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

TL;DR

This paper introduces a novel method using range conditions on distributions, specifically Dirac distributions, to improve geometric calibration in 2D tomography systems, especially under data truncation conditions.

Contribution

It develops new range conditions on distributions for tomography and applies them to address data truncation in geometric calibration of 2D parallel and fan-beam geometries.

Findings

Effective calibration formulas derived for truncated data

Applicable to both parallel and fan-beam geometries

Demonstrated potential for improved self-calibration methods

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.