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Region-of-Interest reconstruction from truncated cone-beam projections

Region-of-Interest (ROI) tomography aims at reconstructing a region of interest $C$ inside a body using only x-ray projections intersecting $C$ with the goal to reduce overall radiation exposure when only a small specific region of the body needs to be examined. We consider x-ray acquisition from sources located on a smooth curve $Γ$ in $\mathbb{R}^3$ verifying classical Tuy's condition. In this situation, the {\it non-trucated} cone-beam transform $D f$ of smooth densities $f$ admits an explicit inverse $Z$; however $Z$ cannot directly reconstruct $f$ from ROI-truncated projections. To deal with the ROI tomography problem, we introduce a novel reconstruction approach. For densities $f$ in $L^{\infty}(B)$ where $B$ is a bounded ball in $\mathbb{R}^3$, our method iterates an operator $U$ combining ROI-truncated projections, inversion by the operator $Z$ and appropriate regularization operators. Assuming only knowledge of projections corresponding to a spherical ROI $C \subset B$, given $ε>0$, we prove that if $C$ is sufficiently large our iterative reconstruction algorithm converges uniformly to an $ε$-accurate approximation of $f$, where the accuracy depends on the regularity of $f$ quantified in the Sobolev norm $W^5(B)$. This result shows the existence of a critical ROI radius ensuring the convergence of the ROI reconstruction algorithm to $ε$-accurate approximations of $f$. We numerically verified these theoretical results using simulated acquisition of ROI-truncated cone-beam projection data for multiple acquisition geometries. Numerical experiments indicate that the critical ROI radius is fairly small with respect to the support region~$B$.