# Simultaneous Coded Plane-Wave Imaging Using an Advanced Ultrasound Forward Model
## 2 Materials and Methods
### Derivation of the Model for Plane-Wave Imaging  
PWs are unfocused waves emitted using an array composed of \(N_{\text{el}}\) separate transducers equally spaced by a pitch \(p\). To emit a PW with a tilt angle \(\theta^{(k)}\), the emission delays for each element \(j\) must be computed using the following expression:  
\[t_{j}=(j-1)p\frac{\sin(\theta^{(k)})}{c}, \tag{9}\]  
where \(j=[1...N_{\text{el}}]\), \(p\) is the pitch, and \((k)\) is the \(k^{\text{th}}\) wave. From Equations (8) and (A3) in Appendix B, the PW equivalent of Equation (8) is given by:  
\[\begin{split}\mathbf{y}_{i}=w_{i}\mathbf{E}\Bigg{[}\sum\limits_{ j=1}^{N_{\text{el}}}w_{j}\mathbf{C}_{ji}\Bigg{]}\boldsymbol{\gamma}=w_{i} \mathbf{EC}_{i}\boldsymbol{\gamma}=\boldsymbol{\Lambda}_{i}\boldsymbol{\gamma},\\ \text{with: }\boldsymbol{\Lambda}_{i}=w_{i}\mathbf{EC}_{ji} \end{split} \tag{10}\]  
where \(w_{i}\) are the apodization weights at reception. Considering a uniform spatial apodization \(w_{j}=w_{i}=1\), \(\forall(j,i)\), we can use Equations (10) and (A6) from Appendix B to write the received signal at the \(i^{\text{th}}\) element of the array as:  
\[\begin{split}\mathbf{y}_{i}^{(k)}=\mathbf{E}^{(k)}\Bigg{[}\sum \limits_{j=1}^{N_{\text{el}}}\mathbf{C}_{ji}^{(k)}\Bigg{]}\boldsymbol{\gamma}= \mathbf{E}^{(k)}\mathbf{C}_{i}^{(k)}\boldsymbol{\gamma}=\boldsymbol{\Lambda} _{i}^{(k)}\boldsymbol{\gamma},\\ \text{with: }\begin{cases}\mathbf{C}_{i}^{(k)}=\sum\limits_{j=1}^{N_{ \text{el}}}\mathbf{C}_{ji}^{(k)},\\ \boldsymbol{\Lambda}_{i}^{(k)}=\mathbf{E}^{(k)}\mathbf{C}_{i}^{(k)}.\end{cases} \end{split} \tag{11}\]  
where \(a^{(k)}(t)\) is the excitation signal for the \(k^{\text{th}}\) PW.  
As shown in Section 2.2, \(\mathbf{E}^{(k)}\) is a square matrix of size \(N_{y}\times N_{y}\), where \(N_{y}\) is the length of the recorded signals \(\mathbf{y}_{i}\). The size of the matrix \(\mathbf{C}_{i}^{(k)}\) is \(N_{y}\times N_{x}N_{x}\). For a single pair of planar transducers, a good approximation of the recorded signal is achieved when the medium is sampled along the z axis with a step \(\Delta z=c/f_{s}\), which implies that \(N_{z}=N_{y}\), and thus \(N_{y}<<N_{z}N_{x}\). The linear system in Equation (11) contains a greater number of unknowns than equations, which yields an ill-posed problem. In order to move closer to a well-posed problem, a simple approach is to increase the number of equations in Equation (11). To achieve this, the \(N_{\text{el}}\) observation vectors \(\mathbf{y}_{i}^{(k)}\) obtained for each PW with tilt angle \(\theta^{(k)}\) are concatenated. Therefore, similar to in Equation (11), \(\mathbf{\gamma}\) is the same \(\forall i\in 1...N_{\text{el}}\), and the new concatenated kernel \(\mathbf{\Lambda^{(k)}}\) will have the same number of columns as each \(\mathbf{\Lambda_{i}^{(k)}}\); however, it will contain \(N_{\text{el}}\) times more rows. The new concatenated system can be written as follows:  
\[\mathbf{y}^{(k)}=\mathbf{\Lambda^{(k)}}\mathbf{\gamma}, \tag{12}\] \[\text{with: }\left\{\begin{aligned} \mathbf{y}^{(k)}& =\big{[}\mathbf{y}_{1}^{(k)}\|\mathbf{y}_{2}^{(k)}\|\cdots\|\mathbf{y}_{N_{ \text{el}}}^{(k)}\big{]},\\ \mathbf{\Lambda}^{(k)}&=\big{[}\mathbf{\Lambda_{1}^{(k)}}\| \mathbf{\Lambda_{2}^{(k)}}\|\cdots\|\mathbf{\Lambda_{N_{\text{el}}}^{(k)}}\big{]}.\end{aligned}\right.\]  
where \(\|\) represents the vector concatenation along the rows. \(\mathbf{y}^{(k)}\) and \(\mathbf{\Lambda^{(k)}}\) are of size \(N_{\text{el}}N_{y}\times 1\) and \(N_{\text{el}}N_{y}\times N_{z}N_{x}\), respectively. As for the discretization of the medium \(\Omega\), we have \(\Delta x<p\) (i.e., the spatial resolution along the x axis is smaller than the pitch of the linear array), which results in \(N_{x}>N_{\text{el}}\). In other words, the concatenated kernel \(\mathbf{\Lambda}^{(k)}\) is still underdetermined.