Computed tomography (CT) entails the reconstruction of a function f from line integrals of f. This mathematical problem is encountered in a growing number of diverse settings in medicine, science, and technology. This introductory article is divided into three parts. The first part is concerned with general theory and explores questions of uniqueness, stability and inversion, as well as detection of singularities. The second part is devoted to local tomography and is centered around a discussion of recently developed methods for computing jumps of a function from local tomographic data. The third part treats optimal sampling and has at its core a detailed error analysis of the parallel-beam filtered backprojection algorithm. Matlab source code for the filtered backprojection algorithm and the Feldkamp-Davis-Kress algorithm is included in an appendix.

1. Introduction

Computed tomography (CT) entails the reconstruction of a function f from line integrals of f. This mathematical problem is encountered in a growing number of diverse settings in medicine, science, and technology, ranging from the famous application in diagnostic radiology to research in quantum optics. As a consequence, many aspects of CT have been extensively studied and are now well understood, thus providing an interesting model case for the study of other inverse problems. Other aspects, notably three-dimensional reconstructions, still provide numerous open problems.

The purpose of this article is to give an introduction to the topic, treat some aspects in more detail, and to point out references for further study. The reader interested in a broader overview of the field, its relation to various branches of pure and applied mathematics, and its development over the years may wish to consult the monographs [6; 31; 32; 36; 62; 67; 78], the volumes [21; 22; 28; 33; 34; 76; 77], and review articles [42; 49; 56; 58; 66; 84; 89].