The ray transform of symmetric tensor fields with incomplete projection data, I: The kernel of the ray transform

V. A. Sharafutdinov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 1219-1237.

Résumé

We consider the ray transform $I_\Gamma$ that integrates symmetric rank $m$ tensor fields on ${\mathbb{R}}^n$ supported in a bounded convex domain $D\subset{\mathbb{R}}^n$ over lines. The integrals are known for the family $\Gamma$ of lines $l$ such that endpoints of the segment $l\cap D$ belong to a given part $\gamma=\partial D\cap{\mathbb{R}}^n_+$ of the boundary, for some half-space ${\mathbb{R}}^n_+\subset{\mathbb{R}}^n$. We prove that the kernel of the operator $I_\Gamma$ coincides with the space of $\gamma$-potential tensor fields.

Détail

Keywords: tomography with incomplete data, ray transform, tensor analysis.

Bibliographie

[1] A.S. Blagoveshchenskii, “Reconstruction of a function from known integrals of it taken along linear manifolds”, Mat. Notes, 39 (1986), 457–461 | DOI | Zbl

[2] A.S. Denisyuk, “Inversion of the generalized Radon transform”, Am. Math. Soc. Trans., 162 (1994), 19–32 | Zbl

[3] 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

[4] I.M. Gel'fand, S.G. Gindikin, M.I. Graev, “Integral geometry in affine and projective spaces”, J. Sov. Math., 18 (1982), 39–167 | DOI | Zbl

[5] A. Greenleaf, G. Uhlmann, “Nonlocal inversion formulas for the X-ray transform”, Duke Math. J., 58:1 (1989), 205–240 | DOI | Zbl

[6] S. Helgason, The Radon transform, 2nd ed., Birkhäuser, Boston, 1999 | Zbl

[7] A.A. Kirillov, “A problem of I.M. Gel'fand”, Sov. Math. Dokl., 2 (1961), 268–269 | MR

[8] V. Krishnan, P. Stefanov, “A support theorem for the geodesic ray transform of symmetric tensor fields”, Inverse Probl. Imaging, 3:3 (2009), 453–464 | DOI | Zbl

[9] W. Lionheart, V. Sharafutdinov, “Reconstruction algorithm for the linearized polarization tomography problem with incomplete data”, Imaging microstructures. Mathematical and computational challenges, Proceedings of a research conference (Paris, France, June 18–20, 2008), Contemporary Mathematics, 494, ed. Ammari Habib et al., 2009, 137–159 | DOI | Zbl

[10] F. Natterer, The mathematics of computerized tomography, Gohn Wiley Sons, Chichester etc, 1986 | Zbl

[11] V.P. Palamodov, Reconstructive integral geometry, Monographs in Mathematics, 98, Birkhäuser, Basel, 2004 | Zbl

[12] K. Ramaseshan, “Microlocal analysis of the Doppler transform on ${\mathbb R}^3$”, J. Fourier Anal. Appl., 10:1 (2004), 73–82 | DOI | Zbl

[13] V.A. Sharafutdinov, Integral geometry for tensor fields, VSP, Utrecht, 1994 | Zbl

[14] V.A. Sharafutdinov, “Integral geometry of a tensor fields on a surface of revolution”, Sib. Math. J., 38:3 (1997), 603–620 | DOI | Zbl

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

[16] T. Shuster, “The 3D Doppler transform: Elementary properties and computation of reconstruction kernels”, Inverse Probl., 16:3 (2000), 701–722 | DOI | Zbl

[17] P. Stefanov, G. Uhlmann, A. Vasy, “Inverting the local geodesic X-ray transform on tensors”, J. Anal. Math., 136:1 (2018), 151–208 | DOI | Zbl

[18] H.K. Tuy, “An inversion formula for cone-beam reconstruction”, SIAM J. Appl. Math., 43 (1983), 546–552 | DOI | Zbl

[19] L. Vertgeim, “Integral geometry problems for symmetric tensor field with incomplete data”, J. Inverse Ill-posed Problems, 8:3 (2000), 355–364 | DOI | Zbl