# Path Planning for Localization of Radiation Sources Based on Principal Component Analysis
## 2 Proposed Method
### Reconstruction of Radiation Source Distribution via Simple Back-Projection  
This section describes how to use the all-around view Compton camera to reconstruct the measurement data into a 3D radiation source distribution via simple back-projection, which is a general reconstruction method [21]. Several studies have been conducted to propose a methodology for imaging the distribution of radioactivity [22; 23; 24]. Among them, maximum likelihood expectation maximization (MLEM) is widely used to reconstruct data obtained from a radiation detector [25; 26]. The MLEM can estimate the position of the radiation source with high accuracy via iterative calculations. [PERSON] et al. proposed a system to enable real-time online reconstruction through MLEM [19]. This approach can be applied to our system and can significantly improve the performance. Future work may include employing the MLEM in our system. In this study, simple back-projection is adopted as a basic method for reconstructing radiation source distributions. In the case of using a normal Compton camera with scatterer and absorber layers, let \(\theta_{i}\) be the scattering angle at the measurement point index \(i\) when \(N\) denotes all the measurement points. Then, let \(\mathbf{k}\) be the vector connecting the interacting elements on the absorber and scatterer layers, as illustrated in Figure 5. In addition, let \(\mathbf{m}\) be the vector connecting the interacting element on the scatterer and voxels in the projection space. The pixel value \(f(V)\) of the voxels can be defined by the delta function in the simple back-projection method as follows:  
\[f(V)=\sum_{i=1}^{N}\delta(\frac{\mathbf{k}\cdot\mathbf{m}}{|\mathbf{k}||\mathbf{m}|}-\cos \theta_{i}). \tag{2}\]  
As described in Section 2.2, it is difficult to distinguish between an absorber and a scatterer because the process, from the incident to their absorption, takes place at a very high speed. Therefore, for each incident event, two types of calculations are performed for each interacting element in the case of an absorber and a scatterer. The measurement data G can be represented by the following matrix:  
\[G=\left( \begin{array}{cccc}P_{s1}&P_{a1}&E_{s1}&E_{a1}\\ P_{s2}&P_{a2}&E_{s2}&E_{a2}\\ \vdots&\vdots&\vdots&\vdots\\ P_{sN}&P_{aN}&E_{sN}&E_{aN}\end{array} \right), \tag{3}\]  
where each row represents the data measured in one incident event, while \(P_{s}\) and \(P_{a}\) are the positions of the interacting elements on the scatterer and absorber, respectively. \(E_{s}\) and \(E_{a}\) represent the deposited energies at \(P_{s}\) and \(P_{a}\), respectively. Because the intensity of  
Figure 4: Obtaining Compton cones using an all-around view Compton camera: the all-around view Compton camera projects Compton cones on both sides of a straight line connecting the two interacting elements. the radiation source is constant, the measurement data that do not satisfy the following equation can be ignored:  
\[|E_{s}+E_{a}-E_{rad}|<\Delta E, \tag{4}\]  
where \(\Delta E\) is a margin that considers the energy resolution of the Compton camera. Based on the equations above, a simple back-projection method is implemented as described in Algorithm 1.