## 2 Materials and Methods
### Imaging Performance Evaluation  
First, to evaluate the robustness of the optimized element array against positional errors, US imaging of the wire and cyst scatterers was performed under conditions where the positional error ranged from \(-5\) mm to 5 mm in 1 mm increments. Next, to evaluate the robustness against angular errors, imaging of these scatterers was performed under conditions where the angular error ranged from \(-30^{\circ}\) to \(30^{\circ}\) in \(10^{\circ}\) increments. Then, to evaluate the imaging performance when both positional and angular errors are present, US imaging of wire, cyst, and blood vessel scatterers was performed with the element arrays under nine different conditions: three types of position errors (\(-3\) mm, 0 mm, 3 mm) and three types of angle errors (\(-20^{\circ}\), \(0^{\circ}\), \(20^{\circ}\)). The imaging performance was evaluated by the quality of the obtained US images. To compare the imaging performance of the optimized array with conventional arrays, the same evaluations were conducted with the 1D linear and the 2D matrix arrays. In the linear array, 128 elements with a side lengths of 0.2 mm in the \(x\)-direction and 5 mm in the \(y\)-direction were arranged with an interval of 0.24 mm. In the matrix array, elements with a side lengths of 0.5 mm in both the \(x\)- and \(y\)-directions were arranged with an interval of 2.0 mm. The number of elements in the \(x\)- and \(y\)-directions were 16 and 8, respectively, resulting in a total of 128. The image quality evaluation metrics for the US images of the wire, cyst, and blood vessel are described below. The image quality of the wire's US images was evaluated using the SNR of the wires, as outlined below. The initial step involved the division of the US image into nine 10 mm \(\times\) 10 mm regions, with each region centered on each wire. Subsequently, the pixel of the wire with the greatest brightness within each region was identified, and the region at a distance of at least one wavelength from the aforementioned pixel was defined as the background region. The SNR of each wire was calculated using the following equation [24]:  
\[SNR=20\,log_{10}\frac{s_{max}}{\sigma_{noise}} \tag{21}\]  
where \(s_{max}\) represents the greatest brightness value, and \(\sigma_{noise}\) represents the standard deviation of the brightnesses in the background region. The image quality of the cysts was evaluated based on the contrast-to-noise ratio (CNR) between the inside and outside regions of the cysts using the following procedure. First, the US image was divided into four 15 mm \(\times\) 15 mm regions, with each region centered on each cyst. Subsequently, each image region was subdivided into inside and outside area of a 5 mm radius circle. The CNR of the cyst was calculated using the following equation [25]:  
\[CNR=20\,log_{10}\frac{\mu_{out}-\mu_{in}}{\sqrt{\sigma_{out}^{2}+\sigma_{in}^{ 2}}} \tag{22}\]  
where \(\mu_{in}\) and \(\mu_{out}\) are the average brightnesses of the inside and outside area of the circle, respectively. \(\sigma_{in}\) and \(\sigma_{out}\) are the standard deviations of the brightnesses in the inside and outside regions of the circle, respectively. The image quality of the blood vessel was evaluated based on the error in the measurement of the vessel inner diameter. The inner diameter of the vessel was measured along five lines at \(x=-10,-5,\ 0,\ 5,\text{and}\ 10\ \ \text{mm}\) in the US image using the following procedure [26]. First, a Gaussian filter with \(\sigma=(2,\ 30)\) pixels was applied to the US image in order to prevent misdetection of the vessel walls due to speckle noise. Subsequently, the \(z\)-direction gradient was calculated for each measurement line. Then, within the range of \(z=20\) to 25 mm, the pixel exhibiting the minimum gradient was identified as the potential candidate for the upper inner vessel wall, and within the range of \(z=25\) to 30 mm, the pixel exhibiting the maximum gradient was identified as the potential candidate for the lower inner vessel wall.