System matrix analysis for sparse-view iterative image reconstruction in X-ray CT

Iterative image reconstruction (IIR) with sparsity-exploiting methods, such as total variation (TV) minimization, used for investigations in compressive sensing (CS) claim potentially large reductions in sampling requirements. Quantifying this claim for computed tomography (CT) is non-trivial, as both the singularity of undersampled reconstruction and the sufficient view number for sparse-view reconstruction are ill-defined. In this paper, the singular value decomposition method is used to study the condition number and singularity of the system matrix and the regularized matrix. An estimation method of the empirical lower bound is proposed, which is helpful for estimating the number of projection views required for exact reconstruction. Simulation studies show that the singularity of the system matrices for different projection views is effectively reduced by regularization. Computing the condition number of a regularized matrix is necessary to provide a reference for evaluating the singularity and recovery potential of reconstruction algorithms using regularization. The empirical lower bound is helpful for estimating the projections view number with a sparse reconstruction algorithm.