## 2 Materials and Methods
### Total Focusing Method
#### 2.2.1 Matrix Formulation  
For brevity of notation we now limit ourselves to reconstructions which are performed on an equispaced grid, i.e., to \(\mathbf{r}\ =\ \mathbf{r}(n_{z},n_{x})\ =\ [n_{z}d_{z}+z_{0},n_{x}d_{x}+x_{0}]^{T}\) where \(n_{z}\) and \(n_{x}\) are integers in the half-open intervals \([0,N_{z}]\) and \([0,N_{x}]\), respectively, \(d_{z}\) and \(d_{x}\) denote the sampling intervals and \(z_{0}\) and \(x_{0}\) a start point in the according directions. Furthermore, we denote the number of transducer elements as \(N_{el}\) and, in the case of using fewer transmit cycles, the number of transmit cycles \(N_{tx}\). The A-scans are considered to be time-discrete equivalents of the continuous definition used before, holding \(N_{t}\) samples acquired at a sampling frequency \(f_{S}\).  
Equation (2) can be rewritten as matrix-vector product [8; 13; 14]  
\[\mathbf{i}_{tfm}\ =\ |\mathbf{Mg}|, \tag{3}\]  
where \(\mathbf{g}\ =\ \left[\mathbf{g}_{0,0}^{(a)^{T}},\mathbf{g}_{0,1}^{(a)^{T}}, \ldots,\mathbf{g}_{0,N_{el}-1}^{(a)^{T}},\mathbf{g}_{1,0}^{(a)^{T}},\ldots, \mathbf{g}_{N_{tx}-1,N_{el}-1}^{(a)^{T}}\right]^{T}\in\mathbb{C}^{N_{tx}N_{el}N _{t}}\) are the stacked analytic representations of the measurement data with \(\mathbf{g}_{m_{tx},m_{rx}}^{(a)}\in\mathbb{R}^{N_{t}}\) being a single A-scan, \(\mathbf{i}_{tfm}\ =\ \left[I(\left[z_{0},x_{0}\right]^{T}),I(\left[z_{0}+d_{z},x_{0} \right]^{T}),\ldots,I(\left[z_{0},x_{0}+d_{x}\right]^{T}),\ldots\right]^{T} \in\mathbb{R}^{N_{x}N_{x}}\) is the vectorized image on the equispaced grid, \(\mathbf{M}\in\mathbb{R}^{N_{x}N_{x}N_{tx}N_{t}}\) is the model matrix, and \(|.|\) denotes the elementwise absolute value.  
The matrix is constructed from blocks \(\mathbf{M}_{(n_{z},n_{z}),(m_{tx},m_{rx})}\in\mathbb{R}^{1,N_{t}}\) as  
\[\mathbf{M}\ =\ \left[\begin{array}{cccccc}\mathbf{M}_{(0,0),(0,0)}&\mathbf{M}_{ (0,0),(0,1)}&\cdots&\mathbf{M}_{(0,0),(1,0)}&\cdots\mathbf{M}_{(0,0),(N_{tx}-1, N_{d}-1)}\\ \mathbf{M}_{(1,0),(0,0)}&\cdots&&&&\\ \cdots&&&&\\ \mathbf{M}_{(0,1),(0,0)}&&&&\\ \cdots&&&&\\ \mathbf{M}_{(N_{t}-1,N_{t}-1),(0,0)}&&&&\end{array}\right], \tag{4}\]  
with each block being defined as  
\[\left[\mathbf{M}_{(n_{z},n_{x}),(m_{tx},m_{rx})}\right]_{m_{t}}\ =\ \left\{\begin{array}{cc}\alpha_{m_{tx},m_{rx}}(\mathbf{r}(n_{z},n_{x}))& \mathrm{if}[f_{5}\tau_{m_{tx},m_{tx}}(\mathbf{r}(n_{z},n_{x}))]\ =\ m_{t}\\ 0&\mathrm{otherwise}\end{array}\right., \tag{5}\]  
where \(m_{t}\in[0,N_{t}]\) denotes an integer index, \([.]_{m}\) denotes access to the particular element of the column vector, and \(\lfloor.\rfloor\) is the floor operation. In contrast to the state of the art TFM formulations e.g., in [1,11] the times of flights are rounded to the temporal sampling grid here. Formulating the problem this way allows for an efficient precalculation of the matrix, reducing the reconstruction to a simple matrix vector multiplication [14] and opening the door for the application of algorithms like FISTA.  
The model matrix \(\mathbf{M}\) can be calculated for different transmit schemes. In the remainder of the paper, we assume that \(\mathbf{M}\) was calculated according to the given transmit pattern, furthermore, as a special case, we denote \(\mathbf{M}_{fmc}\) with \(N_{tx}\ =\ N_{el}\) as the matrix that corresponds to the reconstruction of a FMC dataset. Reducing the size of the matrix also consequently reduces the computation time of the reconstruction, which is a simple matrix vector product here.