Function Reconstruction from Reflection Symmetric Radon Data

Abstract

In many areas of two-dimensional imaging science, data acquisition arises as an integral process. The inverse process—or image reconstruction—means the solving of a Radon problem mathematically. It may happen that there exists classes of integral data which are mirror symmetric with respect to a line. Common sense suggests that the occurrence of a symmetry usually provides significant help in the search of the problem solution. Here, we showed an example of the contrary to this popular belief. In fact, to solve such a Radon problem with inherent reflection symmetry, there is a need to split it into two new Radon problems on half-spaces, which do not have solutions at hand. In this paper, a full solution is obtained via geometric inversion mapping of the two original half-spaces Radon problems to the disk interior/exterior Radon problem arising in recent modalities of Compton scattering tomography, which fortunately has explicit worked out inverse formulas.