Gel’fand–Graev’s reconstruction formula in the 3D real space

摘要

Purpose: Gel’fand and Graev performed classical work on the inversion of integral transforms in different spaces [Gel’fand and Graev, Funct. Anal. Appl. 25(1) 1–5 (1991)]. This paper discusses their key results for further research and development.Methods: The Gel’fand–Graev inversion formula reveals a fundamental relationship between projection data and the Hilbert transform of an image to be reconstructed. This differential backprojection (DBP)∕backprojection filtration (BPF) approach was rediscovered in the CT field, and applied in important applications such as reconstruction from truncated projections, interior tomography, and limited-angle tomography. Here the authors present the Gel’fand–Graev inversion formula in a 3D setting assuming the 1D x-ray transform.Results: The pseudodifferential operator is a powerful theoretical tool. There is a fundamental mathematical link between the Gel’fand–Graev formula and the DBP (or BPF) approach in the case of the 1D x-ray transform in a 3D real space.Conclusions: This paper shows the power of mathematics for tomographic imaging and the value of a pure theoretical finding, which may appear quite irrelevant to daily healthcare at the first glance.