  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 $\Gamma$ 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
$\epsilon >0$, we prove that if $C$ is sufficiently large our iterative
reconstruction algorithm converges uniformly to an $\epsilon$-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 $\epsilon$-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~$B$.
