Mathematics for computer tomography

Computerized tomography requires not only fast computers, bur also analysis of mathematical models and construction of numerical algorithms. Classical mathematical theory is combined with modem numerical analysis to form the basis for efficient implementation on fast computers. The solution of the inverse problem of finding the image from given X-ray projections is theoretically obtained by the inverse Radon transform. Since only a finite number of projections are available, some approximation must be found, and this leads to a discrete counterpart of the continuous problem. There are three major classes of numerical solution methods: the Algebraic Reconstruction Method, the Filtered Back projection Method and the Direct Fourier Method. Much research is devoted to making the methods faster and more robust. The first one was used for the original tomography machine, the second one is used on almost all current machines in use. The third one has great potential for the future, since almost ail computation is done by using the fast discrete Fourier transform. We shall describe the basic mathematical problem in computer tomography and the computational methods mentioned above for solving it. In particular we shall emphasize the special difficulties that are built into the problem. However, this is not a review article. Instead, it is intended to describe the influence of modern numerical methods on a fundamental problem of great significance for the society. We shall also indicate how computerized tomography has initiated new important research in central fields of numerical analysis, that can be used for problems in many other applications.