Geodesic
Reconstruction of three-dimensional vector fields based on values of normal, longitudinal, and weighted Radon transforms
Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 4, pp. 125-142.

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The paper considers the vector tomography problem of reconstructing a three-dimensional vector field based on the values of unweighted (normal and longitudinal) and weighted Radon transforms. Using the detailed decomposition of vector fields obtained in the paper, connections are established between the unweighted and weighted Radon transforms acting on vector fields and the Radon transform acting on functions. In particular, the kernels of tomographic integral operators acting on vector fields are described. Some statements of tomography problems for the reconstruction of vector fields are considered, and inversion formulas for their solution are obtained.
Keywords: vector tomography, decomposition of vector field, weighted Radon transform
Mots-clés : normal Radon transform, longitudinal Radon transform, inversion formula. 
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I. E. Svetov; A. P. Polyakova. Reconstruction of three-dimensional vector fields based on values of normal, longitudinal, and weighted Radon transforms. Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 4, pp. 125-142. http://geodesic.mathdoc.fr/item/SJIM_2023_26_4_a8/

[1] Ludwig D., “The Radon transform on Euclidean space”, Commun. Pure Appl. Math., 19 (1966), 49–81 | DOI | MR | Zbl

[2] Helgason S., The Radon Transform, Progress in Mathematics, 5, Springer, N. Y., 1999 | DOI | MR

[3] Louis A. K., “Uncertainty, ghosts, and resolution in Radon problems”, The Radon Transform, 2019, 169–188 | DOI | MR | Zbl

[4] Natterer F., The Mathematics of Computerized Tomography, Wiley, Chichester, 1986 | DOI | MR | MR | Zbl

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

[6] Sparr G., Strahlen K., Lindstrem K., Persson H. W., “Doppler tomography for vector fields”, Inverse Probl., 11:5 (1995), 1051–1061 | DOI | MR | Zbl

[7] Schuster T., “20 years of imaging in vector field tomography: a review”, Mathematical Methods in Biomedical Imaging and Intensity-Modulated Dariation Therapy (IMRT), 2008, 389–424 | MR

[8] 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 | Zbl

[9] I. E. Svetov, “Approximate inversion method for the Radon transform operators of functions and the normal Radon transform of vector and symmetric 2-tensor fields in $\mathbb{R}^3$”, Sib. Elektron. Mat. Izv., 17 (2020), 1073–1087 (in Russian) | DOI | MR | Zbl

[10] Kunyansky L., “A mathematical model and inversion procedure for magneto-acousto-electric tomography”, Inverse Probl., 28:3 (2012), 035002 | DOI | MR | Zbl

[11] Ammari H., Grasland-Mongrain P., Millien P., Seppecher L, Seo J.-K., “A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography”, J. Math. Pures Appl., 103:6 (2015), 1390–1409 | DOI | MR | Zbl

[12] Kunyansky L., McDugald E., Shearer B., “Weighted Radon transforms of vector fields, with applications to magnetoacoustoelectric tomography”, Inverse Problems, 39:6 (2023), 065014 | DOI | MR | Zbl

[13] Derevtsov E. Yu., Svetov I. E., “Tomography of tensor fields in the plain”, Eurasian J. Math. Comput. Appl., 3:2 (2015), 24–68

[14] Louis A. K., “Inversion formulae for ray transforms in vector and tensor tomography”, Inverse Probl., 38:6 (2022), 065008 | DOI | MR | Zbl

[15] Borchers W., Sohr H., “On the equations $\mathrm{rot}\, v = g$ and $\mathrm{diver}\, u = f$ with zero boundary conditions”, Hokkaido Math. J., 19 (1990), 67–87 | DOI | MR | Zbl

[16] Backus G. E., “Poloidal and toroidal fields in geomagnetic field modeling”, Rev. Geophys., 24:1 (1986), 75–109 | DOI | MR

[17] Kazantsev S. G., Kardakov V. B., “Poloidal-Toroidal Decomposition of Solenoidal Vector Fields in the Ball”, Journal of Applied and Industrial Mathematics, 13:3 (2019), 480–499 | DOI | DOI | MR | Zbl

[18] I. E. Svetov and A. P. Polyakova, “Decomposition of symmetric tensor fields in $\mathbb{R}^3$”, Sib. Zh. Ind. Mat., 26:1 (2023), 161–178 | DOI | MR

[19] Weyl N., “The method of ortogonal projection in potential theory”, Duke Math. J., 7:1 (1940), 411–444 | DOI | MR | Zbl

[20] Svetov I. E., Derevtsov E. Yu., Volkov Yu. S., Schuster T., “A numerical solver based on $B$-splines for 2D vector field tomography in a refracting medium”, Math. Comput. Simul., 97 (2014), 207–223 | DOI | MR | Zbl