Ayrık cebirsel geriçatma tekniği için sıkıştırılmış algılama esaslı bir yaklaşım

Image reconstruction from incomplete projections has a crucial meaning in tomographic imaging field, due to some restrictions and requirements. Although the analytical methods, such as filtered backprojections (FBP), are preferable because of their low computational cost; they are not good at reconstructing satisfying images in case of limited number of projections and limited view. On the other hand, iterative methods (e.g. algebraic reconstruction technique (ART), norm optimization) makes the reconstruction from incomplete projection data possible. The ART (as well as its variations) models the reconstruction problem as a system of linear equations where the discretization points (i.e. pixels) of the image are variables and the equations represent the projections. For these algebraic reconstruction methods (abbreviated ARM), there is no unique solution due to the under-determined characteristic of the system, when the incomplete projection data is the case. Many iterative methods take some constraints into consideration and some of those methods suggest to exploit prior knowledge, if exists, in order to find the best approximation to the exact solution. The field of discrete tomography (DT) assumes that the variables have a range (and sometimes domain) of a finite and discrete set, whose element count is few and known a priori; and it aims to find a good quality solution even if the projection samples are highly reduced. Compressed sensing (CS) based methods, in the other respect, aims to find the sparsest solution by assuming the image is sparse in a known domain. Both approaches are used to be able to recover images from the projection data which doesn't satisfy the Nyquist-Shannon criterion. Discrete algebraic reconstruction technique (DART), which is a technique used in DT field and lies at the core of this study, accomplishes the goal stated above by combining a continuous ARM and a discretization scheme, in an iterative manner. In this study, the DART algorithm is investigated and it is combined with an initial total variation minimization (TvMin) technique, which is used to solve CS problems, to ensure a better initial guess. Also, the algorithm is extended with a segmentation procedure in which the threshold value, which simultaneously minimize both the projection error and the total variation (TV), is selected from a finite set of candidates, obtained using a histogram based thresholding scheme. Furthermore, the algorithm is extended with a gray level estimation procedure, which serves as an automatic determination of the gray levels to be used in the discretization step. A formulation is presented in order to approximate the exact gray levels and it is shown that the gray levels can almost be computed, even though they are not known in advance. All implementations are done using MATLAB environment. The proposed algorithm is compared to the DART and the FBP algorithms by the simulation experiments which are done under the conditions of limited number of projections, limited view and noisy projections, and the computational results are presented visually, either via the reconstructed images or the graphics.