Abstract
We study the geodesic X-ray transform X on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and, therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of X and relate it to the conjugate locus. We present numerical examples illustrating the cancellation of singularities. We also show that the attenuated X-ray transform is well posed if the attenuation is positive and there are no more than two conjugate points along each geodesic; but it is still ill-posed if there are three or more conjugate points. Those results follow from our analysis of the weighted X-ray transform.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arnol’d V.I.: Singularities of caustics and wave fronts, volume 62 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1990)
Bao G., Zhang H.: Sensitivity analysis of an inverse problem for the wave equation with caustics. J. Am. Math. Soc. 27, 953–981 (2014)
Belishev M.I., Kurylev Y.V.: To the reconstruction of a Riemannian manifold via its spectral data (BC-method). Comm. Partial Differ. Equ. 17(5–6), 767–804 (1992)
Bellassoued M., DosSantos Ferreira D.: Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Probl. Imaging 5(4), 745–773 (2011)
Boman J., Quinto E.T.: Support theorems for real-analytic Radon transforms. Duke Math. J. 55(4), 943–948 (1987)
Croke C.B.: Rigidity and the distance between boundary points. J. Differ. Geom. 33(2), 445–464 (1991)
Croke, C.B.: Rigidity theorems in Riemannian geometry. In: Geometric methods in inverse problems and PDE control, vol. 137 of IMA Vol. Math. Appl., pp. 47–72. Springer, New York (2004)
Croke C.B., Dairbekov N.S., Sharafutdinov V.A.: Local boundary rigidity of a compact Riemannian manifold with curvature bounded above. Trans. Am. Math. Soc. 352(9), 3937–3956 (2000)
Duistermaat J.J., Guillemin V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1), 39–79 (1975)
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)
Gelfand I.M., Graev M.I., Shapiro Z.J.: Differential forms and integral geometry. Funkcional. Anal. I Priložen. 3(2), 24–40 (1969)
Golubitsky M., Guillemin V. (1973) Stable mappings and their singularities. Graduate Texts in Mathematics, vol. 14. Springer, New York (1973)
Greenleaf A., Uhlmann G.: Nonlocal inversion formulas for the X-ray transform. Duke Math. J. 58(1), 205–240 (1989)
Guillemin, V.: On some results of Gel’fand in integral geometry. In: Pseudodifferential operators and applications (Notre Dame, Ind., 1984), vol. 43 of Proc. Sympos. Pure Math., pp. 149–155. Amer. Math. Soc., Providence (1985)
Guillemin, V., Sternberg, S.: Geometric asymptotics. American Mathematical Society, Providence, R.I., Mathematical Surveys, No. 14 (1977)
Helgason, S.: The Radon transform, vol. 5 of Progress in Mathematics, 2nd edn. Birkhäuser Boston Inc., Boston (1999)
Holman, S.: Microlocal analysis of the geodesic X-ray transform. private communication
Hörmander, L.: The analysis of linear partial differential operators, III, vol. 274. Pseudodifferential operators. Springer, Berlin (1985)
Hörmander, L.: The analysis of linear partial differential operators, IV, vol. 275. Fourier integral operators. Springer, Berlin (1985)
Krishnan V.: On the inversion formulas of Pestov and Uhlmann for the geodesic ray transform. J. Inv. Ill-Posed Problems 18, 401–408 (2010)
Krishnan V.P.: A support theorem for the geodesic ray transform on functions. J. Fourier Anal. Appl. 15(4), 515–520 (2009)
Monard F.: Numerical implementation of two-dimensional geodesic X-ray transforms and their inversion. SIAM J. Imaging Sci. 7(2), 1335–1357 (2014)
Montalto C.: Stable determination of a simple metric, a covector field and a potential from the hyperbolic dirichlet-to-neumann map. Commun. Partial Differ. Equ. 39(1), 120–145 (2014)
Pestov, L., Uhlmann, G.: On characterization of the range and inversion formulas for the geodesic X-ray transform. Int. Math. Res. Not. (80), 4331–4347 (2004)
Pestov L., Uhlmann G.: Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. of Math. (2) 161(2), 1093–1110 (2005)
Quinto, E.T.: Radon transforms satisfying the Bolker assumption. In: 75 years of Radon transform (Vienna, 1992), Conf. Proc. Lecture Notes Math. Phys., IV, pp. 263–270. Int. Press, Cambridge (1994)
Sharafutdinov V.: Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds. J. Geom. Anal. 17(1), 147–187 (2007)
Sharafutdinov V.A. (1994) Integral geometry of tensor fields. Inverse and Ill-posed Problems Series. VSP, Utrecht
Stefanov P., Uhlmann G.: Rigidity for metrics with the same lengths of geodesics. Math. Res. Lett. 5(1-2), 83–96 (1998)
Stefanov P., Uhlmann G.: Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media. J. Funct. Anal. 154(2), 330–358 (1998)
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)
Stefanov P., Uhlmann G.: Boundary rigidity and stability for generic simple metrics. J. Am. Math. Soc. 18(4), 975–1003 (2005)
Stefanov P., Uhlmann G.: Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map. Int. Math. Res. Not. 17(17), 1047–1061 (2005)
Stefanov, P., Uhlmann, G.: Boundary and lens rigidity, tensor tomography and analytic microlocal analysis. In: Algebraic Analysis of Differential Equations. Springer (2008)
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)
Stefanov P., Uhlmann G.: Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds. J. Differ. Geom. 82(2), 383–409 (2009)
Stefanov P., Uhlmann G.: The geodesic X-ray transform with fold caustics. Anal. PDE 5(1-2), 219–260 (2012)
Taylor M.E.: Pseudodifferential operators, vol. 34 of Princeton Mathematical Series. Princeton University Press, Princeton (1981)
Uhlmann, G., Vasy, A.: The inverse problem for the local geodesic ray transform. arxiv:1210.2084
Warner F.W.: The conjugate locus of a Riemannian manifold. Am. J. Math. 87, 575–604 (1965)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. Zelditch
First author partly supported by NSF Grant No. 1265958.
Second author partly supported by a NSF Grant DMS-1301646.
Third author partly supported by NSF Grant No. 1265958 and a Simons Fellowship.
Rights and permissions
About this article
Cite this article
Monard, F., Stefanov, P. & Uhlmann, G. The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points. Commun. Math. Phys. 337, 1491–1513 (2015). https://doi.org/10.1007/s00220-015-2328-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-015-2328-6