## 3 Results
### Evaluation Results with Wire Scatterers  
Figure 4a shows the results of evaluating the robustness against the positional errors in the imaging of wire scatterers. The linear element array exhibited the highest SNR when the positional error was within a range of \(\pm 1\) mm. However, as the positional error increased, the SNR rapidly decreased, becoming lower than that of the matrix and optimized arrays. When comparing the matrix and optimized arrays, the optimized array showed higher SNR except when the positional error was \(\pm 5\) mm. Figure 4b shows the results of evaluating robustness against the angular errors. The SNR of the linear array was higher than the other two arrays when the angular error was within the range of \(\pm 10^{\circ}\), but it significantly decreased as the error exceeded this range. When comparing the matrix and optimized arrays, the optimized array had a higher SNR under all angular error conditions.  
Figure 5 shows the examples of US images of wire scatterers obtained with the linear, matrix, and optimized arrays under four different conditions: two types of positional errors (\(0\) mm and \(3\) mm) and two types of angular errors (\(0^{\circ}\) and \(20^{\circ}\)). US images obtained under all conditions where both positional and angular errors were present are illustrated in Figures S1-S3. In the image of the linear array with no positional or angular errors, all the  
Figure 3: Imaging performance comparison of the optimized element arrays with different cost function weights. (**a**) Representative examples of the optimized element arrays. The one on the right achieved the highest signal-to-noise ratio (SNR) in the imaging of wire scatterers. (**b**) Ultrasound images of the wire scatterers obtained with the optimized element arrays shown in (**a**) with an angle error of \(20^{\circ}\) around the \(z\)-axis and a positional error of \(3\) mm in the \(y\)-direction. The dynamic range is \(25\) dB. (**c**) Average SNRs of the nine wires, each imaged under nine different positional or angular error conditions: three angular errors (\(-20^{\circ}\), \(0^{\circ}\), \(20^{\circ}\)) and three positional errors (\(-3\) mm, \(0\) mm, \(3\) mm). The optimized array with the cost function weights of \(\beta_{1}=10^{-2}\), \(\beta_{2}=10^{0}\) achieved the highest average SNR of the wires.  
wires were clearly visualized with the highest average SNR of 33.9 dB. Some wires, however, were blurred in the lateral direction and disappeared with positional and angular errors. In the matrix array images, all wires were visualized with little variation in image quality regardless of positional and angular errors. However, the speckle noise level in the regions other than the wires was higher than that of the linear and optimized arrays, measuring 27.1 dB even under conditions with no positional or angular errors. In the optimized array images, all wires were clearly visualized regardless of positional or angular errors, with low blurring and high SNR.  
Figure 4: Evaluation of the robustness of the element arrays against positional and angular errors in the imaging of wire scatterers. (**a,b**) The average SNR of the nine wires in US images obtained under various positional and angular errors, respectively.  
Figure 5: Examples of ultrasound images of wire scatterers obtained with the linear, matrix, and optimized arrays. The left column shows the element arrays, and the top row shows the positional relationship between the imaging plane and the element arrays. The dynamic range of the ultrasound images is 25 dB. The average SNR of the wires are displayed on each image.