Range conditions on distributions and their possible application to geometric calibration in 2D parallel and fan-beam geometries

Abstract: In tomography, range conditions or data consistency conditions (DCCs) on functions have proven useful for geometric self-calibration, which involves identifying geometric parameters of acquisition systems based only on acquired radiographic images. These self-calibration methods using range conditions on functions typically require non-truncated data. In this work, we derive range conditions on distributions and demonstrate their application in addressing data truncation issues during the calibration process. We propose a novel approach based on range conditions on distributions, employing Dirac distributions to model markers within the field-of-view of an X-ray system. Our calibration methods are based on the local geometric information from non-truncated projections of a marker set. By applying range conditions to projections of sums of Dirac distributions, combined with specific calibration marker sets, we derive analytical formulas that enable the identification of geometric calibration parameters. We aim to present DCCs on distributions in tomography and explore the potential of DCCs on distributions as a possible tool in calibration. This approach represents one possible application, demonstrating how DCCs on distributions can effectively address challenges such as data truncation and incomplete marker set information. We present results for the 2D parallel geometry (Radon transform) and the 2D fan-beam geometry with sources on a line.