In the preceding sections we have given the fundamentals of the most widely used algorithms in tomography. In many ways these fundamentals are quite different from traditional numerical analysis, the main difference being the consistent use of sampling theory and Fourier analysis. The developement of algorithms is still very lively, in particular in 3D and in Fourier based algorithms.

We dealt only with the most simple problems and with standard situations. Practical problems deviate in many ways from the simple ones we considered. Often the data is incomplete (see Louis (1980)), leading to non uniqueness and instability. Sometimes the integral equation to be solved are not completely specified, be it that the weight function (as in emission tomography, Welch et al. (1998)) or the directions (as in electron microscopy, Wuschke (1990), Gelfand and Goncharov (1990)) are unknown. In particular in technical applications, the number of data is often so small (see e.g. Sielschott and Derichs (1995)) that full reconstruction is impossible and special algorithms have to be developed, usually taylored to the specific application. Sometimes only certain features of the object, such as boundaries between regions of different densities are searched for (Faridani et al. (1997)), Ramm and Katsevich (1996)), calling for special algorithms.

At present we have an adequate understanding of the fundamentals of tomographic reconstruction algorithm. However, new applications of tomography are coming up almost daily, each one presenting new challenges to the numerical analyst. So I guess that research in this field will go on for ever.

Thu Sep 10 10:51:17 MET DST 1998