Roi reconstruction from truncated cone-beam projection

Region-of-Interest (ROI) tomography aims at reconstructing a region of interest C inside a body using only x-ray projections intersecting C and it is useful to reduce overall radiation exposure when only a small specific region of a body needs to be examined. We consider x-ray acquisition from sources located on a smooth curve G in Gamma mathbb R-3 verifying the classical Tuy condition. In this generic situation, the non-trucated cone-beam transform of smooth density functions f admits an explicit inverse Z as originally shown by Grangeat. 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-infinity(B) where B is a bounded ball in 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 of subset B, given epsilon > 0, we prove that if C is sufficiently large our iterative reconstruction algorithm converges at exponential speed to an epsilon-accurate approximation of f in L-infinity. The accuracy depends on the regularity of f quantified by its Sobolev norm in W-5(B). Our result guarantees the existence of a critical ROI radius ensuring the convergence of our ROI reconstruction algorithm to an epsilon-accurate approximation of f. We have 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.