Krol, Andrzej; Li, Si; Shen, Lixin; Xu, Yuesheng Preconditioned alternating projection algorithms for maximum a posteriori ECT reconstruction. (English) Zbl 1270.94018 Inverse Probl. 28, No. 11, Article ID 115005, 34 p. (2012). Summary: We propose a preconditioned alternating projection algorithm (PAPA) for solving the maximum a posteriori (MAP) emission computed tomography (ECT) reconstruction problem. Specifically, we formulate the reconstruction problem as a constrained convex optimization problem with the total variation (TV) regularization. We then characterize the solution of the constrained convex optimization problem and show that it satisfies a system of fixed-point equations defined in terms of two proximity operators raised from the convex functions that define the TV-norm and the constraint involved in the problem. The characterization (of the solution) via the proximity operators that define two projection operators naturally leads to an alternating projection algorithm for finding the solution. For efficient numerical computation, we introduce to the alternating projection algorithm a preconditioning matrix (the EM-preconditioner) for the dense system matrix involved in the optimization problem. We prove theoretically convergence of the PAPA. In numerical experiments, performance of our algorithms, with an appropriately selected preconditioning matrix, is compared with performance of the conventional MAP expectation-maximization (MAP-EM) algorithm with TV regularizer (EM-TV) and that of the recently developed nested EM-TV algorithm for ECT reconstruction. Based on the numerical experiments performed in this work, we observe that the alternating projection algorithm with the EM-preconditioner outperforms significantly the EM-TV in all aspects including the convergence speed, the noise in the reconstructed images and the image quality. It also outperforms the nested EM-TV in the convergence speed while providing comparable image quality. Cited in 25 Documents MSC: 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 65J22 Numerical solution to inverse problems in abstract spaces 65K10 Numerical optimization and variational techniques Keywords:image reconstruction; preconditioned alternating projection algorithm; optimization problem; image quality; efficient numerical computation; system of fixed-point equations; dense system matrix; emission computed tomography; convergence speed PDFBibTeX XMLCite \textit{A. Krol} et al., Inverse Probl. 28, No. 11, Article ID 115005, 34 p. (2012; Zbl 1270.94018) Full Text: DOI