## 2 Materials and Methods
### Multi-Line Transmission  
MLT consists of transmitting \(N_{TX}\) beams simultaneously in different focusing directions. After each transmission (TX), the backscattered signals are collected by the array elements in reception (RX) and beamformed in parallel along the considered steering directions. Then, the beams are moved by an angular step \(\theta_{STEP}=\theta_{SECT}\)/\(N_{LINES}\) in order to cover the full \(\theta_{SECT}\)-wide image field of view with \(N_{LINES}\) scan lines. Therefore, the simultaneous TX beams are separated by an angular distance \(\theta_{TX}=\theta_{SECT}\)/\(N_{TX}\)[22].  
To achieve multiple beam TX, a set of focusing delays has to be computed for each of the simultaneous TX focusing directions. Thus, the MLT excitation pulse is obtained by summing up the excitation signals that would be used to focus the TX beam along each one of the considered steering directions in the classic single-line transmission (SLT) case.  
As explained in [22], the main problem with MLT is the presence, in the final image, of the so called cross-talk artifacts due to interferences between the multiple beams.  
To better understand how such artifacts generate, we have to recall the array theory [24], according to which the TX (\(h_{TX}\)) and RX (\(h_{RX}\)) ultrasound beams at the focal depth can be generally expressed as follows:  
\[h_{TX}(u)=sinc(\frac{p_{x}u}{\lambda})sinc[\frac{Np_{x}}{\lambda}(u-\sum \limits_{j=-\infty}^{+\infty}\frac{j\lambda}{p_{x}}-u^{TX})] \tag{3}\]  
\[h_{RX}(u)=sinc(\frac{p_{x}u}{\lambda})sinc[\frac{Np_{x}}{\lambda}(u-\sum \limits_{j=-\infty}^{+\infty}\frac{j\lambda}{p_{x}}-u^{RX})] \tag{4}\]where \(u=sin(\theta)\), \(u^{TX}\) refers to the focusing direction of the TX beam (\(\theta^{TX}\)) and \(u^{RX}\) to the focusing direction of the RX beam (\(\theta^{RX}\)). \(N\) is the number of array elements, \(p_{x}\) is the array pitch, and \(\lambda\) is the wavelength. Then, in the classic SLT case (i.e., each time one beam is transmitted and received), the pulse-echo (\(h_{PE}\)) response is given by:  
\[h_{PE}(u)=h_{TX}(u)h_{RX}(u). \tag{5}\]  
In MLT imaging, instead, several beams are transmitted in parallel, thus we have a summation in the TX beam expression:  
\[h_{TX}(u)=sinc(\frac{p_{x}u}{\lambda})\sum_{i=1}^{N_{TX}}sinc[\frac{Np_{x}}{ \lambda}(u-\sum_{j=-\infty}^{+\infty}\frac{j\lambda}{p_{x}}-u_{i}^{TX})] \tag{6}\]  
where \(u_{i}^{TX}\) refers to the focusing direction of the \(i\)-th TX beam (\(\theta_{i}^{TX}\)). Hence, in this case the pulse-echo beam in (5) is made of \(N_{TX}\) terms, where one term is the one generated when \(\theta_{i}^{TX}=\theta^{RX}\), i.e., the TX and RX responses are equal (as in a classic SLT scan), and the other terms with \(\theta_{i}^{TX}\
eq\theta^{RX}\) are the so called cross-talk contributions that originate from interferences among the TX and RX beams. Consequently, in the SLT case, when \(N_{TX}=1\), the pulse-echo beam has a \(sinc^{2}\) shape, while in MLT with \(N_{TX}>1\), the beam has a more complex shape, with \(N_{TX}\) peaks in the simultaneous TX focusing directions, as shown in Figure 1 for MLT with four or 12 TX beams (i.e., 4-MLT and 12-MLT).  
In particular, interferences generate two types of cross-talk, i.e., the so called TX cross-talk, caused by interferences among TX beam side lobes and the RX beam main lobe, and RX cross-talk, caused by  
Figure 1: Examples of theoretical beampatterns in 4-MLT and 12-MLT: in the panels on the **left**, the TX (gray) and RX (black) beampatterns are plotted in each MLT case, when the RX beam is steered in one of the TX beams’ direction (e.g., at 0°); on the **right**, the resulting pulse-echo beam shapes are shown.  
interferences between the TX beam main lobe and RX beam side lobes [22], which appear as artifacts around the beam main lobe along the axial or lateral direction, respectively.