## 2 Materials and Methods
### Ultrasound Imaging Simulation  
The performances of the optimized element arrays were evaluated through US imaging in simulations. The simulations were conducted under the following assumptions: far-field, linear propagation, uniform sound speed and attenuation, and Rayleigh scattering. The US parameters used in the simulations were identical to those presented in Table 1. Three different scatterers were employed to represent wire, cyst, and blood vessel (Figure 2). All scatterers were constructed from small scattering points. The details of the scatterers are described in the following paragraphs. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Optimization algorithm & Adam [21] \\ Initial learning rate & Element coordinates : \(10^{-3}\) \\ Element side lengths : \(10^{-5}\) \\ Exponential learning decay rate & 0.9 every 10 steps \\ Number of iterations & 100 \\ \(\beta_{1}=10^{-3.0},\ 10^{-2.5},\ \dots,10^{0.0}\) \\ Cost function weights & \(\beta_{2}=10^{-1.0},\ 10^{-0.5},\ \dots,10^{2.0}\) \\ \(\beta_{3}=10^{-3}\) \\ Minimum element distance (\(D_{min}\)) & Half wavelength (0.375 mm) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters for the element array optimization. Figure 2: Simulated scatterer distributions used in the ultrasound imaging experiments. (**a**) Simulated wire scatterers. Nine wires, each 5 mm in length, are arranged with 10 mm intervals. Scattering points are placed on the wires at \(1/10\) wavelength intervals. In regions other than the wires, scattering points are randomly placed at a density of one per cubic volume with sides of one wavelength. (**b**) Simulated cyst scatterers. Four cysts, each with a diameter of 10 mm, are arranged with 15 mm intervals. There are no scattering points inside the cysts, while scattering points are randomly placed outside the cysts at a density of one per cubic volume with sides of one wavelength. (**c**) Simulated blood  
vessel scatterer. A blood vessel with an inner diameter of 6 mm and an outer diameter of 7 mm is positioned at a depth of 25 mm. There are no scattering points inside the blood vessel, while scattering points are randomly placed in the blood vessel wall and outside the blood vessel at densities of four and one per cubic volume with sides of one wavelength, respectively. (**d-f**) Ground truth maps of the US images for the wire, cyst, and blood vessel scatterers, respectively. In the scattering points distribution of the wire scatterer, nine wires were positioned at the following coordinates: \(x=-10,\ 0,\ 10\) mm and \(z=15,\ 25,\ 35\) mm, with 10 mm intervals between each wire. The wires were oriented along the \(y\)-axis, centered in the \(xz\)-plane, and had a length of 5 mm. The scattering points were positioned on the wires at intervals corresponding to one-tenth of a wavelength. Furthermore, to incorporate noise from regions external to the wires, scattering points were randomly placed at a density of one point per wavelength cube within the region \(x=-20\) to 20 mm, \(y=-20\) to 20 mm, \(z=0\) to 50 mm. In the scattering points distribution of the cyst scatterer, four cysts with a radius of 5 mm were placed with center coordinates of \(x=-7.5,\ 7.5\) mm, \(y=0\) mm, \(z=17.5\), 32.5 mm. First, the scattering points were randomly placed at a density of one point per wavelength cube in the region \(x=-20\) to 20 mm, \(y=-20\) to 20 mm, \(z=0\) to 50 mm. Subsequently, the points within the cysts were removed. The blood vessel scatterer was used to evaluate the imaging performance for targets resembling human tissues. In the scattering points distribution, a model with an inner and outer diameters of 6 and 7 mm, respectively, was positioned at the center coordinate of \(x=0\) mm, \(y=0\) mm, \(z=25\) mm. The orientation of the blood vessel was alligned with the \(x\)-axis.