Convergence analysis of SART: optimization and statistics

Abstract

Simultaneous algebraic reconstruction technique (SART) [A. Andersen, Algebraic reconstruction in CT from limited views, IEEE Trans. Med. Imaging 8 (1989), pp. 50–55; A. Andersen and A. Kak, Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm, Ultrason. Imaging 6 (1984), pp. 81–94] is an iterative method for solving inverse problems of 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.

Keywords:

• simultaneous algebraic reconstruction technique
• Bregman iteration
• dual gradient descent
• expectation maximization
• alternating minimization
• image reconstruction
• component averaging
• Cimmino's algorithm

2010 AMS Subject Classifications :

• 49M29
• 65F10
• 65K10
• 92C55
• 94A08

Acknowledgements

I would like to thank Professor Luminita A. Vese for valuable discussions and suggestions. This work is supported by the Center for Domain-Specific Computing (CDSC) funded by the NSF Expedition in Computing Award CCF-0926127.