INVERSION OF k-PLANE TRANSFORMS AND APPLICATIONS IN COMPUTER TOMOGRAPHY∗

The mathematics behind Computerized Tomography (CT) is based on the study of the parallel beam transform P and the divergent beam transform D. Both of these map a function f in Rn into a function defined on the set of all lines in Rn, by integrating f along these lines. The parallel and divergent k-plane transforms are defined in a similar fashion by integration over k-planes (i.e., translates of k-dimensional subspaces) and are also denoted P and D, respectively. A related transform is the spherical k-plane transform S, which maps a function f on the sphere Sn−1 into its integrals over k-dimensional great circles. This paper discusses the properties of the k-plane transforms P , D, and S and their inverses, emphasizing relationships and similarities between the operators, and their relation to CT. Some new results are included. Most notable are more general conditions under which the inversion formulas hold.