Empirical average-case relation between undersampling and sparsity in x-ray CT
For the CT imaging community, this work provides empirical guidance on the number of projections required for sparsity-based reconstruction, though it is incremental as it does not provide theoretical guarantees.
The paper empirically establishes a quantitative relation between image sparsity and the number of projections needed for accurate reconstruction in x-ray CT using 1-norm minimization, showing a sharp phase transition similar to compressed sensing, and demonstrates that this relation is independent of image size and robust to small noise.
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 reconstruction methods is motivated by recent progress in compressed sensing (CS). However, the CS framework provides neither guarantees of accurate CT reconstruction, 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 empirical 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 exhibits 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 the relation holds independently of image size and is robust to small amounts of additive Gaussian noise.