# The inverse problem for the local geodesic ray transform

Article

First Online:

- 1.2k Downloads
- 20 Citations

## Abstract

Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces the geodesic X-ray transform is globally injective. In addition we prove stability estimates and propose a layer stripping type algorithm for reconstruction.

## Mathematics Subject Classification

53C65 35R30 35S05 53C21## References

- 1.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 (in Russian)Google Scholar
- 2.Boman, J.: Local non-injectivity for weighted Radon transforms. Contemp. Math.
**559**, 39–47 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Eberlein, P.: When is a geodesic flow of Anosov type? II. J. Differ. Geom.
**8**, 437–463 (1973)MathSciNetzbMATHGoogle Scholar - 4.Frigyik, B., Stefanov, P., Uhlmann, G.: The X-ray transform for a generic family of curves. J. Geom. Anal.
**18**, 81–97 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Greene, R.E., Wu, H.: \(C^{\infty }\) convex functions and manifolds of positive curvature. Acta Math.
**137**, 209–245 (1976)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Helgason, S.: Integral Geometry and Radon Transforms. Springer, New York (2010)zbMATHGoogle Scholar
- 7.Herglotz, G.: Über die Elastizität der Erde bei Berücksichtigung ihrer variablen Dichte. Z. Math. Phys.
**52**, 275–299 (1905)zbMATHGoogle Scholar - 8.Holman, S., Uhlmann, G.: On the geodesic ray transform with conjugate points. arXiv:1502.06545 (preprint)
- 9.Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. 1–4. Springer, New York (1983)zbMATHGoogle Scholar
- 10.Ivanov, S.: Volume comparison via boundary distances. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 769–784, New Delhi (2010)Google Scholar
- 11.Krishnan, V.: A support theorem for the geodesic ray transform on functions. J. Fourier Anal. Appl.
**15**, 515–520 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Mazzeo, R., Melrose, R.B.: Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal.
**75**, 260–310 (1987)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Melrose, R.B.: Spectral and Scattering Theory for the Laplacian on Asymptotically Euclidian Spaces. Marcel Dekker, New York (1994)zbMATHGoogle Scholar
- 14.Mukhometov, R.G.: The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (Russian). Dokl. Akad. Nauk SSSR
**232**(1), 32–35 (1977)MathSciNetzbMATHGoogle Scholar - 15.Mukhometov, R.G.: On a problem of reconstructing Riemannian metrics. Sib. Math. J.
**22**(3), 420–433 (1982)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Mukhometov, R.G., Romanov, V.G.: On the problem of finding an isotropic Riemannian metric in an \(n\)-dimensional space (Russian). Dokl. Akad. Nauk SSSR
**243**(1), 41–44 (1978)MathSciNetGoogle Scholar - 17.Parenti, C.: Operatori pseudo-differenziali in \(R^{n}\) e applicazioni. Ann. Mat. Pura Appl. (4)
**93**, 359–389 (1972)Google Scholar - 18.Shubin, M.A.: Pseudodifferential operators in \(R^{n}\). Dokl. Akad. Nauk SSSR
**196**, 316–319 (1971)MathSciNetGoogle Scholar - 19.Stefanov, P., Uhlmann, G.: Stability estimates for the X-ray transform of tensor fields and boundary rigidity. Duke Math. J.
**123**, 445–467 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 20.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
- 21.Stefanov, P., Uhlmann, G.: Integral geometry of tensor fields for a class of non-simple Riemannian manifolds. Am. J. Math.
**130**, 239–268 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Stefanov, P., Uhlmann, G.: The geodesic X-ray transform with fold caustics. Anal. PDE
**5**, 219–260 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Stefanov, P., Uhlmann, G.: Recovery of a source term or a speed with one measurement and applications. Trans. AMS
**365**, 5737–5758 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Wiechert, E., Zoeppritz, K.: Über Erdbebenwellen. Nachr. Koenigl. (Geselschaft Wiss, Goettingen)
**4**, 415–549 (1907)zbMATHGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2015