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Convergence analysis of SART: optimization and statistics. (English) Zbl 1279.94043
Summary: Simultaneous algebraic reconstruction technique (SART) [A. Andersen, Algebraic reconstruction in CT from limited views, IEEE Trans. Med. Imaging 8, 50–55(1989)] [A. Andersen and A. Kak, Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm, Ultrason. Imaging 6, 81–94 (1984)] is an iterative method for solving inverse problems of the form \(Ax(+n)=b\). These kinds of problems arise, for example, in computed tomography (CT) reconstruction, in which case \(A\) is obtained from the discrete Radon transform. In this paper, we provide several methods for derivation of SART and connections between SART and other methods. Using these connections, we also prove the convergence of SART in different ways. These approaches are from optimization and statistical points of view and can be applied to other Landweber-like schemes such as Cimmino’s algorithm and component averaging. Furthermore, the noisy case is considered and the error estimation is given. Several numerical experiments for CT reconstruction are provided to demonstrate the convergence results in practice.

MSC:
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
92C55 Biomedical imaging and signal processing
65F10 Iterative numerical methods for linear systems
65K10 Numerical optimization and variational techniques
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