## 2 XS Detector Targeting for Passive Imaging Detection
Similar to the center of gravity algorithm calculation [20], we adopt Formula (6) for computing the center of mass without considering the noise:  
\[\hat{x}_{1}=\frac{P\sum\limits_{i=1}^{M}iQ_{i}}{\sum\limits_{i=1}^{M}Q_{i}} \tag{7}\]  
In the M-point center of gravity algorithm, we can consider only the electrode with the largest charge and its M \(-\) 1 nearest-neighbor electrodes [20]. Combined with Equation (7), we have:  
\[\hat{x}_{1}=\frac{P\sum\limits_{i=1}^{M}i\exp[-\frac{(iP-x_{1})^{2}}{2\sigma_{e }^{2}}]}{\sum\limits_{i=1}^{M}\exp[-\frac{(iP-x_{1})^{2}}{2\sigma_{e}^{2}}]} \tag{8}\]  
The conventional detection methods have to deal with aliased data, and they distinguish and remove these data according to waveform shape analysis since they cannot obtain the correct position of the photons. Specifically, it is well known that aliased signals may occur, except that the interval between the arrival times is greater than the dead time [30]. More than one photon may arrive during the dead time, and an aliased signal occurs when the positions of these arrived photons are close. This aliased signal interferes with the subsequent signal processing and leads to incorrect photon positions. The conventional detection processes usually remove these aliased data and have to increase the detection time in order to obtain enough non-aliased signals. Assume that \(K_{1}\) non-aliased data and \(K_{2}\) aliased data are captured in one detection.\(\hat{x}_{1}\), \(\hat{x}_{2}\),..., \(\hat{x}_{K_{1}}\) denotes the \(K_{1}\) non-aliased data. \(\hat{x}_{1}^{\prime}\), \(\hat{x}_{2}^{\prime}\),..., \(\hat{x}_{K_{2}}^{\prime}\), denotes the \(K_{2}\) aliased data. The removal method removes the \(K_{2}\) aliased data, and the mean value \(\overline{X}\) of \(\hat{x}_{1}\), \(\hat{x}_{2}\),..., \(\hat{x}_{K_{1}}\) is used to represent the estimate of the object position \(x_{a}\). Indeed, these aliased data are useful for center of mass detection. The signal superposition of two photons is analyzed as an example. (\(x_{1}\),\(y_{1}\)) and (\(x_{2}\),\(y_{2}\)) denote the arrival positions of the photons that are received in dead time, which is the minimum time required for the detector to separate two photons. In this paper, dead time refers to the shaping time. Under some conditions, the count rate [31; 32] of the XS detector can be \(\frac{1}{t_{d}}\), where \(t_{d}\) is the dead time of the detector. The anode collects the charge generated by the two photons. The charges from the two photons collected by the \(i\)-th electrode in the X direction are \(Q_{i1}\) and \(Q_{i2}\), respectively:  
\[Q_{i1}=\frac{Q_{total}(1)}{\sqrt{2\pi}\sigma_{e}}\exp[-\frac{(iP-x_{1})^{2}}{ 2\sigma_{e}^{2}}]d \tag{9}\]  
\[N\sim Poisson(st\eta) \tag{1}\]0  
Aliasing occurs when the time interval between the arrival of two photons is less than the shaping time. Superposition occurs during the subsequent signal processing. The signal processing system includes a charge-sensitive amplifier, a Gaussian shaping amplifier, an analog-to-digital converter and an FPGA for signal acquisition, peak extraction, and data transmission. The charge-sensitive amplifier converts the output of each anode into a negative exponential voltage signal. The charge-sensitive amplifier output is amplified and rectified by the shaping amplifier to approximate a Gaussian voltage signal with most of the noise filtered out. An analog-to-digital converter digitizes it to complete the peak extraction. The signal processing process is discussed in detail in [30]. Gaussian-shapedsignals usually have the same waveform shape, which is approximated in this paper using the following function:  
\[f(t)= \begin{cases}0\quad,\,t<-l_{2}\\ l_{1}\frac{(t+l_{2})}{l_{2}^{2}}\exp[-\frac{(t+l_{2})^{2}}{2l_{2}^{2}}]\,\,t\geq-l_{2}\end{cases}  \tag{11}\]  
where \(l_{1}\) and \(l_{2}\) are two hardware parameters, determined by the specific amplification circuitry of the detector, which can be measured during the detection process. The constant \(l_{1}\) is the scale parameter.