## 2 Materials and Methods
### Imaging Geometry and Reconstruction  
Previous preliminary studies have shown that placing the probe outside the center of the imaging system allows us to expand the imaging area and leaves space in the imaging device to install the holder responsible for keeping the breast in the same position during the examination procedure [36]. The widening of the imaging area is important in the context of smaller-size ultrasonic probes or those without the capabilities to create a sectoral image. However, moving the probe outside the vertical axis has limitations. Even if the sectoral view is used, the amount by which the probe can be moved depends on the width of the probe, the maximum angle of scanning, and the distance of the breast from the probe surface. If the situation occurs when the object is too close to the ultrasound probe surface and its part is not visible on the reconstructed image, the probe can be adjusted by an angle to cover this area. This situation is shown in Figure 6.  
Figure 5: Breast phantoms used in measurements: (**a**) ATS Laboratories Model BB-1, (**b**) CSDS Blue Phantom Breast Masses.  
Figure 6: Limitation of the probe offset and the probe adjustment to cover object area.  
When taking into account the above facts, the placement of the probes has been taken arbitrarily, and the same for each probe, to present the results for different probes in the situation where the mounting point of the probe is fixed. The positions of the center of the probe and its angle have been set to the following pairs: (3 cm, 0\({}^{\circ}\)), (4 cm, 10\({}^{\circ}\)), (5 cm, 20\({}^{\circ}\)). Later in the text, these positions will be marked as (3, 0), (4, 10), and (5, 20) for simplicity. The number of scans on the circle has been set to 5\({}^{\circ}\), resulting in 72 scans available for reconstruction. In the present study, due to the examination of submerged breast phantoms in a water tank using a robotic arm, and to simplify the scanning procedure, ultrasound probes were placed inversely, from the water surface.  
Reconstruction of the 3D image using the MACUI method is a straightforward operation. As the probe is placed radially and rotates around the vertical axis, the polar coordinates of the individual scans can be simply converted to cartesian coordinates. However, a simple concatenation of the opposite images may exhibit artifacts in the center of the image, especially when the sectoral scan is used. In such a situation, the ultrasound beam reaches the structure through different acoustic paths and incidents with it at different angles. The case has been studied by the authors and different compounding methods have been evaluated to minimize unwanted effects [49]. The proposed weighted average method has been used in this work during reconstruction. Having the polar coordinates converted to cartesian, all values can be converted into the stack of images that can be subsequently reconstructed using software, such as 3D Slicer, or converted to DICOM format and evaluated by any software that supports DICOM, i.e., Onis Viewer. The 3D Slicer has been used to present the results. Figure 7 presents an example 3D view of the reconstructed structure from the data collected for this study.