## 2 Methods  
Figure 1 illustrates diagrams of volumetric imaging using a 2-D array (also called \"aperture\") that receives pulse-echo signals. The previous studies [7; 8] showed that an X-shaped array (Figure 1a) can have an equivalent point spread function (PSF) as the full array (Figure 1a). Using the geometry in Figure 1b, the PSFs of the full array and X-shaped array can be expressed in the Fourier domain [1; 8] as  
\[H_{\text{Full}}(\alpha,\beta)\propto\left(\frac{\text{sin}\alpha N}{\text{sin} \alpha}\frac{\text{sin}\beta M}{\text{sin}\beta}\right) \tag{1}\]  
\[H_{\text{X-shaped}}(\alpha,\beta)\propto\left(\frac{\text{sin}((\alpha- \beta)N)}{\text{sin}(\alpha-\beta)}+\frac{\text{sin}((\alpha+\beta)M)}{\text{ sin}(\alpha+\beta)}\right) \tag{2}\]  
Figure 1: Volumetric ultrasound imaging: (**a**) full and cross-shaped sparse arrays, (**b**) geometry for the point spread functions (PSFs) of coarrays (\(\theta_{a}\) = azimuthal angle and \(\theta_{e}\) = elevational angle), and (**c**) cross-sectional imaging planes. Here, \(\alpha=\sin(\theta_{a})\cdot\pi d/\lambda\), \(\beta=\sin(\theta_{e})\cdot\pi d/\lambda\), the 2-D array is \(N\times M\), \(\lambda\) is the wavelength of the emitted ultrasound wave, \(d\) is the pitch of the elements, and \(\theta_{a}\) and \(\theta_{e}\) are the angles in azimuthal and elevational planes, respectively. For volumetric imaging, cross-sectional planes can be selected in the elevational (\(y\)-\(z\) plane, \(\alpha=0\)), in the azimuthal (\(x\)-\(z\) plane, \(\beta=0\)), and in the diagonal (\(xy\)-\(z\) plane, \(\alpha\
eq 0\) and \(\beta\
eq 0\)) planes, as shown in Figure 1c. While the response of an X-shaped array (\(H_{\text{X-shaped}}\)) is equivalent to that of a full array (\(H_{\text{Full}}\)) for both the azimuthal and the elevational planes, Equations (1) and (2) are not equivalent in the diagonal plane, as neither \(\alpha\) nor \(\beta\) are zero. The PSFs of X-shaped sparse array with unfocused ultrasound waves was demonstrated in ultrasound fields of 2-D array (64 \(\times\) 64 elements, 250 \(\upmu\)m pitch, and a center frequency at 3.0 MHz). A Field-II program was used to simulate behaviors of the unfocused (i.e., plane wave) transmit beams for this study [16; 17]. Two-way PSFs of full-transmit and full-receive (FT-FR) arrays and full-transmit and X-shaped receive (FT-XR) arrays were calculated at a 30 mm depth. The envelopes of the PSFs were shown, and the side-lobe levels were calculated at the positions of the side-lobe peaks [18]. In addition, 3-D point target phantom was simulated to evaluate image quality from the array configurations for the volumetric imaging. As shown in Figure 2, the 3-D phantom contained regularly-positioned grid points. Each grid scatter point was equally spaced in the azimuthal, elevational and depth directions with an interval of 10 mm. RF data were collected from the Field-II simulations [16; 17] using the same parameters of the PSF simulations, by transmitting a plane wave beam using the full aperture (i.e., 16 mm \(\times\) 16 mm). To cover a pyramidal volumetric imaging area, delay-and-sum beamforming [19] was used to reconstruct 90 \(\times\) 90 scan lines. For the unfocused FT-FR and FT-XR, we simulated B-scan images in the diagonal planes and the simulated images were displayed in a 40 dB dynamic range. To enhance the quality of images generated from unfocused FT-XR, a dynamic transmit focusing (DTxF) method was applied to reconstruct 2-D imaging planes by utilizing coherent angular compounding [20; 21; 22; 23]. Figure 3 presents a geometrical description of the coherent angular compounding imaging applied in the simulation. Transmit time delay at each element was calculated as \(x\sin\theta_{x}+y\sin\theta_{y}\), where \(\theta_{x}\) and \(\theta_{y}\) represent the inclination angles of the plane wave from \(x\)- and \(y\)-axis, respectively [22].