We consider the integral geometry problem of finding a symmetric 2-tensor field in a unit disk provided that the ray transforms of this field are known. We construct singular value decompositions of the operators of longitudinal, transversal, and mixed ray transforms that are the integrals of projections of a field onto the line where they are computed. We essentially use the results on decomposition of tensor fields and their representation in terms of potentials. The singular value decompositions are constructive and can be used for creating an algorithm for recovering a tensor field from its known ray characteristics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. E. Davison, “A singular value decomposition for the Radon transform in n-dimensional Euclidean space,” Numer. Funct. Anal. Optimization 3, 321–340 (1981).
A. K. Louis, “Orthogonal function series expansions and the null space of the Radon transform,” SIAM J. Math. Anal. 15, 621–633 (1984).
E. T. Quinto, “Singular value decomposition and inversion methods for the exterior Radon transform and a spherical transform,” J. Math. Anal. Appl. 95, 437–448 (1985).
P. Maass, “The X-ray transform: singular value decomposition and resolution,” Inverse Probl. 3, 727–741 (1987).
A. K. Louis, “Incomplete data problems in X-ray computerized tomography. I: Singular value decomposition of the limited angle transform,” Numer. Math. 48, 251–262 (1986).
E. Yu. Derevtsov, S. G. Kazantsev, and T. Schuster, “Polynomial bases for subspaces of vector fields in the unit ball. Method of ridge functions,” J. Inverse Ill-Posed Probl. 15, No. 1, 1–38 (2007).
E. Yu. Derevtsov, A. V. Efimov, A. K. Louis, and T. Schuster, “Singular value decomposition and its application to numerical inversion for ray transforms in 2D vector tomography,” J. Inverse Ill-Posed Probl. 19, No. 4–5, 689–715 (2011).
S. G. Kazantsev and A. A. Bukhgeim, “Singular value decomposition for the 2D fan-beam radon transform of tensor fields,” J. Inverse Ill-Posed Probl. 12, No. 4, 1–35 (2004).
N. E. Kochin, Vector Calculus and Foundations of Tensor Calculus [in Russian], ONTI, Moscow etc. (1934).
H. Weyl, “The method of orthogonal projection in potential theory,” Duke Math. J., No. 7, 411–444 (1940).
E. Yu. Derevtsov, “An approach to direct reconstruction of a solenoidal part in vector and tensor tomography problems,” J. Inverse Ill-Posed Probl. 13, No. 3–6, 213–246 (2005).
E. Yu. Derevtsov, “Some problems of nonscalar tomography” [in Russian] In: Proceedings of Conferences, pp. 81–111, Siberian. Elect. Math. Reports 7 (2010).
V. A. Sharafutdinov, Integral Geometry for Tensor Fields [in Russian], Nauka, Novosibirsk (1993); English transl.: VSP, Utrecht (1994).
S. Deans, The Radon Transform and Some of its Applications, John Wiley & Sons, New York (1983).
F. Natterer, Mathematics of Computerized Tomography, SIAM, Philadelphia (2001).
I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in a Hilbert Space [in Russian], Nauka, Moscow (1965).
A. K. Louis, Inverse and Ill-Posed Problems [in German], Teubner, Stuttgart (1989).
T. Schuster, The Method of Approximate Inverse: Theory and Applications, Springer, Heidelberg (2007).
A. K. Louis, Feature Reconstruction in Inverse Problems // Inverse Probl. 27, No. 6, Article ID 065010 (2011).
G. Szegö, Orthogonal Polynomials, Am. Math. Soc., Providence RI (1975).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Vestnik Novosibirskogo Gosudarstvennogo Universiteta: Seriya Matematika, Mekhanika, Informatika 12, No. 3, 2012, pp. 73–94.
Rights and permissions
About this article
Cite this article
Derevtsov, E.Y., Polyakova, A.P. Solution of the Integral Geometry Problem for 2-Tensor Fields by the Singular Value Decomposition Method. J Math Sci 202, 50–71 (2014). https://doi.org/10.1007/s10958-014-2033-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-2033-6