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Solution of the Problem on Image Reconstruction in Computed Tomography

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This paper describes a new solution of the problem of two-dimensional (2-D) image f(xy) reconstruction from a finite number of line-integrals. The reconstruction is a discrete image \(f_{n,m}\) on the Cartesian lattice lying in the XY plane as is required in all practical applications including imaging in medicine, electron microscopy, radio astronomy, geophysics, etc. The image is reconstructed from a finite number of projections. The values of the image are calculated exactly from the line-integrals. To achieve such a reconstruction of the image, we describe a method of calculation of all required ray-sums of the discrete image \(f_{n,m}\) from the line-integrals of f(xy). The paired representation of the image is also considered. The set of projections and number of measurements for each projection are determined (defined) by the complete set of 2-D discrete paired functions which allows composition of the discrete image as a sum of direction images. The model of reconstruction is described, and examples and experimental results of implementation of the proposed method are given. In the framework of the proposed model, the reconstruction is exact. The reconstruction of the image with noise and the reconstruction from a limited range of angles are also described.

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Correspondence to Artyom M. Grigoryan.

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Grigoryan, A.M. Solution of the Problem on Image Reconstruction in Computed Tomography. J Math Imaging Vis 54, 35–63 (2016).

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