The Attenuated Geodesic Ray Transform on Tensors: Generic Injectivity and Stability

  • Yernat M. AssylbekovEmail author


We consider the attenuated geodesic ray transform defined on pairs of symmetric 2-tensors and 1-forms on a simple Riemannian manifold. We prove injectivity and stability results for a class of generic simple metrics and attenuations containing real analytic ones. In fact, methods used in this paper can be modified to generalize our results for a class of non-simple manifolds similar to Stefanov and Uhlmann Am J Math 130(1):239–268 (2008).


Inverse problems Integral geometry Attenuated geodesic ray transform Tensors 

Mathematics Subject Classification

53C65 35R30 



The author is grateful to Professor Plamen Stefanov for his suggestions on an earlier version of this paper. The work was partially supported by AMS-Simons travel grant.


  1. 1.
    Abhishek, A., Mishra, R.K.: Support theorems and an injectivity result for integral moments of a symmetric \(m\)-tensor field (2017). arXiv:1704.02010
  2. 2.
    Ainsworth, G.: The attenuated magnetic ray transform on surfaces. Inverse Probl Imaging 7(1), 27–46 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ainsworth, G., Assylbekov, Y.M.: On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Probl. Imaging 9(2) (2015)Google Scholar
  4. 4.
    Assylbekov, Y.M., Yang, Y.: Determining the first order perturbation of a polyharmonic operator on admissible manifolds. J. Differ. Equ. 262(1), 590–614 (2017)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Assylbekov, Y.M., Zhou, T.: Direct and inverse problems for the nonlinear time-harmonic Maxwell equations in Kerr-type media (2017). arXiv:1709.07767
  6. 6.
    Assylbekov, Y.M., Monard, F., Uhlmann, G.: Inversion formulas and range characterizations for the attenuated geodesic ray transform. J. Math. Pures Appl. 111, 161–190 (2018)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Budinger, T., Gullberg, G., Huesman, R.: Emission computed tomography. Image reconstruction from projections. 147–246 (1979)Google Scholar
  8. 8.
    Chung, F.J., Salo, M., Tzou, L.: Partial data inverse problems for the Hodge Laplacian. Anal. PDE 10(1), 43–93 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dairbekov, N.S., Sharafutdinov, V.A.: On conformal Killing symmetric tensor fields on Riemannian manifolds. Sib. Adv. Math. 21(1), 1–41 (2011)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dairbekov, N.S., Paternain, G.P., Stefanov, P., Uhlmann, G.: The boundary rigidity problem in the presence of a magnetic field. Adv. Math. 216(2), 535–609 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, New York (1976)zbMATHGoogle Scholar
  12. 12.
    Ferreira, D.D.S., Kenig, C.E., Salo, M., Uhlmann, G.: Limiting Carleman weights and anisotropic inverse problems. Invent. Math. 178(1), 119–171 (2009)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Frigyik, B., Stefanov, P., Uhlmann, G.: The X-ray transform for a generic family of curves and weights. J. Geom. Anal. 18(1), 89–108 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ghosh, T., Bhattacharyya, S.: Inverse boundary value problem of determining up to a second order tensor appear in the lower order perturbation of a polyharmonic operator. J. Fourier Anal. Appl. (2018).
  15. 15.
    Guillarmou, C., Paternain, G.P., Salo, M., Uhlmann, G.: The X-ray transform for connections in negative curvature. Commun. Math. Phys. 343(1), 83–127 (2016)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Guillemin, V., Sternberg, S.: Some problems in integral geometry and some related problems in micro-local analysis. Am. J. Math. 101(4), 915–955 (1979)zbMATHGoogle Scholar
  17. 17.
    Holman, S.: Generic local uniqueness and stability in polarization tomography. J. Geom. Anal. 1(23), 229–269 (2013)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Holman, S., Stefanov, P.: The weighted Doppler transform. Inverse Probl. Imaging 4(1), 111–130 (2010)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Juhlin, P.: Principles of Doppler tomography. LUTFD2/(TFMA-92)/7002 P, 17 (1992)Google Scholar
  20. 20.
    Kenig, C.E., Salo, M., Uhlmann, G.: Inverse problems for the anisotropic Maxwell equations. Duke Math. J. 157(2), 369–419 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Krupchyk, K., Uhlmann, G.: Inverse problems for magnetic schrödinger operators in transversally anisotropic geometries. Commun. Math. Phys. 361(2), 525–582 (2018)zbMATHGoogle Scholar
  22. 22.
    Krupchyk, K., Uhlmann, G.: Inverse problems for advection diffusion equations in admissible geometries. Commun. Partial Differ. Equ. 43(4), 585–615 (2018). MathSciNetGoogle Scholar
  23. 23.
    Melrose, R.B.: Spectral and scattering theory for the laplacian on asymptotically euclidian spaces. In: Lecture Notes in Pure and Applied Mathematics, pp. 85–85 (1994)Google Scholar
  24. 24.
    McLean, W.C.H.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  25. 25.
    Michel, R.: Sur la rigidité imposée par la longueur des géodésiques. Invent. Math. 65(1), 71–83 (1981)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Monard, F.: Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces. SIAM J. Math. Anal. 48(2), 1155–1177 (2016)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Monard, F.: Efficient tensor tomography in fan-beam coordinates. II: attenuated transforms. Inverse Probl. Imaging 12(2), 433–460 (2018)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Morrey, C., Nirenberg, L.: On the analyticity of the solutions of linear elliptic systems of partial differential equations. Commun. Pure Appl. Math. 10(2), 271–290 (1957)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Paternain, G.P., Salo, M.: Carleman estimates for geodesic X-ray transforms (2018). arXiv:1805.02163
  30. 30.
    Paternain, G.P., Salo, M., Uhlmann, G.: The attenuated ray transform for connections and Higgs fields. Geom. Funct. Anal. 22(5), 1460–1489 (2012)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Paternain, G.P., Salo, M., Uhlmann, G.: Tensor tomography on surfaces. Invent. Math. 193(1), 229–247 (2013)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Paternain, G.P., Salo, M., Uhlmann, G.: Tensor tomography: progress and challenges. Chin. Ann. Math. Ser. B 35(3) (2014)Google Scholar
  33. 33.
    Paternain, G.P., Salo, M., Uhlmann, G.: Invariant distributions, Beurling transforms and tensor tomography in higher dimensions. Math. Ann. 363(1–2), 305–362 (2015)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Paternain, G.P., Salo, M., Uhlmann, G., Zhou, H.: The geodesic X-ray transform with matrix weights-ray transform with matrix weights (2016). arXiv:1605.07894
  35. 35.
    Sadiq, K., Scherzer, O., Tamasan, A.: On the X-ray transform of planar symmetric 2-tensors. J. Math. Anal. Appl. 442(1), 31–49 (2016)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Salo, M., Uhlmann, G.: The attenuated ray transform on simple surfaces. J. Differ. Geom. 88(1), 161–187 (2011)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Sharafutdinov, V.A.: Integral Geometry of Tensor Fields. VSP, Utrecht (1994)zbMATHGoogle Scholar
  38. 38.
    Sharafutdinov, V.: Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds. J. Geom. Anal. 17(1), 147–187 (2007)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Shubin, M.: Pseudodifferential Operators and Spectral Theory. Springer, New York (2001)zbMATHGoogle Scholar
  40. 40.
    Sjöstrand, J.: Singularités analytiques microlocales, vol. 82. Société Mathématique de France (1982)Google Scholar
  41. 41.
    Stefanov, P.: Microlocal approach to tensor tomography and boundary and lens rigidity. Serdica Math. J. 34(1), 67p–112p (2008)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Stefanov, P.: A sharp stability estimate in tensor tomography. In: Journal of Physics: Conference Series, vol. 124, p. 012007. IOP Publishing (2008)Google Scholar
  43. 43.
    Stefanov, P., Uhlmann, G.: Stability estimates for the X-ray transform of tensor fields and boundary rigidity. Duke Math. J. 123(3), 445–467 (2004)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Stefanov, P., Uhlmann, G.: Boundary rigidity and stability for generic simple metrics. J. Am. Math. Soc. 18(4), 975–1003 (2005)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Stefanov, P., Uhlmann, G.: Boundary and lens rigidity, tensor tomography and analytic microlocal analysis. In: Aoki, T., Majima, H., Katei, Y., Tose, N. (eds.) Algebraic Analysis of Differential Equations, Fetschrift in Honor of Takahiro Kawai. pp. 275–293 (2008)Google Scholar
  46. 46.
    Stefanov, P., Uhlmann, G.: Integral geometry of tensor fields on a class of non-simple Riemannian manifolds. Am. J. Math. 130(1), 239–268 (2008)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Stefanov, P., Uhlmann, G., Vasy, A.: Inverting the local geodesic X-ray transform on tensors. J. Anal. Math. 136(1), 151–208 (2018)MathSciNetGoogle Scholar
  48. 48.
    Taylor, M.E.: Pseudodifferential Operators. Princeton Mathematical Series, vol. 34. Princeton University Press, Princeton (1981)zbMATHGoogle Scholar
  49. 49.
    Taylor, M.: Partial Differential Equations I: Basic Theory. Applied Mathematical Sciences, vol. 115. Springer, New York (2011)zbMATHGoogle Scholar
  50. 50.
    Trèves, F.: Introduction to Pseudodifferential and Fourier Integral Operators. Springer, New York (1980)zbMATHGoogle Scholar
  51. 51.
    Uhlmann, G., Vasy, A.: The inverse problem for the local geodesic ray transform. Invent. Math. 205(1), 83–120 (2016)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Zhou, H.: Generic injectivity and stability of inverse problems for connections. Commun. Partial Differ. Equ. 42(5), 780–801 (2017)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Computational Mathematics, Science and EngineeringMichigan State UniversityEast LansingUSA

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