Finite series-expansion reconstruction methods

TLDR

Series‑expansion reconstruction methods, first introduced in the 1970s, are algebraic/iterative optimization techniques that discretize the image domain before analysis and differ fundamentally from transform‑based approaches, yet many questions remain unanswered. The paper aims to explain the model setup, methodology, and role of optimization theory in series‑expansion reconstruction algorithms, and to justify their study despite the speed advantage of transform methods. The paper answers the posed questions about model setup, methodology, optimization theory, algorithm design, and the rationale for studying series‑expansion methods.

Abstract

Series-expansion reconstruction methods made their first appearance in the scientific literature and in the CT scanner industry around 1970. Great research efforts have gone into them since but many questions still wait to be answered. These methods, synonymously known as algebraic methods, iterative algorithms, or optimization theory techniques, are based on the discretization of the image domain prior to any mathematical analysis and thus are rooted in a completely different branch of mathematics than the transform methods which are discussed in this issue by Lewitt [51]. How is the model set up? What is the methodology of the approach? Where does mathematical optimization theory enter? What do these reconstruction algorithms look like? How are quadratic optimization, entropy optimization, and Bayesian analysis used in image reconstruction? Finally, why study series expansion methods if transform methods are so much faster? These are some of the questions that are answered in this paper.