## Appendix A  
In Appendix A, we give details about the reconstruction algorithm. In particular, how we computed: (1) the two-way travel times between the US transducers and the center of each imaged voxel and (2) the apodization windows. The 3-D image voxel grid was defined in the fixed coordinate system (O, \(\mathbf{e_{x}}\), \(\mathbf{e_{y}}\), \(\mathbf{e_{z}}\)) presented in Figure 1b. In the xz-plane, the origin O corresponded to the rotation axis when \(l_{trans}=0\). Along the y-axis, O corresponded to the center of the active array aperture. The mobile Cartesian coordinate system (O\({}_{a}\), \(\mathbf{u}\), \(\mathbf{v}\), \(\mathbf{w}\)) was attached to the transducer array (Figure 1a). In 3-D, the origin O\({}_{a}\) was set to the center of the array, and the elements of the array were uniformly distributed along the v-axis. For computation of the ultrasound pulse travel times, transformation from (O\({}_{a}\),\(\mathbf{u}\),\(\mathbf{v}\),\(\mathbf{w}\)) to (O, \(\mathbf{e_{x}}\), \(\mathbf{e_{y}}\), \(\mathbf{e_{z}}\)) was performed in two steps. First, the coordinates of the basis (\(\mathbf{u}\),\(\mathbf{v}\),\(\mathbf{w}\)) were computed by taking into account potential misalignments of \(\mathbf{v}\) with \(\mathbf{e_{y}}\) and the rotation angle \(\alpha_{rot}\). Then, the coordinates of O\({}_{a}\) in (O, \(\mathbf{e_{x}}\), \(\mathbf{e_{y}}\), \(\mathbf{e_{z}}\)) were computed given a possible mismatch between O\({}_{a}\) and O when \(l_{trans}=0\) and \(\alpha_{rot}=0\), the rotation angle \(\alpha_{rot}\), and the signed translation length \(l_{trans}\). Misalignment parameters were determined with a calibration process: a known object (thread) was imaged and parameters were adjusted to maximize the amplitude of the reconstructed image. For the pathway from the US array to the voxel, we assumed that cylindrical waves were emitted. The distance between the emitting axis and the center P of one voxel was computed and divided by the speed of sound to get the travel time. The location of the emitting axis depended on the tomographic position and the steering angle \(\gamma_{st}\). For the straight transmit (\(\gamma_{st}=0\)), the emitting axis was the v-axis. For \(\gamma_{st}\
eq 0\), the first firing element (at one edge of the emitting aperture) stayed on the emitting axis, but the direction of the axis was rotated by \(\gamma_{st}\) around \(\mathbf{u}\). The u-coordinate of P in the mobile coordinate system was computed and compared with \(d_{u}\) (Figure 1b) to implement the apodization detailed in Section 2.2. This apodization accounted for the focusing of the transducer elements in the elevation direction (u-axis) by limiting the contribution of the signals to a slice of width \(2\times d_{u}\), which was taken to be equal to the pulse-echo \(-12\) dB full width of the ultrasound beam [21] (Section 2.2). This slice width was then related to the angular aperture \(\beta\) by the equation \(d_{u}=0.3\times\lambda_{c}/\tan(\beta)\). For the pathway from the voxel to the US array, we assumed that the echo was a spherical wave emitted by P and detected by point detectors located at the center of each element of the array. For each tomographic position, a dynamic aperture approach was implemented to keep the angle of acceptance in the lateral direction (along the v-axis) as constant as possible and thereby obtain a homogenous resolution in the \(\mathrm{e_{y}}\)-direction for all the points in the 3-D image [47]. A lateral f-number of 1.3 was considered. It corresponded to the ratio between the elevation focus of the array \(F\) and the reception aperture (64 elements) and to a compromise between the resolution value and its homogeneity. This dynamic aperture was apodized with a Hamming window to avoid strong side lobes.