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The X-Ray Transform for a Generic Family of Curves and Weights

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Abstract

We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves Γ. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of Γ covers T * M.

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References

  1. Anikonov, Y., Romanov, V.: On uniqueness of determination of a form of first degree by its integrals along geodesics. Inverse Ill-posed Probl. Ser. 5(6), 487–480 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bernstein, I.N., Gerver, M.L.: Conditions on distinguishability of metrics by hodographs. In: Methods and Algorithms of Interpretation of Seismological Information. Computerized Seismology, vol. 13, pp. 50–73. Nauka, Moscow (1980) (in Russian)

    Google Scholar 

  3. Boman, J.: An example of non-uniqueness for a generalized Radon transform. J. Anal. Math. 61, 395–401 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. Boman, F., Quinto, E.: Support theorems for real-analytic Radon transforms. Duke Math. J. 55(4), 943–948 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Coddington, E., Levinson, N.: Theory of Ordinary Differential Equations. R.E. Krieger, Malabar (1984)

    Google Scholar 

  6. Dairbekov, N.: Integral geometry problem for nontrapping manifolds. Inverse Probl. 22(2), 431–445 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dairbekov, N., Paternain, G., Stefanov, P., Uhlmann, G.: Boundary rigidity problem in the presence of a magnetic field. arXiv:math/0611788v2

  8. Greenleaf, A., Uhlmann, G.: Nonlocal inversion formulas for the X-ray transform. Duke Math. J. 58(1), 205–240 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Guillemin, V.: On some results of Gel’fand in integral geometry. In: Pseudodifferential Operators and Applications, Notre Dame, IN, 1984, pp. 149–155. Am. Math. Soc., Providence (1985)

    Google Scholar 

  10. Guillemin, V., Sternberg, S.: Geometric Asymptotics. Mathematical Surveys and Monographs, vol. 14. Am. Math. Soc., Providence (1977)

    MATH  Google Scholar 

  11. Guillemin, V., Sternberg, S.: Some problems in integral geometry and some related problems in micro-local analysis. Am. J. Math. 101, 915–955 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kenig, C., Sjöstrand, J., Uhlmann, G.: The Calderón problem with partial data. Ann. Math. 165(2), 567–591 (2007)

    MATH  Google Scholar 

  13. Mikhlin, S., Prössdorf, S.: Singular Integral Operators. Springer, New York (1986)

    Google Scholar 

  14. Mukhometov, R.: Inverse kinematic problem of seismic on the plane. In: Math. Problems of Geophysics, Akad. Nauk. SSSR, Sibirsk. Otdel., Vychisl. Tsentr, Novosibirsk 6(2), 243–252 (1975) (in Russian)

  15. Mukhometov, R.: The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry. Dokl. Akad. Nauk SSSR 232(1), 32–35 (1977) (Russian)

    MathSciNet  Google Scholar 

  16. Mukhometov, R.: On a problem of reconstructing Riemannian metrics. Sib. Math. J. 22(3), 420–433 (1982)

    Article  MathSciNet  Google Scholar 

  17. Quinto, E.: Radon transforms satisfying the Bolker assumption. In: Proceedings of Conference Seventy-Five Years of Radon Transforms, pp. 263–270. International Press Co., Hong Kong (1994)

    Google Scholar 

  18. Romanov, V.: Integral geometry on geodesics of an isotropic Riemannian metric. Sov. Math. Dokl. 19(4), 847–851 (1978)

    MATH  Google Scholar 

  19. Sjöstrand, J.: Singularités analytiques microlocales. Astérisque 95, 1–166 (1982)

    MATH  Google Scholar 

  20. Stefanov, P., Uhlmann, G.: Stability estimates for the X-ray transform of tensor fields and boundary rigidity. Duke Math. J. 123, 445–467 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Stefanov, P., Uhlmann, G.: Boundary rigidity and stability for generic simple metrics. J. Am. Math. Soc. 18, 975–1003 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Stefanov, P., Uhlmann, G.: Integral geometry of tensor fields on a class of non-simple Riemannian manifolds. Am. J. Math. (to appear), arXiv:math/0601178v3

  23. Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  24. Taylor, M.: Pseudodifferential Operators. Princeton Mathematical Series, vol. 34. Princeton University Press, Princeton (1981)

    MATH  Google Scholar 

  25. Treves, F.: Introduction to Pseudodifferential and Fourier Integral Operators, vol. 1, Pseudodifferential Operators. The University Series in Mathematics. Plenum, New York (1980)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Bela Frigyik & Plamen Stefanov

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Gunther Uhlmann

Authors
  1. Bela Frigyik
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  2. Plamen Stefanov
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  3. Gunther Uhlmann
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Corresponding author

Correspondence to Bela Frigyik.

Additional information

Second author was partly supported by NSF Grant DMS-0400869; third author was partly supported by NSF and a Walker Family Endowed Professorship.

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Frigyik, B., Stefanov, P. & Uhlmann, G. The X-Ray Transform for a Generic Family of Curves and Weights. J Geom Anal 18, 89–108 (2008). https://doi.org/10.1007/s12220-007-9007-6

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  • DOI: https://doi.org/10.1007/s12220-007-9007-6

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