A discretized tomographic image reconstruction based upon total variation regularization

Tomographic image reconstruction problem has an ill-posed nature like many other linear inverse problems in the image processing domain. Discrete tomography (DT) techniques are developed to cope with this drawback by utilizing the discreteness of an image. Discrete algebraic reconstruction technique (DART) is a DT technique that alternates between an inversion stage, employed by the algebraic reconstruction methods (ARM), and a discretization (i.e. segmentation) stage. Total variation (TV) minimization is another popular technique that deals with the ill-posedness by exploiting the piece-wise constancy of the image and basically requires to solve a convex optimization problem. In this paper, we propose an algorithm which also performs the successive sequences of inversion and discretization, but it estimates the continuous reconstructions under TV-based regularization instead of using ARM. Our algorithm incorporates the DART's idea of reducing the number of unknowns through the subsequent iterations, with a 1-D TV-based setting. As a second contribution, we also suggest a procedure to be able to select the segmentation parameters automatically which can be applied when the gray levels (corresponding to the different densities in the scanned object) are not known a priori. We performed various experiments using different phantoms, to show the proposed algorithm reveals better approximations when compared to DART, as well as three other continuous reconstruction techniques. While investigating the performances, we considered limited number of projections, limited-view, noisy projections and lack of prior knowledge on gray levels scenarios.