This paper presents a numerical method for reconstructing a tomographic image for a region of interest (RoI) within a section using narrowed radiation beams to reduce radiation exposure. The RoI image reconstruction is formulated as a discrete problem to avoid the truncated (incomplete) problem associated with the conventional analytic filtered backprojection method. This in turn allows local reconstruction of RoI images without any prior information or constraints. A coarse image of the entire section is first reconstructed with the aid of a modified convex maximum likelihood (MCML) algorithm. The coarse image is then used to account for the effect of the RoI surroundings. With RoI-specific projections, an RoI image is then reconstructed, with either the MCML or conjugate gradient methods, at the desired pixel size. The proposed method is evaluated using an anthropomorphic FORBILD head phantom, showing an image quality comparable to that of conventional computed tomography, but with a ~69% reduction in radiation exposure. Unlike existing approaches that require some prior knowledge of a segment of the image, this approach does not require any prior image information and imposes no constraints on the solution.