Tomographic image reconstruction via estimation of sparse unidirectional gradients

Since computed tomography (CT) was developed over 35 years ago, new mathematical ideas and computational algorithms have been continuingly elaborated to improve the quality of reconstructed images. In recent years, a considerable effort can be noticed to apply the sparse solution of underdetermined system theory to the reconstruction of CT images from undersampled data. Its significance stems from the possibility of obtaining good quality CT images from low dose projections. Among diverse approaches, total variation (TV) minimizing 2D gradients of an image, seems to be the most popular method. In this paper, a new method for CT image reconstruction via sparse gradients estimation (SGE), is proposed. It consists in estimating 1D gradients specified in four directions using the iterative reweighting algorithm. To investigate its properties and to compare it with TV and other related methods, numerical simulations were performed according to the Monte Carlo scheme, using the Shepp-Logan and more realistic brain phantoms scanned at 9-60 directions in the range from 0 to 179°, with measurement data disturbed by additive Gaussians noise characterized by the relative level of 0.1%, 0.2%, 0.5%, 1%, 2% and 5%. The accuracy of image reconstruction was assessed in terms of the relative root-mean-square (RMS) error. The results show that the proposed SGE algorithm has returned more accurate images than TV for the cases fulfilling the sparsity conditions. Particularly, it preserves sharp edges of regions representing different tissues or organs and yields images of much better quality reconstructed from a small number of projections disturbed by relatively low measurement noise.