Proceedings of Symposia in Applied Mathematics Volume 63, 2006 An Introduction to X-ray Tomography and Radon Transforms Eric Todd Quinto ABSTRACT. This article provides an introduction to the mathematics behind X-ray tomography. After explaining the mathematical model, we will consider some of the fundamental theoretical ideas in the field, including the projection slice theorem, range theorem, inversion formula, and microlocal properties of the underlying Radon transform. We will use this microlocal analysis to predict which singularities of objects will be well reconstructed from limited tomographic data. We will introduce specific limited data problems: the exte- rior problem, region of interest tomography, and limited angle region of interest tomography, and we use some of the author's reconstructions for these prob- lems to illustrate the microlocal predictions about singularities. The appendix includes proofs of the basic microlocal properties of the Radon transform. Our overarching goal is to show some of the ways integral geometry and microlocal analysis can help one understand limited data tomography. 1. Introduction The goal of tomography is to recover the interior structure of a body using external measurements, and tomography is based on deep pure mathematics and numerical analysis as well as physics and engineering. In this article, we will in- troduce some of the fundamental mathematical concepts in X-ray tomography and microlocal analysis and apply them to limited data problems. In the process, we will outline how the problems come up in practice and show what the microlocal 2000 Mathematics Subject Classification. Primary: 92C55, 44A12 Secondary: 35S30, 58J40. Key words and phrases. Tomography, Radon Transform, Microlocal Analysis. I am indebted to many researchers including, but not limited to the following people. Allan Cormack was a mentor to me, and he introduced me to the field of tomography and the exterior problem (§3.1). Allan and I enjoyed solving pure mathematical problems, too. Carlos Berenstein, Larry Shepp, and Larry Zalcman helped me as I began my career. Frank Natterer and Alfred Louis have been invaluable as I learned more, showing how deep analysis and numerical work go hand in hand. I learned about and appreciated Lambda CT from Kennan Smith and Adel Faridani. Victor Guillemin and Sig Helgason gave me a love of microlocal analysis and integral geometry. Peter Kuchment provided very helpful comments on material that appeared in a related article [38], and Matthias Hahn and Gestur Olafsson corrected some misprints. I thank these folks, the other speakers in this short course, and other friends for making it all fun. This research is based upon work supported by the National Science Foundation under grants DMS-0200788 and DMS-0456858 and Tufts University FRAC. ©2006 American Mathematical Society 1
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