Newton‐SOR method for fast statistical tomographic image reconstruction

Abstract

AbstractThere are recent studies on image reconstruction for emission CT and transmission CT, using the maximum likelihood (ML) estimation or the maximum a posteriori (MAP) estimation, aiming at quality improvement of the reconstructed image. Those methods in general have the problem that minimization of the convex cost function by iterations is required, which increases the computational complexity. This paper proposes a new image reconstruction method, which requires less computation per iteration and realizes a fast convergence. The proposed method is based on the principle that the convex cost function is approximated in the iteration by a quadratic function, and the value of the quadratic function is reduced by SOR method, which is the solution procedure for the linear equation. The method can be called the Newton‐SOR method in the sense that both methods are combined. In order to guarantee global convergence of the Newton‐SOR method, mathematical techniques, called trust region and line search methods, are used, and the projection SOR method with the nonnegative condition is used, where a negative value appearing in the iteration is replaced by zero. The proposed method satisfies all of the desired properties of the iteration method as follows. (1) The convergence to the true solution minimizing the cost function is guaranteed. (2) The computational complexity per iteration is less, and a fast convergence is realized. (3) The method can be applied in a unified way to a wide range of convex cost functions. (4) The nonnegative condition can be handled easily. The effectiveness of the proposed method is demonstrated by a simulation experiment. The result of application to the clinical PET data is also shown. © 2003 Wiley Periodicals, Inc. Syst Comp Jpn, 34(4): 1–11, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.10178