Geodesic
Recovery of a vector field in the cylinder by its jointly known NMR images and ray transforms
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 86-103.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper we consider a problem of recovering a 3D vector field given in cylinder by means of jointly known nuclear magnetic resonance (NMR) images and ray transforms. The NRM images and 2D longitudinal and transverse ray transforms are known in every plane orthogonal to the cylinder axis. The 3D ray transforms of new type connected with a family of the parallel planes are defined. Simulation confirms the legitimacy and further perspective of the proposed approach.
Keywords: vector field, cylindrical domain, NMR image, ray transform, boundary value problem, numerical simulation.
Mots-clés : inversion formula 
@article{SEMR_2021_18_1_a34,
     author = {E. Yu. Derevtsov and S. V. Maltseva},
     title = {Recovery of a vector field in the cylinder by its jointly known {NMR} images and ray transforms},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {86--103},
     publisher = {mathdoc},
     volume = {18},
     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a34/}
}
TY  - JOUR
AU  - E. Yu. Derevtsov
AU  - S. V. Maltseva
TI  - Recovery of a vector field in the cylinder by its jointly known NMR images and ray transforms
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2021
SP  - 86
EP  - 103
VL  - 18
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a34/
LA  - en
ID  - SEMR_2021_18_1_a34
ER  - 
%0 Journal Article
%A E. Yu. Derevtsov
%A S. V. Maltseva
%T Recovery of a vector field in the cylinder by its jointly known NMR images and ray transforms
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2021
%P 86-103
%V 18
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a34/
%G en
%F SEMR_2021_18_1_a34
E. Yu. Derevtsov; S. V. Maltseva. Recovery of a vector field in the cylinder by its jointly known NMR images and ray transforms. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 86-103. http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a34/

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

[2] I.E. Svetov, “Reconstruction of solenoidal part of a three-dimensional vector field by its ray transforms along straight lines, parallel to the coordinate planes”, Numer. Analysis Appl., 5:3 (2012), 271–283 | Zbl

[3] I. Svetov, “Slice-by-slice numerical solution of $3D$-vector tomography problem”, Journal of Physics: Conference Series, 410 (2013), 012042

[4] I.E. Svetov, S.V. Maltseva, A.K. Louis, “The method of approximate inverse in slice-by-slice vector tomography problems”, Lecture Notes in Computer Science, 11974, 2020, 487–494 | Zbl

[5] E. Yu. Derevtsov, “Numerical solution of a problem of refractive tomography in a tube domain”, Sib. Zh. Ind. Mat., 18:4 (2015), 30–41 | Zbl

[6] S.V. Maltseva, I.E. Svetov, A.P. Polyakova, “Reconstruction of a function and its singular support in a cylinder by tomographic data”, Eurasian J. Math. and Comp. Appl., 8:2 (2020), 86–97

[7] L. Alvarez, F.P. Guichard, P.-L. Lions, J.-M. Morel, “Axioms and fundamental equations of image processing”, Arch. Rat. Mech. Anal., 123:3 (1993), 199–257 | Zbl

[8] N. Beckmann, “High resolution magnetic resonance angiography non-invasively reveals mouse strain differences in the cerebrovascular anatomy in vivo”, Magn. Reson. Med., 44:2 (2000), 252–258

[9] S.V. Maltseva, A.A. Cherevko, A.K. Khe, A.E. Akulov, A.A. Savelov, A.A. Tulupov, E. Yu. Derevtsov, M.P. Moshkin, A.P. Chupakhin, “Reconstruction of unbroken vasculature of mouse by varying the slope of the scan plane in MRI”, Journal of Physics. Conference Series, 677:1 (2016), 012003

[10] S.V. Maltseva, A.A. Cherevko, A.K. Khe, A.E. Akulov, A.A. Savelov, A.A. Tulupov, E. Yu. Derevtsov, M.P. Moshkin, A.P. Chupakhin, “Reconstruction of Complex Vasculature by Varying the Slope of the Scan Plane in High-Field Magnetic Resonance Imaging”, Applied Magnetic Resonance, 47:1 (2016), 23–39

[11] J.L. Kinsey, “Fourier transform Doppler spectroscopy: A new means of obtaining velocity-angle distributions in scattering experiments”, J. Chem. Phys., 66:6 (1977), 2560–2565

[12] S.P. Juhlin, “Doppler tomography”, Proc. 15th Annual Intern. Conf. IEEE, EMBS, 1993, 212–213

[13] E. Yu. Derevtsov, V.V. Pikalov, “Reconstruction of vector fields and their singularities from ray transform”, Numer. Analysis Appl., 4:1 (2011), 21–35 | Zbl

[14] E. Yu. Derevtsov, I.E. Svetov, “Tomography of tensor fields in the plane”, Eurasian J. Math. Comp. Applications, 3:2 (2015), 24–68

[15] N.E. Kochin, Vector calculus and the beginnings of tensor calculus, Izdat. Akad. Nauk SSSR, M., 1951 | MR

[16] H. Weyl, “The method of orthogonal projection in potential theory”, Duke Math. J., 7 (1940), 411–444 | Zbl

[17] S. Deans, The Radon Transform and Some of its Applications, John Wiley Sons, New York etc, 1983 | Zbl