## 2 Materials and Methods
### Experimental Implementation
#### 2.3.1 Experimental Setup  
The experimental setup presented in Figure 3 can be divided in three main parts: (1) the US acquisition system consisting of an US linear array driven by a programmable US platform, (2) the scanning system comprising two motorized stages and their motion controller, and (3) the synchronization system piloted with a programmable trigger generator. The acquisition process was fully automated. Figure 2: Successive motor positions for a rotate-translate scan with the angular sampling period \(\Delta\alpha_{2}\approx 4^{\circ}\). Positions used for reconstruction are shown with red markers. Blue markers designate positions that are necessary for the continuous mechanical scan but are discarded prior to volume reconstruction. The black crosses represent targeted endpoints for the motor trajectories. Because of the density taper, the effective angular step is \(4^{\circ}\) around \(\alpha_{ret}\) = 0 and increases with the value of \(\mid\alpha_{ret}\mid\). The bounds of the angular range \([-45^{\circ},45^{\circ}]\) are also not reached. The first four endpoints are notated A, B, C, and D. The corresponding video of the scanner in action is displayed in Video S1. The ultrasound transducer array was a 128-element linear array (L7-4, Linear Array, Philips Medical Systems, Seattle, WA, USA) with a 5-MHz center frequency and 298-\(\upmu\)m inter-element spacing. In the elevation direction, the elements of the array had an aperture of \(D=7.5\) mm and were cylindrically focused to a focal distance of \(F=25\) mm, resulting in a f-number of 3.3. Therefore, as described in Equations (1) and (2), \(\Delta\alpha_{1}\approx 8^{\circ}\), \(\Delta\alpha_{2}\approx 4^{\circ}\), and \(\Delta\alpha_{3}\approx 2^{\circ}\). Additionally, for this array, \(\lambda_{\rm c}=0.3\) mm and \(H=1\) mm with a speed of sound of 1500 ms\({}^{-1}\). This linear array was chosen because it is widely used in the ultrasound community. Acoustic coupling between the sample and the ultrasound array was obtained by immersion in a water tank. The array was driven by a programmable, 64-channel Vantage ultrasound system (Verasonics, Kirkland, WA, USA). For all the transmit events and all the receive events, only the 64 central elements of the array were used. Upon the arrival of each external trigger event, all these 64 array elements were fired to produce a \"plane wave\" emission (beam that is unfocused in the lateral direction of the array or the v-direction shown in Figure 1) [26]. Transmitted pulses were 1 cycle long at 5.2 MHz. For most acquisitions, all the elements in the transducer array were fired at the same time to produce an untilted plane wave (\(\gamma_{\rm st}=0\)) referred to as \"straight transmit\". We also implemented an emission mode in which tilted plane waves were generated with different angles, \(\gamma_{\rm st}\), for successive trigger events. The steering angles were controlled electronically by introducing a small delay between the firing of adjacent elements according to the desired inclination angle. In this mode, named \"steered transmit\", a cyclic emission sequence was used and, for each cycle, the tilted plane waves were emitted with increasing angular steering (Table 1). The steered angles were chosen to ensure uncorrelated transmitted wave vectors. Given the active  
Figure 3: Annotated picture of the experimental setup. The rotate-translate scanner is comprised of two motorized stages that scan an ultrasound (US) array. The rotation stage is mounted on the translate stage. The US array is fixed on the rotation stage with a custom-built holder. The array axis (axis along the row of its elements) was manually aligned to correspond to the rotation axis. Triggers synchronize (1) recording the motion-controlled stage positions and (2) the US acquisition with the programmable US system. The imaged sample is placed in the water tank below the array. The axes of the fixed-coordinate system (\(\mathbf{e_{x}}\), \(\mathbf{e_{y}}\), \(\mathbf{e_{z}}\)) are represented.