Region-of-interest reconstructions from truncated 3D x-ray projections

This paper introduces a method of region-of-interest (ROI) reconstruction from truncated 3D X-ray projections, consisting of a waveletbased regularized iterative reconstruction procedure that, under appropriate conditions, converges within the ROI to an exact or highly accurate solution. ROI tomography is motivated by the goal to reduce the overall radiation exposure when primarily the reconstruction of a specified region rather than the entire object is required. Our approach assumes that only the 3D truncated X-ray projections, i.e., the projection data restricted to the image of the ROI, are known and does not assume any previous knowledge about the density function, except for standard assumptions about integrability and regularity needed to ensure that forward and backward transforms are well defined. We provide rigorous theoretical justification for the convergence of our regularized reconstruction algorithm in the continuous setting and prove the existence of a critical radius of a spherical ROI that ensures the convergence of the algorithm. Theoretical results are validated numerically using simulated acquisition and truncation of projection data for various acquisition geometries and ROI sizes and locations. We provide a numerical analysis of the ROI reconstruction stability as a function of the ROI size, showing that our algorithm converges also for ROI sizes which are rather small with respect to Department of Mathematics, University of Houston, Houston, Texas 77204, USA. Biomedical Engineering, University of Houston, Houston, Texas 77204, USA.