Some approaches to a~reconstruction of a~singular support of scalar, vector and tensor fields by their known tomographic data
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 5 (2008), pp. 632-646.

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The problem of singular support reconstruction of scalar, vector and tensor fields by their given tomographic data is formulated in the paper. Main definitions and properties of differential and integral operators which were used for the problem of the singular support visualization are described, as well as the results of numerical simulations.
Keywords: tensor field, ray transform, back projection operator, singular support.
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E. Yu. Derevtsov. Some approaches to a~reconstruction of a~singular support of scalar, vector and tensor fields by their known tomographic data. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 5 (2008), pp. 632-646. http://geodesic.mathdoc.fr/item/SEMR_2008_5_a54/

[1] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin, Obobschennye funktsii, Integralnaya geometriya i svyazannye s nei voprosy teorii predstavlenii, vypusk 5, GIFML, M., 1962, 656 pp.

[2] I. M. Gelfand, A. B. Goncharov, “Vosstanovlenie finitnoi funktsii, iskhodya iz ee integralov po pryamym, peresekayuschim dannoe mnozhestvo tochek v prostranstve”, DAN SSSR, 290:5 (1986), 1037–1040 | MR

[3] V. P. Palamodov, “Nekotorye singulyarnye zadachi tomografii”, Matematicheskie problemy tomografii. Voprosy kibernetiki, eds. I. M. Gelfand, S. G. Gindikin, M., 1990, 132–140 | MR | Zbl

[4] E. I. Vainberg, I. A. Kazak, M. L. Faingoiz, “$X$-ray computerized back projection tomography with filtration by double differentiation. Procedure and information features”, Soviet J. Nondest. Test., 21 (1985), 106–113

[5] I. M. Gelfand, S. G. Gindikin, M. I. Graev, Izbrannye zadachi integralnoi geometrii, Dobrosvet, M., 2000, 208 pp. | MR

[6] A. Faridani, E. L. Ritman, and K. T. Smith, “Local tomography”, SIAM J. Appl. Math., 52:2 (1992), 459–484 | DOI | MR | Zbl

[7] A. Faridani, D. V. Finch, E. L. Ritman, and K. T. Smith, “Local tomography II”, SIAM J. Appl. Math., 57:4 (1997), 1095–1127 | DOI | MR | Zbl

[8] A. K. Louis, P. Maass, “Contour Reconstruction in 3-D $X$-Ray CT”, IEEE Trans. Med. Imag., 12:4 (1993), 764–769 | DOI | MR

[9] E. T. Quinto, “Singularities of the $X$-ray transform and limited data tomography in $\mathbb R^2$ and $\mathbb R^3$”, SIAM J. Math. Anal., 24 (1993), 1215–1225 | DOI | MR | Zbl

[10] A. G. Ramm, “New methods for finding discontinuities of functions from local tomographic data”, J. Inverse and Ill-Posed Problems, 5:2 (1997), 165–175 | DOI | MR

[11] A. G. Ramm and A. I. Katsevich, The Radon transform and local tomography, CRC Press, Boca Raton, 1996 | MR | Zbl

[12] S. G. Mikhlin, Mnogomernye singulyarnye integraly i integralnye uravneniya, GIFML, M., 1962, 256 pp.

[13] D. S. Anikonov, A. E. Kovtanyuk, I. V. Prokhorov, Ispolzovanie uravneniya perenosa v tomografii, Logos, M., 2000, 224 pp.

[14] D. S. Anikonov, “Spetsialnaya zadacha integralnoi geometrii”, Doklady RAN, 415:1 (2007), 7–9 | MR | Zbl

[15] E. Yu. Derevtsov, V. V. Pickalov., T. Schuster, A. K. Louis, “Reconstruction of singularities in local vector and tensor tomography”, International Conference “Inverse Problems: Modelling and Simulation”. Abstracts (May 29–June 02, 2006), Fethiye, Turkey, 2006, 38–40

[16] Derevtsov E. Yu., Pickalov V. V., Schuster T., “Application of local operators for numerical reconstruction of a singular support of a vector field by its known ray transforms”, VI International Conference on Inverse Problems in Engineering: Theory and Practice (June 15–19, 2008, Dourdan, Paris, France), Journal of Physics: Conference Series, 8 (to appear) | Zbl

[17] V. A. Sharafutdinov, Integralnaya geometriya tenzornykh polei, Nauka, Novosibirsk, 1993, 233 pp. | MR | Zbl