Gel'fand-Graev's reconstruction formula in the 3D real space

Abstract

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. © 2011 American Association of Physicists in Medicine.