# A Spatial Compounding Method for Non-Delayed Sequential Beamforming
## 2 Materials and Methods
### Pressure Field Analysis  
The spatiotemporal impulse response of the transmit sub-aperture is used to extract excitation signals which correspond to different transmit configurations [6]. For simplicity, we consider only the region where the spherical wave assumption is valid. We make some assumptions: all elements of the transducer share the same electromechanical impulse response with uniform directivity function, and the discrete transmit sub-aperture is modeled as a continuous curved aperture. The pressure field of the imaging area can be obtained through a spatiotemporal impulse response [16]. Assuming a sinusoidal excitation \(v(t)\) and a rigid baffle, the pressure field for point \(x_{p}\) at time \(t\) can be written as  
\[p(\overrightarrow{x_{p}},t)=\rho v(t)*\frac{\partial h(\overrightarrow{x_{p}},t)}{\partial t}, \tag{4}\]  
where \(\rho\) is the equilibrium density of the medium, and \(h(\overrightarrow{x_{p}},t)\) in the equation can be given by [16]. \[h(\overrightarrow{x_{p}},t)=\frac{1}{2\pi}\int_{R_{min}}^{R_{max}}\beta(R) \delta\left(t-\frac{R}{c}\right)dR, \tag{5}\]  
where \(R\) is the distance from the point \(x_{p}\) to the aperture, and \(\beta(R)\) is the corresponding angle in the polar coordinates. Integrating the above equation over the aperture surface, the pressure field for the point above the focus in Figure 2a can be expressed as  
\[p(\overrightarrow{x_{p}},t)=\frac{\rho c\beta_{i}}{\pi}v\left(t-\frac{R_{min} }{c}\right)-\frac{\rho c}{2\pi}\left[\beta_{1}v\left(t-\frac{R_{1}}{c}\right) +\beta_{2}v\left(t-\frac{R_{2}}{c}\right)\right] \tag{6}\]  
where \(\beta_{1},\ \beta_{2}\), and \(\beta_{i}\) are associated with \(R_{1},\ R_{2},\ \text{and}\ R_{min}\). For the point below the focus in Figure 2b, the pressure can be expressed as  
\[p(\overrightarrow{x_{p}},t)=\frac{\rho c}{2\pi}\left[\beta_{1}v\left(t-\frac {R_{1}}{c}\right)+\beta_{2}v\left(t-\frac{R_{2}}{c}\right)\right]-\frac{\rho c \beta_{i}}{\pi}v\left(t-\frac{R_{max}}{c}\right), \tag{7}\]  
where \(\beta_{i}\) is associated with \(R_{max}\). Figure 2: Geometries for calculating pressure field of transmit sub-aperture for spatial point \(P\) in front of the focus (**a**) and behind the focus (**b**). Equations (7) and (8) show that the transmit wave shape can be characterized by three spherical waves. The total field is a combination of the three pulses with different transmit origins and different time delays. NDSB selects the excitation signal with the highest energy to do the reconstruction which is denoted as \(v(t-\frac{R_{min}}{c})\) in Equation (7) and \(v(t-\frac{R_{max}}{c})\) in Equation (8). On the other hand, beamforming can also be performed under two other excitation signals from the two ends of the aperture. Because the three excitation signals are from different origins, their corresponding beamformed images are therefore spatially decorrelated. Compounding can be done by summing these images together. To reconstruct the image originated from the two transmit aperture ends, the wave propagation path needs to be adjusted. The TOF corresponds to the excitation signal emitted from the ends of the active aperture can be expressed as  
\[t_{TOF\_END}=\frac{1}{c}(|\overrightarrow{r_{VS}}-\overrightarrow{r_{e}}|-| \overrightarrow{r_{O}}-\overrightarrow{r_{VS}}|+|\overrightarrow{r_{O}}- \overrightarrow{r_{ip}}|+|\overrightarrow{r_{r}}-\overrightarrow{r_{ip}}|). \tag{8}\]  
Figure 2(a) presents the TOF to generate the image corresponds to the left end of the active aperture, and Figure 2(b) corresponds to the right end. To produce HRI at \((x,z)\), the delay time \(\Delta t\) is also adjusted as  
\[\Delta t_{TOF\_END}=\frac{(\sqrt{(z-z_{f})^{2}+(x-x_{f})^{2}}+\sqrt{z_{f}^{2 }+(x_{f}-x_{O})^{2}}-|z-z_{f}|)}{c}, \tag{9}\]  
where \(x_{O}\) denotes the lateral position of the left/right end of the active aperture. The high-resolution image is denoted as \(HRI_{left\_end}\) and \(HRI_{right\_end}\) using Equation (2).