Statistical modelling and Bayesian inversion for a Compton imaging system: application to radioactive source localisation

Abstract

This paper presents a statistical forward model for a Compton imaging system, called Compton imager. This system, under development at the University of Illinois Urbana Champaign, is a variant of Compton cameras with a single type of sensors which can simultaneously act as scatterers and absorbers. This imager is convenient for imaging situations requiring a wide field of view. The proposed statistical forward model is then used to solve the inverse problem of estimating the location and energy of point-like sources from observed data. This inverse problem is formulated and solved in a Bayesian framework by using a Metropolis within Gibbs algorithm for the estimation of the location, and an expectation-maximization algorithm for the estimation of the energy. This approach leads to more accurate estimation when compared with the deterministic standard back-projection approach, with the additional benefit of uncertainty quantification in the low photon imaging setting.

Figures (15)

• Figure 1: Working principle of a Compton camera. The locations and the amount of energy lost by the incoming photon are recorded by each layer. From this information, the position of the source lies on a conical surface of semi-aperture angle $\omega$\ref{['eq:Compton_formula']}.
• Figure 2: Compton imager: setup and working principle. An incoming photon of energy $E_0$ is emitted by a source (in red) at position $\boldsymbol{r}_0$. The photon travels in a first sensor in green before undergoing scattered at position $\boldsymbol{r}_1$ in the sensor in light blue according to the path represented by the green array. The energy deposited during the first interaction is denoted $E_1$. The scattered photon travels in the two sensors in dark blue. Its path is depicted by the pink array. The second interaction position happens at the position $\boldsymbol{r}_2$ with a deposited energy $E_2$. (a) Side view. (b) Top view.
• Figure 3: Physics of the Compton imager. (a) A point-like source $S$ at position $\boldsymbol{r}_0$ emits a photon of energy $E_0$. This photon interacts with one of the sensor of the imager at position $\boldsymbol{r}_1$. This interaction results in a scattered photon of energy $(E_0-E_1)$, where $E_1$ is the part of energy collected by the sensor at $\boldsymbol{r}_1$. The scattered photon interacts then at position $\boldsymbol{r}_2$. According to the Compton kinematics $\boldsymbol{r}_2$ is on a cone of apex $\boldsymbol{r}_1$, axis $(\boldsymbol{r}_1-\boldsymbol{r}_0)$ and semi-aperture $\omega$, that can be computed via the Compton formula \ref{['eq:Compton_formula']}. The sensor records the amount of energy $E_2$ collected at $\boldsymbol{r}_2$, which is equal to $(E_0-E_1)$ if this photon interaction was a photoelectric absorption or less than $(E_0-E_1)$ if it was a Compton interaction. (b) Conversely, given the positions $\boldsymbol{r}_1, \boldsymbol{r}_2$ and energy depositions $E_1, E_2$ of the two interactions and assuming that we know $E_0$, $\boldsymbol{r}_0$ is on the cone of apex $\boldsymbol{r}_1$, axis $(\boldsymbol{r}_1, \boldsymbol{r}_2)$ and semi angle $\omega(E_0, E_1)$.
• Figure 4: Photon trajectory from its emission (top) to the second interaction (bottom) and associated densities of each stage.
• Figure 5: Hierarchical model between variables. From the measurements $\{\tilde{\boldsymbol{r}}_{1n}, \tilde{E}_{1n}, \tilde{\boldsymbol{r}}_{2n}, \tilde{E}_{2n}\}_{n=1}^N$, the true positions and energy depositions $\{{\boldsymbol{r}}_{1n}, {E}_{1n}, {\boldsymbol{r}}_{2n}, {E}_{2n}\}_{n=1}^N$ and the corresponding hidden standard deviations $\sigma_{xy}$ and $\sigma_z$ are estimated, assuming measurements are corrupted by Gaussian noises. $N$ additional virtual source positions, acting as if there were $N$ sources, are also introduced to deal with potential outliers and assign the events to the relevant source $\boldsymbol{r}_0^{(k)}$. The relative intensities of sources $\{w_0^{(k)}\}_{k=1}^K$ are also estimated. Note that $E_0$ is not depicted here since it is considered as a known value for this algorithm. Circular nodes represent the variables to estimate. Measurements are in rectangular nodes. Diamond shaped nodes contain the fixed hyperparameters $\{\alpha_k\}_{k=0}^K$ and $\kappa$ that will be set according to prior knowledge about the experiment.