A general extending and constraining procedure for linear iterative methods

Abstract

Algebraic reconstruction techniques (ARTs), on both their successive and simultaneous formulations, have been developed since the early 1970s as efficient ‘row-action methods’ for solving the image-reconstruction problem in computerized tomography. In this respect, two important development directions were concerned with, first, their extension to the inconsistent case of the reconstruction problem and, second, their combination with constraining strategies, imposed by the particularities of the reconstructed image. In the first part of this paper, we introduce extending and constraining procedures for a general iterative method of an ART type and we propose a set of sufficient assumptions that ensure the convergence of the corresponding algorithms. As an application of this approach, we prove that Cimmino's simultaneous reflection method satisfies this set of assumptions, and we derive extended and constrained versions for it. Numerical experiments with all these versions are presented on a head phantom widely used in the image reconstruction literature. We also consider hard thresholding constraining used in sparse approximation problems and apply it successfully to a 3D particle image-reconstruction problem.

Keywords:

• algebraic reconstruction techniques
• inconsistent least-squares problems
• constraining strategies
• Cimmino algorithm
• Cimmino extended algorithm
• hard thresholding operator

2000 AMS Subject Classifications :

• 65F10
• 65F20
• 65F22

Acknowledgements

We express our gratitude to Prof. Dr Tommy Elfving and Dr Touraj Nikazad from Linköping University, Sweden, for providing the head phantom used in the numerical experiments. We thank the anonymous referees for their comments and advice which helped us improve the presentation.