Empirical average-case relation between undersampling and sparsity in x-ray CT

Abstract. In x-ray computed tomography (CT) it is generally acknowledged that reconstruction methods exploiting image sparsity allow reconstruction from a significantly reduced number of projections. The use of such recon-struction methods is motivated by recent progress in compressed sensing (CS). However, the CS framework provides neither guarantees of accurate CT re-construction, nor any relation between sparsity and a sufficient number of measurements for recovery, i.e., perfect reconstruction from noise-free data. We consider reconstruction through 1-norm minimization, as proposed in CS, from data obtained using a standard CT fan-beam sampling pattern. In em-pirical simulation studies we establish quantitatively a relation between the image sparsity and the sufficient number of measurements for recovery within image classes motivated by tomographic applications. We show empirically that the specific relation depends on the image class and in many cases ex-hibits a sharp phase transition as seen in CS, i.e. same-sparsity image require the same number of projections for recovery. Finally we demonstrate that th