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

Yangbo Ye; Hengyong Yu; Ge Wang

doi:10.1118/1.3577765

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

Journal article

Open access

Peer reviewed

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

Yangbo Ye, Hengyong Yu and Ge Wang

Medical physics (Lancaster), Vol.38(S1), pp.S69-S75

07/20/2011

DOI: 10.1118/1.3577765

PMCID: PMC3172127

PMID: 21978119

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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.

backprojection filtration (BPF)

computed tomography (CT)

differential backprojection (DBP)

image reconstruction

pseudo-differential operator

x-ray transform

Details

Title: Subtitle: Gel'fand-Graev's reconstruction formula in the 3D real space
Creators: Yangbo Ye - University of Iowa
Hengyong Yu - Wake Forest University
Ge Wang - Wake Forest University
Resource Type: Journal article
Publication Details: Medical physics (Lancaster), Vol.38(S1), pp.S69-S75
Publisher: American Association of Physicists in Medicine
DOI: 10.1118/1.3577765
PMID: 21978119
PMCID: PMC3172127
ISSN: 0094-2405
eISSN: 2473-4209
Grant note: EB009275 and EB011785 / UNSPECIFIED
Date published: 07/20/2011
Academic Unit: Mathematics
Record Identifier: 9984241148302771

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