Abstract
We pose and study a rather particular integral geometry problem. In the two-dimensional space we consider all possible straight lines that cross some domain. The known data consist of the integrals over every line of this kind of an unknown piecewise smooth function that depends on both points of the domain and the variables characterizing the lines. The object we seek is the discontinuity curve of the integrand. This problem arose in the author’s previous research in X-ray tomography. In essence, it is a generalization of one mathematical aspect of flaw detection theory, but seems of interest in its own right. The main result of this article is the construction of a special function that can be unbounded only near the required curve. Precisely for this reason we call the function the indicator of contact boundaries. A uniqueness theorem for the solution follows rather easily from the property of indicators.
Similar content being viewed by others
References
Mikhlin S. G., Multidimensional Singular Integrals and Integral Equations [in Russian], Fizmatgiz, Moscow (1962).
Vladimirov V. S., “Mathematical problems of the one-velocity theory of transport of particles,” Trudy Mat. Inst. Akad. Nauk SSSR, 61, 3–158 (1961).
Anikonov D. S., “Integro-differential heterogeneity indicator in a tomography problem,” J. Inverse Ill-Posed Problems, 7, No. 1, 17–59 (1999).
Anikonov D. S., Kovtanyuk A. E., and Prokhorov I. V., Application of the Transport Equation in Tomography [in Russian], Logos, Moscow (2000).
Lavrent’ev M. M. and Savel’ev L. Ya., Theory of Operators and Ill-Posed Problems [in Russian], Sobolev Inst. Mat., Novosibirsk (1999).
Romanov V. G., Inverse Problems of Differential Equations [in Russian], Novosibirsk Univ., Novosibirsk (1973).
Gel’fand I. M. and Goncharov A. B., “Reconstruction of a compactly supported function from its integrals on lines intersecting a set of points in a space,” Dokl. Akad. Nauk SSSR, 290, No. 5, 1037–1040 (1986).
Palamodov V. P., “Some singular problems in tomography,” in: Mathematical Problems of Tomography [in Russian], Nauka, Moscow, 1990, pp. 132–140.
Vainberg È. N., Kazak I. A., and Faingoiz M. L., “X-ray computerized back projection tomography with filtration by double differentiation,” Defektoskopiya, No. 2, 31–39 (1985).
Faridani A., Keinert F., Ritman T. L., and Smith K. T., “Local and global tomography,” in: Signal Processing, Springer-Verlag, New York, 1990, pp. 241–255 (IMA Vol. Math. Appl.; 23).
Louis A. K. and Maass P., “Contour reconstruction in 3-D X-Ray CT,” IEEE Trans. Med. Imaging, 12, No. 4, 109–115 (1993).
Katsevich A. I. and Ramm A. G., “New methods for finding values of the jumps of a function from its local tomographic data,” Inverse Problems, 11, 1005–1023 (1995).
Derevtsov E. Yu., Pickalov V. V., Schuster T., and Louis A. K., “Reconstruction of singularities in local vector and tensor tomography,” in: Abstracts: International Conference “Inverse Problems: Modelling and Simulation,” May 29–June 02, 2006, Fethiye, Turkey, 2006, pp. 38–40.
Sharafutdinov V., Skonan M., and Uhlmann G., Regularity of Ghosts in Tensor Tomography [Preprint, No. 136], Sobolev Inst. Mat., Novosibirsk (2004).
Anikonov D. S., “On the boundedness of a singular integral operator in the space \( C^\alpha (\bar G) \),” Math. USSR-Sb., 33, No. 4, 447–464 (1977).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2008 Anikonov D. S.
The author was supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-1440.2008.1), and the Interdisciplinary Integration Grants (No. 3 and No. 48; 2006) of the Siberian Division of the Russian Academy of Sciences.
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 4, pp. 739–755, July–August, 2008.
Rights and permissions
About this article
Cite this article
Anikonov, D.S. The indicator of contact boundaries for an integral geometry problem. Sib Math J 49, 587–600 (2008). https://doi.org/10.1007/s11202-008-0056-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11202-008-0056-2