# Path Planning for Localization of Radiation Sources Based on Principal Component Analysis
## 2 Proposed Method
### Gamma-Ray Measurement Using All-Around View Compton Camera  
Various gamma-ray detectors have been proposed; however, they can all be classified into three types: gas-filled, scintillation, and semiconductor detectors [16]. In [8], the Geiger-Muller counter, which is a gas-filled detector, was used to obtain the radiation dose rate. However, scintillation and semiconductor detectors have adequate sensitivity and energy resolution for radiation dose rate measurement and gamma spectrometry [12]. In [17], the authors used a CsI(Tl) scintillator and a Silicon Photomultiplier as a detector. By counting the number of incident gamma rays, measurements were conducted. When high-energy resolution measurements are required, semiconductor detectors should be considered [12]. Using CdTe or CZT ensures high-energy-resolution gamma spectroscopy, although such detectors are expensive. A gamma camera consists of one or multiple detectors, and it can be classified into three types: pinhole, coded-aperture, and Compton [12]. Although pinhole cameras have high angular resolution, they are based on heavy collimators. Therefore, they are not appropriate for mobile robots. In [18; 19], a coded-aperture camera was used for gamma-ray imaging. The authors used an active mask and achieved a wide FOV. In [10; 20],  
Figure 2: Schematic of proposed method: the proposed method consists of three parts: gamma-ray measurement, reconstruction of the radiation source distribution, and path planning. The robot repeats the gamma-ray measurement, reconstruction, path planning, and movement until the path is closed. Compton cameras were used to localize radiation sources because of their wide FOV. Thus, because of its advantages, the Compton camera is adopted in our system. As illustrated in Figure 3, the Compton camera consists of two layers: a scatterer and an absorber. The gamma ray, which interacts with the elements in the first and second layers, is recorded as the measurement data. The data contain the positions of the elements interacting with gamma rays at the two layers, respectively, with the deposited energy of radiation. Let \(A\) and \(B\) be the positions where the gamma ray is deposited on the scatterer and absorber, respectively, while \(\theta\) is the scattering angle of the gamma ray with respect to the straight line connecting \(A\) and \(B\). According to the kinematics of Compton scattering, the scattering angle is calculated as follows:  
\[cos\theta=1-m_{e}c^{2}(\frac{1}{E_{B}}-\frac{1}{E_{A}+E_{B}}), \tag{1}\]  
where \(m_{e}\), \(E_{A}\) and \(E_{B}\), and \(c\) represent the mass of an electron, deposited energy at each layer, and speed of light, respectively. The positions at which the two elements interact with the incident radiation are considered the coordinates relative to the internal coordinate system of the Compton camera. A cone representing the positions of the radiation sources on the surface is estimated via two consecutive interaction positions of the elements and scattering angle. As the two interaction positions \(A\) and \(B\) and the scattering angle \(\theta\) are given, the radiation source lies on the cone surface with an axis direction \(\overrightarrow{BA}\) and a half opening angle \(\theta\). This cone is called a Compton cone, and the radiation source position is estimated on the multiple overlapping Compton cone surfaces via back projection. Although the all-around view Compton camera cannot clearly differentiate between the scatterer and absorber, the principle is the same as that of a general Compton camera. A Compton cone can be obtained from the two consecutive interaction positions of the elements, \(A\) and \(B\), with the scattering angle. However, it is difficult to determine whether the two interacting elements are absorbers or scatterers because the time between the scatter and the subsequent absorption ranges from \(p\)s to \(ns\). Therefore, for each incident event, two types of calculations are performed for each interacting element in the case of an absorber and a scatterer.