## 2 Materials and Methods
### Element Array-Optimization Experiment  
The experiments of optimizing a 128-element array were conducted. The US parameters used in the experiments are presented in Table 1. In the initial state of the optimization process, the side lengths of the elements in the \(x\)- and \(y\)-directions were set to 0.5 mm. The initial arrangement was a grid of 16 elements in the \(x\)-direction and 8 elements in the \(y\)-direction, with an interval of 2.0 mm between adjacent elements. The region in which the elements could exist was \(-20\) to \(20\) mm in the \(x\)-direction and \(-10\) to \(10\) mm in the \(y\)-direction. The side lengths of the elements ranged from \(0.2\) mm to \(5\) mm.  
The imaging region was defined as the \(xz\) plane, with the \(x\) coordinates ranging from \(-15\) to \(15\) mm and the \(z\) coordinates ranging from \(10\) to \(40\) mm. The number of focus points \(n_{focus}\) was set to \(1024\), and their arrangement was randomized within the imaging region in each iteration. To mitigate potential the bias in the arrangement, the imaging region was divided into \(32\times 32\) subregions, with a single point randomly allocated to each section. The parallel shift distance in the \(y\)-direction was randomly assigned within the range of \(\pm 5\) mm. The rotation angle of the element array around the \(z\)-axis was randomly assigned within the range of \(\pm 30^{\circ}\). The number of nearby points \(n_{near}\) was set to \(8\). These points were arranged at equal intervals on an ellipse centered on the focus points in the plane with the same \(z\) coordinate. The major axis of the ellipse was five wavelengths in the \(y\)-direction, and the minor axis was one wavelength in the \(x\)-direction. The reason for the discrepancy between the \(x\)- and \(y\)-directions is that the elevation focus level required for US imaging is typically lower than the lateral fucus. The number of peripheral points \(n_{periph}\) was set to \(1000\). The area in which these points were placed has a z-coordinate in the range of \(\pm 5\) mm from the focal point, and an \(xy\)-coordinate that is equal to the area in which the element can exist. However, the inside of the ellipse, on which the nearby points were located, was excluded. A schematic of these arrangements is provided in Figure 1.  
The parameters used for the optimization process are presented in Table 2. The optimizations were performed with variety of \(\beta_{1}\) and \(\beta_{2}\) combinations to identify the optimal combination. These parameters were determined empirically. The optimization results were evaluated through the experiment of the US imaging of the simulated wires. The details of the US imaging experiment and performance evaluation metrics are described in Sections 2.5 and 2.6, respectively. The optimization processes were implemented using  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Number of cycles \(\left(n_{cycles}\right)\) & \(5\) \\ Center frequency \(\left(f_{c}\right)\) & \(5\) MHz \\ Sound speed \(\left(\varepsilon\right)\) & \(1500\) m\(\cdot\) s\({}^{-1}\) \\ Attenuation Coefficient \(\left(\alpha\right)\) & \(0.5\) dB \(\cdot\) cm\({}^{-1}\cdot\) MHz\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Ultrasound parameters used in the element array-optimization experiments.  
Figure 1: The arrangement of the focus point, nearby points, and peripheral points. The focus point for evaluating the main lobe level was randomly placed within the imaging region. The nearby points for evaluating the main lobe width were arranged at equal intervals on an ellipse centered on the focus point. The peripheral points for evaluating the side lobe level were randomly distributed within the region where the depth is close to the focus point.  
Python 3.6.9 and TensorFlow 2.0.4 [20]. The time required for optimization was about 60 min when using CPU: Intel Xeon Gold 6230 (Santa Clara, CA, USA), and GPU: NVIDIA GeForce RTX 2080 Ti (Santa Clara, CA, USA).