Integral Geometry and Tomography

  • Victor Isakov
Part of the Applied Mathematical Sciences book series (AMS, volume 127)


The problems of integral geometry are to determine a function given (weighted) integrals of this function over a “rich” family of manifolds. These problems are of importance in medical applications (tomography), and they are quite useful for dealing with inverse problems in hyperbolic differential equations (integrals of unknown coefficients over ellipsoids or lines can be obtained from the first terms of the asymptotic expansion of rapidly oscillating solutions and an information about first-arrival times of a wave). There has been significant progress in the classical Radon problem when manifolds are hyperplanes and the weight function is the unity; there are interesting results in the plane case when a family of curves is regular (resembling locally the family of straight lines) or in case of the family of straight lines with an arbitrary regular attenuation. Still there are many interesting open questions about the problem with local data and simultaneous recovery of density of a source and of attenuation. We give a brief review of this area, referring for more information to the books of Natterer [Nat] and Sharafutdinov [Sh].


Attenuation Coefficient Inversion Formula Integral Geometry Collision Kernel Hyperbolic Differential Equation 
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  1. [Anik1]
    Anikonov, Y.E. The solvability of a certain problem of integral geometry. Mat. Sb., 101 (1976), 271–279.MathSciNetGoogle Scholar
  2. [Anik2]
    Anikonov, Y.E. Multidimensional Inverse and Ill-Posed Problems for Differential Equations. VSP, Netherlands, 1995.CrossRefzbMATHGoogle Scholar
  3. [ABK]
    Arbuzov, E.V., Bukhgeim, A.L., Kazantsev, S.G. Two-dimensional tomography problems and the theory of A-analytic functions. Siberian Adv. Math., 8 (1998), 1–20.MathSciNetGoogle Scholar
  4. [Ba]
    Bal, G. Inverse transport theory and applications. Inverse Problems, 25 (2009), 053001.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Bey]
    Beylkin, G. The inversion problem and applications of the generalized Radon transform. Comm. Pure Appl. Math., 37 (1984), 580–599.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Bo]
    Boman, J. An example of non-uniqueness for a generalized Radon transform. J. d’Analyse Math., 61 (1993), 395–401.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [BS]
    Boman, J., Strömberg, J.-O. Novikov’s Inversion Formula for the Attenuated Radon Transform-A New Approach. J. Geom. Anal., 14 (2004), 185–198.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [Bon]
    Bondarenko, A. The structure of fundamental solution of the time-independent transport equation. J. Math. Anal. Appl., 221 (1998), 430–451.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [ChoS1]
    Choulli, M., Stefanov, P. Inverse scattering and inverse boundary value problems for the linear Boltzman equation. Comm. Part. Diff. Equat., 21 (1996), 763–785.CrossRefzbMATHGoogle Scholar
  10. [ChoS2]
    Choulli, M., Stefanov, P. An inverse boundary value problem for the stationary transport equation. Osaka J. Math., 36(1999), 87–104.MathSciNetzbMATHGoogle Scholar
  11. [Cor]
    Cormack, A.M. Representation of a function by its line integrals, with some radiological applications. J. Appl. Phys., 34 (1963), 2722–2727.CrossRefzbMATHGoogle Scholar
  12. [D]
    Deans, J. Gegenbauer transforms via the Radon Transform. SIAM J. Math. Anal., 10 (1979), 577–585.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [EKP]
    Ehrenpreis, L., Kuchment, P., Panchenko, A. Attenuated Radon transform and F. Joun’s equation I: Range conditions. Contemp. Math. AMS, 251 (2000), 173–188.Google Scholar
  14. [Fri]
    Friedrichs, K.O. Symmetric Positive Linear Differential Equations. Comm. Pure Appl. Math., 11 (1958), 333–418.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [GeGS]
    Gelfand, I.M., Gindikin, S.G., Shapiro, Z.Ya. A local problem of integral geometry in a space of curves. Func. Anal. Appl., 139 (1980), 248–262.Google Scholar
  16. [Hö2]
    Hörmander, L. The Analysis of Linear Partial Differential Operators. Springer Verlag, New York, 1983–1985.Google Scholar
  17. [IsSu1]
    Isakov, V., Sun, Z. Stability estimates for hyperbolic inverse problems with local boundary data. Inverse Problems, 8 (1992), 193–206.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [LaB]
    Lavrentiev, M.M., Bukhgeim, A. A certain class of operator equations of the first kind. Funct. Anal. Appl., 7 (1973), 290–298.MathSciNetCrossRefGoogle Scholar
  19. [Mi]
    Miranda, C. Partial Differential Equations of Elliptic Type. Ergebn. Math., Band 2, Springer-Verlag, 1970.Google Scholar
  20. [Mu1]
    Muhometov, R. The reconstruction problem of a two-dimensional Riemannian metric and integral geometry. Soviet Math. Dokl., 18 (1977), 32–35.MathSciNetGoogle Scholar
  21. [Mu2]
    Muhometov, R. On one problem of reconstruction of a two-dimensional Riemannian metric and integral geometry. Siber. Math. J., 22 (1981), 119–135.MathSciNetGoogle Scholar
  22. [Nat]
    Natterer, F. The Mathematics of Computerized Tomography. Teubner, Stuttgart and Wiley, New York, 1986.zbMATHGoogle Scholar
  23. [No2]
    Novikov, R.G. An inversion formula for the attenuated X-ray transform. Ark. Math., 40 (2002), 145–167.CrossRefzbMATHGoogle Scholar
  24. [PU]
    Pestov, L., Uhlmann, G. Two-dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. Math., 161 (2005), 1089–1106.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [PrOV]
    Prilepko, A.I., Orlovskii, D.G., Vasin, I.A. Methods for solving inverse problems in mathematical physics. Marcel Dekker, New York-Basel, 2000.Google Scholar
  26. [Rom]
    Romanov, V.G.Inverse Problems of Mathematical Physics. VNU Science Press BV, Utrecht, 1987.Google Scholar
  27. [Sh]
    Sharafutdinov, V.A. Integral Geometry of Tensor Fields. VSP, Utrecht, 1994.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Victor Isakov
    • 1
  1. 1.Department of Mathematics and StatisticsWichita State UniversityWichitaUSA

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