## 2 Methods
The coherent angular compounding was performed by coherently summing  
Figure 2: Graphical illustration of 3-D phantom in (**a**) 3-D coordinate, (**b**) azimuthal (\(x\)-\(z\)), (**c**) elevational (\(y\)-\(z\)), and (**d**) diagonal (\(xy\)-\(z\)) planes. the received signals from several plane waves transmitted at different angles (i.e., varying \(\theta_{x}\) and \(\theta_{y}\)). For image reconstruction, the time delay was calculated by using the time delays of both transmit and receive to coherently align the received signals before summation. Then, the plane waves were rotated around the point placed behind the transducer as shown in Figure 3[21; 22]. Thus, a transmit delay offset (\(t_{\text{offset}}\)) was added to the time delay between the array and every imaging point. According to previous studies [20; 21; 22; 23], the coherent summation of the received signals from plane wave transmits at different angles can generate an equal performance of a focal point of the focused transmits. Therefore, coherent angular compounding can achieve DTxF by creating foci at every imaging point with the unfocused transmit beams. The optimum angular increment (\(\Delta\theta=\lambda/D\)) and the number of transmit angles (\(D/\lambda\)-\(F\#\)) for the angular compounding were calculated to cover the imaging angle to avoid grating- or side-lobe artifacts [16]. Then, using the optimum parameters, the simulations for the unfocused FT-XR employed a 31 \(\times\) 31 transmit angles with a step increment of 1.8\({}^{\circ}\) in both azimuthal (\(\theta_{x}\)) and elevational (\(\theta_{y}\)) directions. When \(\kappa\times\kappa\) plane waves are transmitted, frame rate decreases by \(\kappa^{2}\) times. Since the X-shaped array could degrade the image quality in the diagonal plane (i.e., \(\alpha\
eq 0\) and \(\beta\
eq 0\) in Equation (2)), plane waves tilted in the diagonal direction (i.e., \(\theta_{x}=\theta_{y}\)) were selected to investigate the enhancement of the image quality in the diagonal plane using reduced number of transmits. In addition to the experiment with the optimum parameters for coherent angular compounding, the tilted plane waves at 11 different angles with a step increment of 5\({}^{\circ}\) in the diagonal direction (\(\theta_{x}=\theta_{y}\)) were employed. Using the coherent angular compounding, 90 \(\times\) 90 scan lines were reconstructed from the acquired data. Table 1 shows a quantitative summary including numbers of transmit and receive elements, number of transmits per frame, frame rate, and signal-to-noise ratios (SNR) gain. In the current study, SNR gains relative to a single element's SNR were evaluated. Thus, the SNR gains for the unfocused transmit were \(\sqrt{N_{t}N_{r}N_{beam}}\), where \(N_{t}\) is the number of transmit elements, \(N_{r}\) the number of receive elements, and \(N_{beam}\) the number of beams for a frame. To predict the noise susceptibility of an X-shaped array, relative imaging SNR were estimated for unfocused FT-FR and FT-XR while the number of elements (\(N\times N\)) of a 2-D array increased from 8 \(\times\) 8 to 64 \(\times\) 64. To numerically compare with a conventional one-dimensional (1-D) linear array, a SNR relative to 100 elements linear array was calculated by subtracting the SNR gain of a 100 elements array (i.e., 100\(\sqrt{100}\)) from the SNR gains of the FT-FR and the FT-XR. The 1-D linear array had the same element size and pitch as the 2-D arrays did, and all elements were used to transmit as well as to receive with focusing. The relative imaging SNRs of the FT-XR in conjunction with DTxF using \(N\times N\), 31 \(\times\) 31, and 11 tilted plane waves were also evaluated, of which the SNR gain was multiplied by a square root of the number of the transmitted plane waves, to explore variations in the noise susceptibility of the X-shaped array for volumetric imaging. Figure 3: Geometrical description of coherent angular compounding.