## 2 Materials and Methods
### Experimental Implementation
#### 2.3.2 Practical Implementation of the Scanning  
Three different acquisition modes were employed. The first mode was a 3-D, rotate-translate tomographic scan with straight transmit events (\(\gamma_{\rm st}\) = 0). The targeted endpoints of the motion were at \(l_{trans\_endpoints}\) (\(\alpha_{rot}\)) = \(l_{trans\_endre}\)(\(\alpha_{rot}\)) \(\pm\) L/2 and are illustrated in Figure 2 (black crosses). For a time-efficient scanning, we implemented continuous scanning rather than step-by-step scanning [27]. We chose to perform translation scans repetitively with a back-and-forth oscillatory motion. The rotation motion was performed in a single direction (increasing \(\alpha_{rot}\)). To illustrate the stage motion, we notated A, B, C, and D as the first four endpoints of the scan in Figure 2. For two successive rotation angles, \(\alpha_{rot}\) 1 and \(\alpha_{rot}\) 2, the translation endpoints were scanned in one translation direction for \(\alpha_{rot}\) 1 (from A to B for instance) and in the reverse direction for \(\alpha_{rot}\) 2 (from C to D). Thereby, we minimized the scanning time. The rotation motion starts from the second translation endpoint at \(\alpha_{rot}\) 1 (B in the example) and we aimed at reaching the second rotational position \(\alpha_{rot}\) 2 when the targeted first translational endpoint at \(\alpha_{rot}\) 2 was reached (C here). A translation motion was additionally needed between the two endpoint positions (B and C in the example). In the example, the first, positive-direction translation occurred from A to C. The rotational motion started when the translation stage reached B. When C was reached, the rotation motion and the positive-direction translation ended and the translation direction was reversed. \begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{2}{c}{**Scanning Mode**} & \multicolumn{3}{c}{**Rotate-Translate Scan**} & **Translate-Only Scan** \\ \hline \multicolumn{2}{c}{} & Straight transmit & \multicolumn{2}{c}{Sleared transmit} & \multicolumn{2}{c}{Stranglit transmit} \\ Mode \# & & 1 & 2 & 3 & 3 \\ Angular sampling period & \(\Delta_{\rm{A1}}=\beta\) & \(\Delta_{\rm{A2}}=\beta/2\) & \(\Delta_{\rm{A3}}=\beta/4\) & \(\Delta_{\rm{A2}}=\beta/2\) & 0 \\ \multicolumn{2}{c}{} & \(\Delta^{\circ}\) & 8 & 4 & 2 & 4 & 0 \\ Number of angular positions 1 & 6 & 12 & 23 & 12 & 1 \\ Steering angles \(\gamma_{\rm{A}}\) (\({}^{\circ}\)) & 0 & 0 & 0 & \(-4,-2\cdot 0;2\cdot 4\) & 0 \\ Acquisition rate (Hz)\({}^{2}\) & 50 & 50 & 50 & 250 & 50 \\ Number of tomographic positions 2 & 267 & 508 & 947 & 2533 & 244 \\ Acquisition duration (\(\delta\)) & 5.3 & 10.2 & 19 & 10.1 & 4.9 \\ Reconstruction duration (\(\delta\)) 3 & 11.2 & 20.9 & 952 & 14.2 \\ \hline \end{tabular}  
* The number of angular positions results from the angular sampling period \(\Delta_{\rm{A}}\) and the density slope. The acquisition rate corresponds to the pulse repetition frequency (PR) of the trigger generation, and the number of tomographic positions to the number of generated trigger pulses. 3 The reconstruction durations were measured for the largest impact volume \(\Delta\kappa\times\alpha\times\alpha\times\alpha\times\beta\times 30\)\(\times\)\(19\times 30\) mm\({}^{3}\) and a period size of \(\alpha\times\beta\)\(\times\)\(p\)\(\approx\)\(p\)\(\approx\)\(22\times 144\)\(\times 27\) mm\({}^{3}\), resulting in total of \(417\times 133\times 417\)\(\approx\)\(22\) million voxels. The volumetric image reconstruction was performed using a Graphics Processing Unit (GPU) NVIDIA Gefproc RTX 2306 SUPER. For the modes 1 and 2, none tomographic positions were discarded for the reconstruction (see Appendix A).  
\end{table} 
Table 1: Parameters of the different acquisition modes for a translation range \(L=30\) mm. The trajectories of positioning stages had trapezoidal or triangular velocity profiles. Therefore, for each movement, the following parameters had to be set: the endpoints, the targeted velocity, and the acceleration. The deceleration was set equal to the acceleration.