A Formulation in Concordance with the Sampling Theorem for Band-Limited Images Reconstruction from Projections

摘要

A rigorous method for band-limited images reconstruction from sampled projections is presented. The problem is formulated as the minimization of a quadratic criterion expressed in the frequency domain; consequences of spectral support limitation and sampling are taken into account. Then, the optimal reconstruction is obtained in two steps: a classical 1D-convolution/backprojection followed by the computation of the solution of a 2D convolution equation. The proposed solution for this second step is based on a conjugate gradient method. In both steps, the computations are performed in the frequency domain. This reduces the number of computations and allows the execution in a reasonable amount of time. A particular choice of the weights establishes a link between this approach and Radon's classical backprojection method: the classical method is optimal when the number of projections is infinite. In practice, the method proposed here improves the results obtained when computing an estimate of the solution by backprojections from a finite number of samples and projections when the projections are noisefree.