\) axis) (Figure 2a) with IR \(h_{j}(t)\) and \(h_{i}(t)\), respectively. As shown by [41; 42], the total received signal \(y_{ji}(t)\) can be approximated by applying [PERSON]  
Figure 1: a—Schematic representation of the proposed imaging approach. \(\gamma\)—tissue reflectivity function; \(\delta^{(k)}\)—short electrical excitation of the \(k^{th}\) wave; \(g^{(k)}\)—received echoes that correspond to the emission of \(\delta^{(k)}\) (\(k^{th}\) wave); \(a^{(k)}\)—code carried by the \(k^{th}\) wave; \(\mathbf{y}^{(k)}\)—received echoes after emission of \(a^{(k)}\); \(\mathbf{y}\)—recorded echoes when \(N_{\text{pw}}\) plane waves are emitted simultaneously; \(f^{(k)}\)—optimal compression filter that corresponds to the \(k^{th}\) wave; and \(\delta^{(k)}\)—estimation of \(g^{(k)}\) obtained from \(\mathbf{y}\). Finally, an estimation \(\gamma\) of the tissue reflectivity function \(\gamma\) is obtained by solving the inverse problem. principle to both the emitting and receiving surfaces \(S_{i}\) and \(S_{i}\), in other words, subdividing the surfaces \(S_{i}\) and \(S_{j}\) in \(N\) punctual transducers. Thus, the total signal received by \(i\) is equal to the sum of all of the signals generated by the \((u_{j},v_{i})\) pairs.