Microlocal analysis of a non-linear cone transform and applications to Compton camera imaging

Abstract

We present a novel method to recover the source intensity, $f : \mathbb{R}^n \to \mathbb{R}$, and attenuation coefficient, $μ: \mathbb{R}^n \to \mathbb{R}$, in Compton camera imaging. We apply a non-linear model, which accounts for ray attenuation. We show that the data, $h$, can be modeled $h = \mathcal{R}(f,μ) = R(fg)$, where $g = \exp(-Gμ)$ models attenuation, $G$ is a (linear) divergent beam transform, and $R$ is a linear operator which defines the integrals of $fg$ over cones. Commonly in the literature, $μ$ is set to zero, and the data $h = Rf$ is linear. We address the case when $μ\neq 0$ and the transform is non-linear. To simplify the analysis, we first transform the data into weighted line integrals, $\tilde{h} = \mathcal{D}_k(f,μ) = D_k(fg)$, where $D_k$ is a weighted ray transform. Assuming practically reasonable geometric conditions, we show that $\tilde{h} = \exp(-X_{w_1}μ)X_{w_2}f$, where $X_w$ is a weighted X-ray transform, and the $w_i$ are smooth weights. After which, we use the theory of conormal distributions to describe the singularities of $\tilde{h}$. We show that there are artifacts in the reconstruction, and we quantify their strength using Sobolev spaces. We combine this theory with a geometric argument to recover the edges of $f$ and ultimately prove that $f$ and $μ$ are unique to $h$. The recovery of $f$ is notably more stable than that of $μ$, which we also discuss. To validate our theory, we present simulated reconstructions of $f$ and $μ$ using the proposed method.