Quantitative study of undersampled recoverability for sparse images in computed tomography

Image reconstruction methods based on exploiting image sparsity, motivated by compressed sensing (CS), allow reconstruction from a significantly reduced number of projections in X-ray computed tomography (CT). However, CS provides neither theoretical guarantees of accurate CT reconstruction, nor any relation between sparsity and a sufficient number of measurements for recovery. In this paper, we demonstrate empirically through computer simulations that minimization of the image 1-norm allows for recovery of sparse images from fewer measurements than unknown pixels, without relying on artificial random sampling patterns. We establish quantitatively an average-case relation between image sparsity and sufficient number of measurements for recovery, and we show that the transition from non-recovery to recovery is sharp within well-defined classes of simple and semi-realistic test images. The specific behavior depends on the type of image, but the same quantitative relation holds independently of image size.