## 2 Theory: Coherent Multi-Transducer Ultrasound Imaging  
A CoMTUS system consists of \(N\) synchronized US arrays that take turns transmitting PWs into a common FOV. Only one individual array transmits at each time while all \(N\) arrays simultaneously receive. The subwavelength localization accuracy required to coherently combine RF data from multiple probes is achieved without the use of any external tracking device. The method is briefly summarized in this section, but more details can be found in [9; 10; 13]. In this study, we consider a CoMTUS system formed by \(N=2\) identical linear arrays that lie in the same elevational plane (\(y=0\)) and are angled relative to each other so as to share part of the FOV, as shown in Figure 1. In such a configuration, the following nomenclature is used. For a sequence in which array \(i\) transmits and array \(j\) receives, the RF data received on channel \(h\) of array \(j\) at time \(t\) are named \(T_{i}R_{j}(h,t)\). The resulting image and all transducer coordinates are defined in a global coordinate system arbitrarily located in space \((\hat{x}_{0},\hat{z}_{0})\). For a given linear array \(i\), a local coordinate system \((\hat{x}_{i},\hat{z}_{i})\) is defined at the centre of the array surface with the \(\hat{z}\) direction orthogonal to the transducer surface and directed away from array \(i\). The position and orientation of array \(i\) are then characterized in the global coordinate system with three parameters that define a translation vector \(\mathbf{r}_{i}\) and a rotation angle \(\theta_{i}\)[14]. Standard delay and sum beamforming for PW imaging [15] can be extended to the present multi-transducer setup taking into account the full path length between the transmit array and the receive elements. Considering that array \(i\) transmits a PW at certain angle \(\alpha\), the image point to be beamformed located at \(Q_{k}\) and described by the vector \(\mathbf{r}_{k}\) can be computed from the echoes received at transducer \(j\) as:  
\[s_{i,j}(Q_{k};\alpha)=\sum_{h=1}^{H}T_{i}R_{j}\left(h,t_{i,j,h}(Q_{k};\alpha) \right)=\sum_{h=1}^{H}T_{i}R_{j}\left(h,\frac{D_{i,j,h}(Q_{k};\alpha)}{c}\right) \tag{1}\]  
where \(H\) is the total number of elements in the array, \(c\) is the average SOS of the propagation medium and \(t_{i,j,h}(Q_{k};\alpha)\) is the time-of-flight of a PW at angle \(\alpha\) transmitted by array \(i\), backscattered by the point \(Q_{k}\) and received by the element \(h\) of array \(j\). The total distance \(D_{i,j,h}(Q_{k};\alpha)\) travelled by the wave is defined by,  
\[\mathrm{D}_{i,j,h}(Q_{k};\alpha)=d(L_{i}(\alpha),\mathbf{r}_{k})+\|\mathbf{r} _{j,h}-\mathbf{r}_{k}\| \tag{2}\]  
where \(d(\mathbf{r}_{k},L_{i}(\alpha))\) is the distance from \(\mathbf{r}_{k}\) to the line \(L_{i}(\alpha)\) that defines the transmitted PW and \(\|\mathbf{r}_{j,h}-\mathbf{r}_{k}\|\) is the Euclidean distance between \(\mathbf{r}_{k}\) and the receive element \(h\) of array \(j\). These distances are represented in Figure 1.  
Figure 1: Coherent multi-transducer ultrasound (CoMTUS) beamforming scheme. The global and local coordinates of the arrays are shown. In this example, array T1 transmits a plane wave (PW) at angle \(\alpha\), and array T2 receives the resulting echoes backscattered from the scatter \(Q_{k}\).