Numerical solution of reconstruction problem of a potential symmetric 2-tensor field in a ball from its normal Radon transform

A. P. Polyakova ; I. E. Svetov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 154-174.

Résumé

We propose a numerical solution of reconstruction problem of a potential symmetric 2-tensor field in a ball from the known values of the normal Radon transform. The algorithm is based on the method of truncated singular value decomposition. Numerical simulations confirm that the proposed method yields good results of reconstruction of potential symmetric 2-tensor fields.

Détail

Keywords: tensor tomography, potential symmetric 2-tensor field, operator of inner differentiation, approximation, truncated singular value decomposition
Mots-clés : normal Radon transform, orthogonal polynomials.

Bibliographie

[1] Yu. A. Kravtsov, Yu. I. Orlov, Geometrical optics of inhomogeneous media, Wave phenomena, 6, Springer, 1990 | DOI | MR

[2] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, 1994 | MR

[3] R. Novikov, V. Sharafutdinov, “On the problem of polarization tomography, I”, Inverse Problems, 23:3 (2007), 1229–1257 | DOI | MR | Zbl

[4] V. A. Sharafutdinov, “The problem of polarization tomography, II”, Inverse Problems, 24:3 (2008), 035010 | DOI | MR | Zbl

[5] V. V. Pikalov, T. S. Melnikova, Plasma tomography, Nauka, Novosibirsk, 1995 (in Russian)

[6] H. K. Aben, Integral photoelasticity, Valgus, Tallin, 1975 (in Russian)

[7] H. Aben, A. Puro, “Photoelastic tomography for three-dimensional flow birefringence studies”, Inverse Problems, 13:2 (1997), 215–221 | DOI | MR | Zbl

[8] A. Puro, “Magneto-photoelasticity as parametric tensor field tomography”, Inverse problems, 14:5 (1998), 1315–1330 | DOI | Zbl

[9] V. Sharafutdinov, J. Wang, “Tomography of small residual stresses”, Inverse Problems, 28:6 (2012), 065017 | DOI | MR | Zbl

[10] W. Lionheart, P. Withers, “Diffraction tomography of strain”, Inverse Problems, 31:4 (2015), 045005 | DOI | MR | Zbl

[11] V. P. Karassiov, “Polarization Tomography of Quantum Radiation: Theoretical Aspects and Operator Approach”, Theoretical and Mathematical Physics, 145:3 (2005), 1666–1677 | DOI | MR | Zbl

[12] V. Y. Panin, G. L. Zeng, M. Defrise, G. T. Gullberg, “Diffusion tensor MR imaging of principal directions: A tensor tomography approach”, Phys. Med. Biol., 47 (2002), 2737–2757 | DOI

[13] V. M. Gelikonov, G. V. Gelikonov, “New approach to crosspolarized optical coherence tomography based on orthogonal arbitrarily polarized modes”, Laser Phys. Lett., 3:9 (2006), 445–451 | DOI

[14] E. Y. Derevtsov, “An approach to direct reconstruction of a solenoidal part in vector and tensor tomography problems”, Journal of Inverse and Ill-Posed Problems, 13:3–6 (2005), 213–246 | DOI | MR | Zbl

[15] I. E. Svetov, A. P. Polyakova, “Reconstruction of 2-Tensor Fields, Given in a Unit Circle, by Their Ray Transform Based on LSM with $B$-Splines”, Numerical Analysis and Applications, 3:2 (2010), 151–164 | DOI | Zbl

[16] I. E. Svetov, “Reconstruction of solenoidal 2-tensor fields, given in a unit disk, in their longitudinal ray transforms”, Proceedings of first international youth school-conference “Theory and numerical methods of inverse and ill-conditioned problems solving”, Part I, Siberian Electronic Mathematical Reports, 7, 2010, 139–149 (in Russian)

[17] A. Denisjuk, “Inversion of the X-ray transform for 3D symmetric tensor fields with sources on a curve”, Inverse Problems, 22:2 (2006), 399–411 | DOI | MR | Zbl

[18] V. Sharafutdinov, “Slice-by-slice reconstruction algorithm for vector tomography with incomplete data”, Inverse Problems, 23:6 (2007), 2603–2627 | DOI | MR | Zbl

[19] M. Defrise, G. T. Gullberg, $3D$ reconstruction of tensors and vectors, Technical Report No LBNL-54936, LBNL, Berkeley, 2005

[20] A. K. Louis, “Orthogonal function series expansions and the null space of the Radon transform”, Society for industrial and applied mathematics, 15:3 (1984), 621–633 | MR | Zbl

[21] P. Maass, “The X-ray transform: Singular value decomposition and resolution”, Inverse problems, 3:4 (1987), 727–741 | DOI | MR

[22] E. Yu. Derevtsov, A. V. Efimov, A. K. Louis, T. Schuster, “Singular value decomposition and its application to numerical inversion for ray transforms in 2D vector tomography”, Journal of Inverse and Ill-Posed Problems, 19:4–5 (2011), 689–715 | MR | Zbl

[23] E. Yu. Derevtsov, A. P. Polyakova, “Solution of the Integral Geometry Problem for 2-Tensor Fields by the Singular Value Decomposition Method”, Journal of Mathematical Sciences, 202:1 (2014), 50–71 | DOI | MR

[24] Polyakova A. P., “Reconstruction of a vector field in a ball from its normal Radon transform”, Journal of Mathematical Sciences, 205:3 (2015), 418–439 | DOI | MR

[25] I. E. Svetov, A. P. Polyakova, “Comparison of two algorithms for the numerical solution of the two-dimensional vector tomography”, Siberian Electronic Mathematical Reports, 10 (2013), 90–108 (in Russian) | MR | Zbl

[26] I. E. Svetov, A. P. Polyakova, “Approximate solution of two-dimensional 2-tensor tomography problem using truncated singular value decomposition”, Siberian Electronic Mathematical Reports, 12 (2015), 480–499 (in Russian)

[27] Polyakova A. P., Svetov I. E., “Numerical Solution of the Problem of Reconstructing a Potential Vector Field in the Unit Ball from Its Normal Radon Transform”, Journal of Applied and Industrial Mathematics, 9:4 (2015), 547–558 | DOI

[28] S. Deans, The Radon Transform and Some of its Applications, Wiley, New York, 1983 | MR | Zbl

[29] V. A. Tcheverda, V. I. Kostin, “r-pseudoinverse for compact operators in Hilbert spaces: existence and stability”, Journal of Inverse and Ill-Posed Problems, 3:2 (1995), 131–148 | MR

[30] V. A. Tcheverda, V. I. Kostin, “r-pseudoinverse for compact operator”, Proceedings of first international youth school-conference “Theory and numerical methods of inverse and ill-conditioned problems solving”, Part I, Siberian Electronic Mathematical Reports, 7, 2010, 258–282 (in Russian)