MATHEMATICS FOR COMPUTER-TOMOGRAPHY

Computerized tomography requires not only fast computers, bur also ana lysis of mathematical models and construction of numerical algorithms. Classical mathematical theory is combined with modem numerical analys is to form the basis for efficient implementation on fast computers. T he solution of the inverse problem of finding the image from given X-r ay projections is theoretically obtained by the inverse Radon transfor m. Since only a finite number of projections are available, some appro ximation must be found, and this leads to a discrete counterpart of th e continuous problem. There are three major classes of numerical solut ion methods: the Algebraic Reconstruction Method, the Filtered Back pr ojection Method and the Direct Fourier Method. Much research is devote d 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 th e future, since almost ail computation is done by using the fast discr ete Fourier transform. We shall describe the basic mathematical proble m in computer tomography and the computational methods mentioned above for solving it. In particular we shall emphasize the special difficul ties that are built into the problem. However, this is not a review ar ticle. Instead, it is intended to describe the influence of modern num erical methods on a fundamental problem of great significance for the society. We shall also indicate how computerized tomography has initia ted new important research in central fields of numerical analysis, th at can be used for problems in many other applications.