Geodesic
Properties of the ray transforms of two-dimensional $2$-tensor fields given in the unit disk
Sibirskij žurnal industrialʹnoj matematiki, Tome 16 (2013) no. 4, pp. 121-130.

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

We study the longitudinal, transverse, and mixed ray transforms acting on two-dimensional symmetric $2$-tensor fields. Namely, the kernels of the ray transforms are described; the connection between the ray transforms and the Radon transform is established; unconditional estimates of stability for each of the ray transforms are obtained; inversion formulas for a recovery of symmetric $2$-tensor field components and for a recovery of the potential are obtained; projection theorems for the ray transforms are proved.
Keywords: integral geometry, symmetric $2$-tensor field, solenoidal field, potential field, longitudinal ray transform, mixed ray transform, estimate of stability, projection theorem.
Mots-clés : transverse ray transform, inversion formula 
@article{SJIM_2013_16_4_a10,
     author = {I. E. Svetov},
     title = {Properties of the ray transforms of two-dimensional $2$-tensor fields given in the unit disk},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {121--130},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2013_16_4_a10/}
}
TY  - JOUR
AU  - I. E. Svetov
TI  - Properties of the ray transforms of two-dimensional $2$-tensor fields given in the unit disk
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2013
SP  - 121
EP  - 130
VL  - 16
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2013_16_4_a10/
LA  - ru
ID  - SJIM_2013_16_4_a10
ER  - 
%0 Journal Article
%A I. E. Svetov
%T Properties of the ray transforms of two-dimensional $2$-tensor fields given in the unit disk
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2013
%P 121-130
%V 16
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2013_16_4_a10/
%G ru
%F SJIM_2013_16_4_a10
I. E. Svetov. Properties of the ray transforms of two-dimensional $2$-tensor fields given in the unit disk. Sibirskij žurnal industrialʹnoj matematiki, Tome 16 (2013) no. 4, pp. 121-130. http://geodesic.mathdoc.fr/item/SJIM_2013_16_4_a10/

[1] Romanov V. G., “Integralnaya geometriya na geodezicheskikh izotropnoi rimanovoi metriki”, Dokl. AN SSSR, 241:2 (1978), 290–293 | MR | Zbl

[2] Romanov V. G., Obratnye zadachi matematicheskoi fiziki, Nauka, M., 1984 | MR

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

[4] Deans S. R., The Radon Transform and Some of its Applications, John Wiley Sons, N.Y., 1983 | MR | Zbl

[5] 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”, Mathematics and Computers in Simulation, 94 (2013), 15–32 | DOI | MR

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

[7] Kochin N. E., Vektornoe ischislenie i nachala tenzornogo ischisleniya, Nauka, M., 1965

[8] Natterer F., Matematicheskie aspekty kompyuternoi tomografii, Mir, M., 1990 | MR | Zbl