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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Singular value decomposition and its application to numerical inversion for ray transforms in 2D vector tomography

1Sobolev Institute of Mathematics, Acad. Koptyug prosp., 4, 630090 Novosibirsk, Russia.

2Novosibirsk State University, Pirogova St., 2, 630090 Novosibirsk, Russia.

3Institute of Applied Mathematics, Saarland University, 66041 Saarbrücken, Germany.

4Department for Mathematics, Carl von Ossietzky University Oldenburg, 26129 Oldenburg, Germany.

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 19, Issue 4-5, Pages 689–715, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip.2011.047, September 2011

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The operators of longitudinal and transverse ray transforms acting on vector fields on the unit disc are considered in the paper. The goal is to construct SVD-decompositions of the operators and invert them approximately by means of truncated decomposition for the parallel scheme of data acquisition. The orthogonal bases in the initial spaces and the image spaces are constructed using harmonic, Jacobi and Gegenbauer polynomials. Based on the obtained decompositions inversion formulas are derived and the polynomial approximations for the inverse operators are obtained. Numerical tests for data sets with different noise levels of smooth and discontinuous fields show the validity of the approach for the reconstruction of solenoidal or potential parts of vector fields from their ray transforms.

Keywords.: Vector tomography; vector field; Radon transform; ray transform; singular value decomposition; orthogonal polynomials

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