Optimality of the fully discrete filtered backprojection algorithm for tomographic inversion

Although the filtered backprojection algorithm ( FBA) has been the standard reconstruction algorithm in 2D computerized tomography for more than 30 years, its convergence behavior is not completely settled so far. Relying on convergence results by Rieder and Faridani for the semi-discrete FBA [SIAM J. Numer. Anal., 41(3), 869-892, 2003], we show optimality of the fully discrete version for reconstructing sufficiently smooth density distributions. Further, we introduce MFBA, a modified version of FBA, and prove its optimality under weaker smoothness requirements. Remarkably MFBA may have a larger convergence order in the angular than in the lateral variable, thus allowing optimal convergence in case of angular under-sampling. Moreover, MFBA can be seen as a limit of the phantom view method introduced to increase angular resolution.