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Region-of-Interest (ROI) tomography aims at reconstructing a region ofinterest $C$ inside a body using only x-ray projections intersecting $C$ withthe goal to reduce overall radiation exposure when only a small specific regionof the body needs to be examined. We consider x-ray acquisition from sourceslocated on a smooth curve $\Gamma$ in $\mathbb{R}^3$ verifying classical Tuy'scondition. In this situation, the {\it non-trucated} cone-beam transform $D f$of smooth densities $f$ admits an explicit inverse $Z$; however $Z$ cannotdirectly reconstruct $f$ from ROI-truncated projections. To deal with the ROItomography problem, we introduce a novel reconstruction approach. For densities$f$ in $L^{\infty}(B)$ where $B$ is a bounded ball in $\mathbb{R}^3$, ourmethod iterates an operator $U$ combining ROI-truncated projections, inversionby the operator $Z$ and appropriate regularization operators. Assuming onlyknowledge of projections corresponding to a spherical ROI $C \subset B$, given$\epsilon >0$, we prove that if $C$ is sufficiently large our iterativereconstruction algorithm converges uniformly to an $\epsilon$-accurateapproximation 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 acritical ROI radius ensuring the convergence of the ROI reconstructionalgorithm to $\epsilon$-accurate approximations of $f$. We numerically verifiedthese theoretical results using simulated acquisition of ROI-truncatedcone-beam projection data for multiple acquisition geometries. Numericalexperiments indicate that the critical ROI radius is fairly small with respectto the support region~$B$.

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