Abstract:
The problem of recovering of the vector field, which is defined in the ball, by its known normal Radon transform, which is an integral along the planes of the projection of the vector field on the normal to the plane. It is shown that solenoidal fields, which are tangential on the boundary of the ball, are formed the core of the normal Radon transform. It is therefore possible to recover only potential part of the vector field. In this paper, for the subspace of potential fields with the potentials, which are equal to zero at the boundary, an orthogonal basis is constructed and normal Radon transform of these basic vector functions is calculated. The result is a singular value decomposition of the normal Radon transform in this space. The resulting decomposition can be used as a basis for the numerical solution of the problem of recovery of potential part of vector field on the assumption that the harmonic part of the original vector field is absent.
Keywords:
vector tomography, potential field, normal Radon transform, singular value decomposition, orthogonal polynomials.
Citation:
A. P. Polyakova, “Reconstruction of vector field which given in ball by its known the normal Radon transform”, Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 13:4 (2013), 119–142; J. Math. Sci., 205:3 (2015), 418–439
\Bibitem{Pol13}
\by A.~P.~Polyakova
\paper Reconstruction of vector field which given in ball by its known the normal Radon transform
\jour Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform.
\yr 2013
\vol 13
\issue 4
\pages 119--142
\mathnet{http://mi.mathnet.ru/vngu319}
\transl
\jour J. Math. Sci.
\yr 2015
\vol 205
\issue 3
\pages 418--439
\crossref{https://doi.org/10.1007/s10958-015-2256-1}
Linking options:
https://www.mathnet.ru/eng/vngu319
https://www.mathnet.ru/eng/vngu/v13/i4/p119
This publication is cited in the following 13 articles:
Ivan E Svetov, Anna P Polyakova, “Inversion of generalized Radon transforms acting on 3D vector and symmetric tensor fields”, Inverse Problems, 40:1 (2024), 015009
Alfred K Louis, “A unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality”, Inverse Problems, 40:8 (2024), 085007
I. E. Svetov, A. P. Polyakova, “Decomposition of symmetric tensor fields in $\mathbb{R}^3$”, J. Appl. Industr. Math., 17:1 (2023), 199–212
R. Sathesh Raaj, “Breast cancer detection and diagnosis using hybrid deep learning architecture”, Biomedical Signal Processing and Control, 82 (2023), 104558
L Kunyansky, E McDugald, B Shearer, “Weighted Radon transforms of vector fields, with applications to magnetoacoustoelectric tomography”, Inverse Problems, 39:6 (2023), 065014
I. E. Svetov, A. P. Polyakova, “Reconstruction of three-dimensional vector fields based on values of normal, longitudinal, and weighted Radon transforms”, J. Appl. Industr. Math., 17:4 (2023), 842–858
Anna P. Polyakova, Ivan E. Svetov, “A numerical solution of the dynamic vector tomography problem using the truncated singular value decomposition method”, Journal of Inverse and Ill-posed Problems, 2022
A P Polyakova, I E Svetov, “On a singular value decomposition of the normal Radon transform operator acting on 3D 2-tensor fields”, J. Phys.: Conf. Ser., 1715:1 (2021), 012041
I E Svetov, A P Polyakova, “The method of approximate inverse for the normal Radon transform operator”, J. Phys.: Conf. Ser., 1715:1 (2021), 012048
Anna P. Polyakova, Ivan E. Svetov, Bernadette N. Hahn, Lecture Notes in Computer Science, 11974, Numerical Computations: Theory and Algorithms, 2020, 446
Ivan E. Svetov, Svetlana V. Maltseva, Alfred K. Louis, Lecture Notes in Computer Science, 11974, Numerical Computations: Theory and Algorithms, 2020, 487
Sarah Leweke, Volker Michel, Naomi Schneider, “Vectorial Slepian Functions on the Ball”, Numerical Functional Analysis and Optimization, 39:11 (2018), 1120
A. P. Polyakova, I. E. Svetov, “Numerical solution of reconstruction problem of a potential vector field in a ball from its normal Radon transform”, J. Appl. Industr. Math., 9:4 (2015), 547–558
Citation in format AMSBIB
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