## 2 Materials and Methods  
### Presentation of the Scanning Scheme  
The rotate-translate scanning scheme is diagrammed in 2-D (in the axial-elevation plane) for one element of the linear US array (Figure 1). The extension to 3-D is direct when the lateral dimension of the array (i.e., axis along the row of active array elements) is taken into account. We considered here a linear array with a cylindrical lens, which is typical of clinical linear US arrays for 2-D ultrasonography. The focusing power of the lens is usually weak, which ensures a large depth of field but a relatively poor elevation resolution. These depth-of-field and resolution performances are directly linked to the small angular aperture 2.8 of the element (Figure 1a) or the large f-number in the elevation direction. The f-number is equal to \(F/D\) where \(F\) and \(D\) are the focal distance and the width of the array element, respectively (Figure 1a). To benefit from the large depth of field but with a significant improvement in the elevation spatial resolution, we developed a synthetic aperture approach. The synthesis of the aperture increases the angular aperture for each array element over an extended imaged area thanks to a rotate-translate scanning scheme. The rotation increases the angular aperture and the 1-D translation ensures coverage of the imaged zone. To obtain a single-side access, the rotation axis was set to pass through the center of the transducer face O\({}_{\text{a}}\) (Figure 1a) and to be perpendicular to the axial direction of the transducer (w-axis) [20]. This axial direction is named transducer axis. The translation axis is perpendicular to both the rotation axis and the transducer axis when the rotation angle \(\alpha_{rot}\) equals zero (Figure 1b). Translation across the imaged area is performed at a constant angle, \(\alpha_{rot}\), and then repeated for different angles of rotation (Figure 1c). All the translation scans had the same length \(L\). To benefit from the full depth of field around the focus, the imaged area was centered on the focal distance \(F\) in the e\({}_{z}\)-direction (Figure 1c). As a consequence, the translational scan at each angle, \(\alpha_{rot}\), was centered on the point \(l_{trans\_{center}}(\alpha_{rot})=-F\,\times\,\tan(\alpha_{rot})\) in the e\({}_{x}\)-direction (Figure 1c). To have a constant step \(\Delta l\) between successive translational positions of the transducer axis, the translational scan at \(\alpha_{rot}\) was divided into tomographic positions separated by \(\Delta l/\cos(\alpha_{rot})\). The ultrasound pulse-echo signals were recorded for each tomographic position of the transducer element. In this configuration and with an adapted \(\Delta l\) (see Section 2.2), each point of the imaged area was probed with all the different orientations of the transducer. Therefore, for each pixel of the sampled area, the ultrasound signals from the different positions could be combined coherently during the reconstruction process to build a larger synthetic aperture. This process also corresponds to spatial coherent compound imaging in the elevation direction. Figure 1: Schematic diagram of the scanning scheme in 2-D. (**a**) One weakly focused transducer array element of width \(D\) and focal distance \(F\). \(H\) is the pulse-echo full-width half-maximum (FWHM) of the ultrasound beam. The angular full aperture is 2.\(\beta\). The angle bisector is named transducer axis. A Cartesian coordinate system is attached to the transducer element face: **u** corresponds to the elevation direction and **w** to the axial direction. For a linear array, the long axis along the row of its active array elements (lateral direction) is the **v**-axis. The origin O\({}_{\text{a}}\) is the center of the transducer element face; (**b**) the transducer element at one tomographic position.