## 2 XS Detector Targeting for Passive Imaging Detection
The constant \(l_{2}\) determines the width of the waveform, and a large width of the waveform could result in a long detector shaping time. \(t\) represents time. The arrival time of the photon determines the offset of the waveform, and the charge collected by the anode determines the peak of the waveform. The waveform is \(Q_{i1}f(t-t_{1})\), where \(t_{1}\) is the arrival time of the photon, and \(Q_{i1}\) is the charge collected by the anode. In Figure 3, at time \(t_{1}\), the peak of the curve \(Q_{i1}\) is arrived at, followed by \(t_{2}\), the peak of the second curve \(Q_{i2}\) being arrived at. The peak of the superimposed waveform is attained at time \(t_{s}\), which is between \(t_{1}\) and \(t_{2}\). When the signal reaches a predefined threshold, sampling begins, marked as moment 0 in the figure. The sampling period lasts 200 ns, and the waveform is truncated at the end of this duration. In practical detection, the detector can separate two photon events when the time interval between them is greater than the shaping time. We model the peak after superposition as a linear combination of the charges of the two signals:  
\[Q_{i}=Q_{i1}f(t_{s}-t_{1})+Q_{i2}f(t_{s}-t_{2}) \tag{12}\]  
Time \(t_{s}\) may change according to different anode strips. In the theoretical analysis, in order to simplify the model, we only consider the case where the positions of two aliased photons are very close to each other, and we assume that the time \(t_{s}\) is constant. In the simulation experiments, we consider \(t_{s}\) a variable in order to make the simulation closer to the actual detection. The final charge is obtained as a linear combination of the two charges. We denote \(f(t_{s}-t_{1})\), \(f(t_{2}-t_{s})\) as \(k_{1}\), \(k_{2}\), respectively. We use \(\dot{x}^{\prime}\) to denote the result of a single  
Figure 3: The schematic diagram is the waveform superposition process during Gaussian shaping. \(Q_{i1}\) and \(Q_{i2}\) are two waveforms before superposition. \(Q_{sum}\) is the waveform after superposition. The peak of the superimposed waveform is attained at time \(t_{s}\), which is between \(t_{1}\) and \(t_{2}\). acquisition in the case of aliasing. For the aliased signals, the center of gravity position is also calculated using center of mass Formula (7) in the following form:  
\[\hat{x}^{\prime}=\frac{P\sum\limits_{i=1}^{M}iQ_{i}}{\sum\limits_{i=1}^{M}Q_{i}} \tag{13}\]  
Substituting Equations (9), (10) and (12) into Equation (13) gives Equation (14):  
\[\hat{x}^{\prime}=\frac{P\sum\limits_{i=1}^{M}i\bigg{\{}Q_{total}(1)\exp[-\frac{ (iP-x_{1})^{2}}{2\sigma_{e}^{2}}]k_{1}+Q_{total}(2)\exp[-\frac{(iP-x_{2})^{2}}{2 \sigma_{e}^{2}}]k_{2}\bigg{\}}}{\sum\limits_{i=1}^{M}\bigg{\{}Q_{total}(1)\exp[ -\frac{(iP-x_{1})^{2}}{2\sigma_{e}^{2}}]k_{1}+Q_{total}(2)\exp[-\frac{(iP-x_{2} )^{2}}{2\sigma_{e}^{2}}]k_{2}\bigg{\}}} \tag{14}\]  
There exists \(k_{3}\) satisfying Equation (15):  
\[\sum\limits_{i=1}^{M}\exp[-\frac{(iP-x_{2})^{2}}{2\sigma_{e}^{2}}]=k_{3}\sum \limits_{i=1}^{M}\exp[-\frac{(iP-x_{1})^{2}}{2\sigma_{e}^{2}}] \tag{15}\]  
Substituting Equations (8) and (15) into Equation (14) gives Equation (16):  
\[\hat{x}^{\prime}=\frac{Q_{total}(1)k_{1}}{Q_{total}(1)k_{1}+Q_{total}(2)k_{2}k_ {3}}\hat{x}_{1}+\frac{k_{3}Q_{total}(2)k_{2}}{Q_{total}(1)k_{1}+Q_{total}(2)k_{ 2}k_{3}}\hat{x}_{2} \tag{16}\]  
We organize Equation (16) to obtain Equation (17):  
\[\hat{x}^{\prime}=\frac{Q_{total}(1)k_{1}}{Q_{total}(1)k_{1}+Q_{total}(2)k_{2}k_ {3}}\hat{x}_{1}+[1-\frac{Q_{total}(1)k_{1}}{Q_{total}(1)k_{1}+Q_{total}(2)k_{2} k_{3}}]\hat{x}_{2} \tag{17}\]  
The aliased data collected in one experiment are \(\hat{x}^{\prime}_{1}\), \(\hat{x}^{\prime}_{2}\), \(\cdot\), \(\hat{x}^{\prime}_{K_{2}}\). We denote the \(j\)-th data point as \(\hat{x}^{\prime}_{j}\). Even if \(Q_{total}\), \(k_{1}\), \(k_{2}\) and \(k_{3}\) are different for each acquisition, there exists \(c_{j}\) in the range of [0, 1] such that Equation (18) holds.