Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T01:55:08.903Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  15 January 2023

Gabriel P. Paternain ,
Mikko Salo  and
Gunther Uhlmann
Gabriel P. Paternain
Affiliation:
University of Cambridge
Mikko Salo
Affiliation:
University of Jyväskylä, Finland
Gunther Uhlmann
Affiliation:
University of Washington
Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Geometric Inverse Problems
With Emphasis on Two Dimensions
 , pp. 332 - 341
Publisher: Cambridge University Press
Print publication year: 2023

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abel, N. H. 1826. Auflösung einer mechanischen Aufgabe. J. Reine Angew. Math., 1, 153–157.Google Scholar
Ainsworth, Gareth. 2013. The attenuated magnetic ray transform on surfaces. Inverse Probl. Imaging, 7(1), 27–46.Google Scholar
Ainsworth, Gareth, and Assylbekov, Yernat M. 2015. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Probl. Imaging, 9(2), 317–335.Google Scholar
Andersson, Joel, and Boman, Jan. 2018. Stability estimates for the local Radon transform. Inverse Prob., 34(3), 034004, 23.CrossRefGoogle Scholar
Arbuzov, È. V., Bukhgeĭm, A. L., and Kazantsev, S. G. 1998. Two-dimensional tomography problems and the theory of A-analytic functions [translation of Algebra, Geometry, Analysis and Mathematical Physics (Russian) (Novosibirsk, 1996), 6–20, 189, Izdat. Ross. Akad. Nauk Sibirsk. Otdel. Inst. Mat., Novosibirsk, 1997; MR1624170 (99m:44003)]. Siberian Adv. Math., 8(4), 1–20.Google Scholar
Assylbekov, Yernat M., and Dairbekov, Nurlan S. 2018. The X-ray transform on a general family of curves on Finsler surfaces. J. Geom. Anal., 28(2), 1428–1455.Google Scholar
Assylbekov, Yernat M., and Stefanov, Plamen. 2020. Sharp stability estimate for the geodesic ray transform. Inverse Prob., 36(2), 025013, 14.Google Scholar
Assylbekov, Yernat M., Monard, François, and Uhlmann, Gunther. 2018. Inversion formulas and range characterizations for the attenuated geodesic ray transform. J. Math. Pures Appl. (9), 111, 161–190.Google Scholar
Bagby, T., and Gauthier, P. M. 1992. Uniform approximation by global harmonic functions. Pages 15–26 of: Approximation by Solutions of Partial Differential Equations (Hanstholm, 1991). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 365. Kluwer Academic Publishers, Dordrecht.Google Scholar
Bal, Guillaume. 2019. Introduction to Inverse Problems. University of Chicago, lecture notes. Available at: www.stat.uchicago.edu/~guillaumebal/publications.htmlGoogle Scholar
Bateman, Harry. 1910. The solution of the integral equation connecting the velocity of propagation of an earthquake wave in the interior of the Earth with the times which the disturbance takes to travel to the different stations on the Earth's surface. Philos. Mag., 19, 576–587.Google Scholar
Belishev, M. I. 2003. The Calderon problem for two-dimensional manifolds by the BC- method. SIAM J. Math. Anal., 35(1), 172–182.Google Scholar
Bergh, Jöran, and Löfström, Jörgen. 1976. Interpolation Spaces: An Introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York.Google Scholar
Bernšteĭn, I. N., and Gerver, M. L. 1978. A problem of integral geometry for a family of geodesics and an inverse kinematic seismics problem. Dokl. Akad. Nauk SSSR, 243(2), 302–305.Google Scholar
Bers, Lipman. 1948. On rings of analytic functions. Bull. Am. Math. Soc., 54, 311–315.Google Scholar
Betelú, Santiago, Gulliver, Robert, and Littman, Walter. 2002. Boundary control of PDEs via curvature flows: the view from the boundary. II. vol. 46. Special issue dedicated to the memory of Jacques-Louis Lions.Google Scholar
Bohr, Jan. 2021. Stability of the non-abelian X-ray transform in dimension ≥3. J. Geom. Anal., 31, 11226–11269.CrossRefGoogle Scholar
Bohr, Jan, and Paternain, Gabriel P. 2021. The transport Oka-Grauert principle for simple surfaces. arXiv:2108.05125.Google Scholar
Boman, Jan. 1993. An example of nonuniqueness for a generalized Radon transform. J. Anal. Math., 61, 395–401.Google Scholar
Boman, Jan, and Quinto, Eric Todd. 1987. Support theorems for real-analytic Radon transforms. Duke Math. J., 55(4), 943–948.Google Scholar
Boman, Jan, and Sharafutdinov, Vladimir. 2018. Stability estimates in tensor tomography. Inverse Probl. Imaging, 12(5), 1245–1262.Google Scholar
Boman, Jan, and Strömberg, Jan-Olov. 2004. Novikov's inversion formula for the attenuated Radon transform—a new approach. J. Geom. Anal., 14(2), 185–198.Google Scholar
Burago, Dmitri, and Ivanov, Sergei. 2010. Boundary rigidity and filling volume minimality of metrics close to a flat one. Ann. Math. (2), 171(2), 1183–1211.CrossRefGoogle Scholar
Burago, Dmitri, and Ivanov, Sergei. 2013. Area minimizers and boundary rigidity of almost hyperbolic metrics. Duke Math. J., 162(7), 1205–1248.CrossRefGoogle Scholar
Croke, Christopher. 2014. Scattering rigidity with trapped geodesics. Ergodic Theory Dyn. Syst., 34(3), 826–836.Google Scholar
Croke, Christopher B. 1990. Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv., 65(1), 150–169.Google Scholar
Croke, Christopher B. 1991. Rigidity and the distance between boundary points. J. Differ. Geom., 33(2), 445–464.Google Scholar
Croke, Christopher B. 2004. Rigidity theorems in Riemannian geometry. Pages 47–72 of: Geometric Methods in Inverse Problems and PDE Control. IMA Vol. Math. Appl., vol. 137. Springer, New York.CrossRefGoogle Scholar
Croke, Christopher B., and Herreros, Pilar. 2016. Lens rigidity with trapped geodesics in two dimensions. Asian J. Math., 20(1), 47–57.Google Scholar
Croke, Christopher B., and Sharafutdinov, Vladimir A. 1998. Spectral rigidity of a compact negatively curved manifold. Topology, 37(6), 1265–1273.Google Scholar
Dairbekov, N. S., and Sharafutdinov, V. A. 2010. Conformal Killing symmetric tensor fields on Riemannian manifolds. Mat. Tr., 13(1), 85–145.Google Scholar
Dairbekov, Nurlan S. 2006. Integral geometry problem for nontrapping manifolds. Inverse Probl., 22(2), 431–445.Google Scholar
Dairbekov, Nurlan S., Paternain, Gabriel P., Stefanov, Plamen, and Uhlmann, Gunther. 2007. The boundary rigidity problem in the presence of a magnetic field. Adv. Math., 216(2), 535–609.Google Scholar
Daudé, Thierry, Kamran, Niky, and Nicoleau, François. 2019. Non-uniqueness results for the anisotropic Calderón problem with data measured on disjoint sets. Ann. Inst. Fourier (Grenoble), 69(1), 119–170.CrossRefGoogle Scholar
Desai, Naeem M., Lionheart, William R. B., Sales, Morten, Strobl, Markus, and Schmidt, Søren. 2020. Polarimetric neutron tomography of magnetic fields: uniqueness of solution and reconstruction. Inverse Probl., 36(4), 045001, 17.Google Scholar
Donaldson, S. K. 1992. Boundary value problems for Yang-Mills fields. J. Geom. Phys., 8(1–4), 89–122.Google Scholar
Donaldson, Simon. 2011. Riemann Surfaces. Oxford Graduate Texts in Mathematics, vol. 22. Oxford University Press, Oxford.Google Scholar
Dos Santos Ferreira, David, Kenig, Carlos E., Salo, Mikko, and Uhlmann, Gunther. 2009. Limiting Carleman weights and anisotropic inverse problems. Invent. Math., 178(1), 119–171.Google Scholar
Dos Santos Ferreira, David, Kurylev, Yaroslav, Lassas, Matti, and Salo, Mikko. 2016. The Calderón problem in transversally anisotropic geometries. J. Eur. Math. Soc. (JEMS), 18(11), 2579–2626.Google Scholar
Duistermaat, J. J., and Hörmander, L. 1972. Fourier integral operators. II. Acta Math., 128(3-4), 183–269.Google Scholar
Eskin, G. 2004. On non-abelian Radon transform. Russ. J. Math. Phys., 11(4), 391–408.Google Scholar
Eskin, Gregory, and Ralston, James. 2003. Inverse boundary value problems for systems of partial differential equations. Pages 105–113 of: Recent Development in Theories & Numerics. World Scientific Publishing, River Edge, NJ.Google Scholar
Eskin, Gregory, and Ralston, James. 2004. On the inverse boundary value problem for linear isotropic elasticity and Cauchy-Riemann systems. Pages 53–69 of: Inverse Problems and Spectral Theory. Contemp. Math., vol. 348. American Mathematical Society, Providence, RI.Google Scholar
Farkas, H. M., and Kra, I. 1992. Riemann Surfaces. Second edn. Graduate Texts in Mathematics, vol. 71. Springer-Verlag, New York.Google Scholar
Finch, David, and Uhlmann, Gunther. 2001. The x-ray transform for a non-abelian connection in two dimensions. vol. 17. Special issue to celebrate Pierre Sabatier's 65th birthday (Montpellier, 2000).Google Scholar
Finch, David V. 2003. The attenuated x-ray transform: recent developments. Pages 47–66 of: Inside Out: Inverse Problems and Applications. Math. Sci. Res. Inst. Publ., vol. 47. Cambridge University Press, Cambridge.Google Scholar
Folland, Gerald B. 1995. Introduction to Partial Differential Equations. Second edn. Princeton University Press, Princeton, NJ.Google Scholar
Forster, Otto. 1981. Lectures on Riemann Surfaces. Graduate Texts in Mathematics, vol. 81. Springer-Verlag, New York-Berlin. Translated from the German by Bruce Gilligan.Google Scholar
Funk, P. 1913. Über Flächen mit lauter geschlossenen geodätischen Linien. Math. Ann., 74(2), 278–300.Google Scholar
Gallot, Sylvestre, Hulin, Dominique, and Lafontaine, Jacques. 2004. Riemannian Geometry. Third edn. Universitext. Springer-Verlag, Berlin.Google Scholar
Gorenflo, Rudolf, and Vessella, Sergio. 1991. Abel Integral Equations. Lecture Notes in Mathematics, vol. 1461. Springer-Verlag, Berlin. Analysis and Applications.Google Scholar
Graham, C. Robin, Guillarmou, Colin, Stefanov, Plamen, and Uhlmann, Gunther. 2019. X-ray transform and boundary rigidity for asymptotically hyperbolic manifolds. Ann. Inst. Fourier (Grenoble), 69(7), 2857–2919.Google Scholar
Guillarmou, Colin. 2017a. Invariant distributions and X-ray transform for Anosov flows. J. Differ. Geom., 105(2), 177–208.Google Scholar
Guillarmou, Colin. 2017b. Lens rigidity for manifolds with hyperbolic trapped sets. J. Am. Math. Soc., 30(2), 561–599.Google Scholar
Guillarmou, Colin, and Lefeuvre, Thibault. 2019. The marked length spectrum of Anosov manifolds. Ann. Math. (2), 190(1), 321–344.Google Scholar
Guillarmou, Colin, and Mazzucchelli, Marco. 2018. Marked boundary rigidity for surfaces. Ergod. Theory Dyn. Syst., 38(4), 1459–1478.Google Scholar
Guillarmou, Colin, and Monard, François. 2017. Reconstruction formulas for X-ray transforms in negative curvature. Ann. Inst. Fourier (Grenoble), 67(4), 1353–1392.Google Scholar
Guillarmou, Colin, and Sá Barreto, Antônio. 2009. Inverse problems for Einstein manifolds. Inverse Probl. Imaging, 3(1), 1–15.Google Scholar
Guillarmou, Colin, and Tzou, Leo. 2011. Calderón inverse problem with partial data on Riemann surfaces. Duke Math. J., 158(1), 83–120.Google Scholar
Guillarmou, Colin, Paternain, Gabriel P., Salo, Mikko, and Uhlmann, Gunther. 2016. The X-ray transform for connections in negative curvature. Comm. Math. Phys., 343(1), 83–127.Google Scholar
Guillarmou, Colin, Lassas, Matti, and Tzou, Leo. 2020. X-ray transform in asymptotically conic spaces. International Mathematics Research Notices, Nov.Google Scholar
Guillarmou, Colin, Mazzucchelli, Marco, and Tzou, Leo. 2021. Boundary and lens rigidity for non-convex manifolds. Am. J. Math., 143(2), 533–575.Google Scholar
Guillemin, Victor. 1975. Some remarks on integral geometry. Technical Report, MIT.Google Scholar
Guillemin, Victor. 1976. The Radon transform on Zoll surfaces. Adv. Math., 22(1), 85–119.Google Scholar
Guillemin, Victor. 1985. On some results of Gel'fand in integral geometry, in Pseudodifferential Operators and Applications (edited by Tréves, F.), 149–155, Proc. Sympos. Pure Math. 43, Amer. Math. Soc., Providence RI.Google Scholar
Guillemin, Victor, and Kazhdan, David. 1980a. Some inverse spectral results for negatively curved 2-manifolds. Topology, 19(3), 301–312.Google Scholar
Guillemin, Victor, and Kazhdan, David. 1980b. Some inverse spectral results for negatively curved n-manifolds. Pages 153–180 of: Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979). Proc. Sympos. Pure Math., XXXVI. American Mathematical Society, Providence, R.I.Google Scholar
Guillemin, Victor, and Sternberg, Shlomo. 1977. Geometric Asymptotics. Mathematical Surveys, No. 14. American Mathematical Society, Providence, R.I.Google Scholar
Gunning, Robert C., and Rossi, Hugo. 1965. Analytic Functions of Several Complex Variables. Prentice-Hall, Inc., Englewood Cliffs, NJ.Google Scholar
Hall, Brian. 2015. Lie Groups, Lie Algebras, and Representations. Second edn. Graduate Texts in Mathematics, vol. 222. Springer, Cham. An elementary introduction.Google Scholar
Helgason, Sigurdur. 1999. The Radon Transform. Second edn. Progress in Mathematics, vol. 5. Birkhäuser Boston, Inc., Boston, MA.Google Scholar
Helgason, Sigurdur. 2011. Integral Geometry and Radon Transforms. Springer, New York.Google Scholar
Herglotz, G. 1907. Über das Benndorfsche Problem der Fortpflanzungsgeschwindigkeit der Erdbebenstrahlen. Physikalische Zeitschrift, 8, 145–147.Google Scholar
Hilger, A., Manke, I., and Kardjilov, N. et al. 2018. Tensorial neutron tomography of three-dimensional magnetic vector fields in bulk materials. Nat. Commun., 9, 4023.Google Scholar
Hofer, Helmut. 1985. A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem. J. London Math. Soc. (2), 31(3), 566–570.Google Scholar
Hörmander, Lars. 1983–1985. The Analysis of Linear Partial Differential Operators. I–IV. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin.Google Scholar
Hubbard, John Hamal. 2006. Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Vol. 1. Matrix Editions, Ithaca, NY. Teichmüller theory, With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, With forewords by William Thurston and Clifford Earle.Google Scholar
Ilmavirta, Joonas. 2014. On the Broken Ray Transform. Ph.D. thesis, University of Jyväskylä, Department of Mathematics and Statistics, Report 140. advisor: Mikko Salo.Google Scholar
Ilmavirta, Joonas. 2016. Coherent quantum tomography. SIAM J. Math. Anal., 48(5), 3039–3064.Google Scholar
Ilmavirta, Joonas, and Monard, Francois. 2019. 4. Integral Geometry on Manifolds with Boundary and Applications. De Gruyter, Berlin. Pages 43–114.Google Scholar
Ilmavirta, Joonas, and Paternain, Gabriel P. 2020. Broken Ray Tensor Tomography with One Reflecting Obstacle.Google Scholar
Ilmavirta, Joonas, and Salo, Mikko. 2016. Broken ray transform on a Riemann surface with a convex obstacle. Comm. Anal. Geom., 24(2), 379–408.Google Scholar
Ivanov, Sergei. 2010. Volume comparison via boundary distances. Pages 769–784 of: Proceedings of the International Congress of Mathematicians. Volume II. Hindustan Book Agency, New Delhi.Google Scholar
Jakobson, Dmitry, and Strohmaier, Alexander. 2007. High energy limits of Laplacetype and Dirac-type eigenfunctions and frame flows. Comm. Math. Phys., 270(3), 813–833.Google Scholar
Jost, Jürgen. 2017. Riemannian Geometry and Geometric Analysis. Seventh edn. Universitext. Springer, Cham.Google Scholar
Katchalov, Alexander, Kurylev, Yaroslav, and Lassas, Matti. 2001. Inverse Boundary Spectral Problems. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 123. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Kazantsev, S. G., and Bukhgeim, A. A. 2007. Inversion of the scalar and vector attenuated X-ray transforms in a unit disc. J. Inverse Ill-Posed Probl., 15(7), 735–765.Google Scholar
Kenig, Carlos, and Salo, Mikko. 2013. The Calderón problem with partial data on manifolds and applications. Anal. PDE, 6(8), 2003–2048.Google Scholar
Kenig, Carlos, and Salo, Mikko. 2014. Recent progress in the Calderón problem with partial data. Pages 193–222 of: Inverse Problems and Applications. Contemp. Math., vol. 615. American Mathematical Society, Providence, RI.Google Scholar
Knieper, Gerhard. 2002. Hyperbolic dynamics and Riemannian geometry. Pages 453–545 of: Handbook of Dynamical Systems, Vol. 1A. North-Holland, Amsterdam.Google Scholar
Koch, Herbert, Rüland, Angkana, and Salo, Mikko. 2021. On Instability Mechanisms for Inverse Problems. Ars Inven. Anal. 2021, Paper No. 7, 93 pp. 35 (49).Google Scholar
Krishnan, Venkateswaran P. 2009. A support theorem for the geodesic ray transform on functions. J. Fourier Anal. Appl., 15(4), 515–520.Google Scholar
Krishnan, Venkateswaran P. 2010. On the inversion formulas of Pestov and Uhlmann for the geodesic ray transform. J. Inverse Ill-Posed Probl., 18(4), 401–408.Google Scholar
Krishnan, Venkateswaran P., and Quinto, Eric Todd. 2015. Microlocal analysis in tomography. Pages 847–902 of: Handbook of Mathematical Methods in Imaging. Vol. 1, 2, 3. Springer, New York.Google Scholar
Krishnan, Venkateswaran P., and Stefanov, Plamen. 2009. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Probl. Imaging, 3(3), 453–464.Google Scholar
Kuchment, Peter. 2014. The Radon Transform and Medical Imaging. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 85. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.Google Scholar
Lassas, Matti, and Uhlmann, Gunther. 2001. On determining a Riemannian manifold from the Dirichlet-to-Neumann map. Ann. Sci. École Norm. Sup. (4), 34(5), 771–787.Google Scholar
Lassas, Matti, Taylor, Michael, and Uhlmann, Gunther. 2003a. The Dirichlet-to- Neumann map for complete Riemannian manifolds with boundary. Comm. Anal. Geom., 11(2), 207–221.Google Scholar
Lassas, Matti, Sharafutdinov, Vladimir, and Uhlmann, Gunther. 2003b. Semiglobal boundary rigidity for Riemannian metrics. Math. Ann., 325(4), 767–793.Google Scholar
Lassas, Matti, Liimatainen, Tony, and Salo, Mikko. 2020. The Poisson embedding approach to the Calderón problem. Math. Ann., 377(1-2), 19–67.Google Scholar
Lee, John M. 1997. Riemannian Manifolds. Graduate Texts in Mathematics, vol. 176. Springer-Verlag, New York. An introduction to curvature.Google Scholar
Lee, John M., and Uhlmann, Gunther. 1989. Determining anisotropic real-analytic conductivities by boundary measurements. Comm. Pure Appl. Math., 42(8), 1097–1112.Google Scholar
Lefeuvre, Thibault. 2020. Local marked boundary rigidity under hyperbolic trapping assumptions. J. Geom. Anal., 30(1), 448–465.CrossRefGoogle Scholar
Lefeuvre, Thibault. 2021. Geometric Inverse Problems on Anosov Manifolds. Online notes available at: https://thibaultlefeuvre.files.wordpress.com/2021/04/survey-geometric-inverse-problems.pdf.Google Scholar
Lehtonen, Jere, Railo, Jesse, and Salo, Mikko. 2018. Tensor tomography on Cartan- Hadamard manifolds. Inverse Probl., 34(4), 044004, 27.Google Scholar
Leonhardt, Ulf, and Philbin, Thomas. 2010. Geometry and Light. Dover Publications, Inc., Mineola, NY. The science of invisibility.Google Scholar
Lerner, Nicolas. 2019. Carleman Inequalities. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 353. Springer, Cham. An introduction and more.Google Scholar
Manakov, S. V., and Zakharov, V. E. 1981. Three-dimensional model of relativistic- invariant field theory, integrable by the inverse scattering transform. Lett. Math. Phys., 5(3), 247–253.Google Scholar
Markoe, Andrew, and Quinto, Eric Todd. 1985. An elementary proof of local invertibil- ity for generalized and attenuated Radon transforms. SIAM J. Math. Anal., 16(5), 1114–1119.Google Scholar
Mazzucchelli, Marco. 2012. Critical Point Theory for Lagrangian Systems. Progress in Mathematics, vol. 293. Birkhäuser/Springer Basel AG, Basel.Google Scholar
Merry, Will, and Paternain, Gabriel P. 2011. Inverse Problems in Geometry and Dynamics. https://www.dpmms.cam.ac.uk/~gpp24/ipgd(3).pdf.Google Scholar
Michel, René. 1978. Sur quelques problèmes de géométrie globale des géodésiques. Bol. Soc. Brasil. Mat., 9(2), 19–37.Google Scholar
Michel, René. 1981/82. Sur la rigidité imposée par la longueur des géodésiques. Invent. Math., 65(1), 71–83.Google Scholar
Monard, François. 2014. Numerical implementation of geodesic X-ray transforms and their inversion. SIAM J. Imaging Sci., 7(2), 1335–1357.Google Scholar
Monard, François. 2016a. Efficient tensor tomography in fan-beam coordinates. Inverse Probl. Imaging, 10(2), 433–459.Google Scholar
Monard, François. 2016b. Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces. SIAM J. Math. Anal., 48(2), 1155–1177.Google Scholar
Monard, François, and Paternain, Gabriel P. 2020. The geodesic X-ray transform with a GL(n,ℂ)-connection. J. Geom. Anal., 30(3), 2515–2557.Google Scholar
Monard, François, Stefanov, Plamen, and Uhlmann, Gunther. 2015. The geodesic ray transform on Riemannian surfaces with conjugate points. Comm. Math. Phys., 337(3), 1491–1513.Google Scholar
Monard, François, Nickl, Richard, and Paternain, Gabriel P. 2019. Efficient nonparametric Bayesian inference for X-ray transforms. Ann. Statist., 47(2), 1113–1147.Google Scholar
Monard, François, Nickl, Richard, and Paternain, Gabriel P. 2021a. Consistent inversion of noisy non-Abelian x-ray transforms. Comm. Pure Appl. Math., 74(5), 1045–1099.Google Scholar
Monard, François, Nickl, Richard, and Paternain, Gabriel P. 2021b. Statistical guarantees for Bayesian uncertainty quantification in non-linear inverse problems with Gaussian process priors. Ann. Statist. 49 (2021), no. 6, 3255–3298.Google Scholar
Muhometov, R. G. 1977. The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry. Dokl. Akad. Nauk SSSR, 232(1), 32–35.Google Scholar
Muhometov, R. G. 1981. On a problem of reconstructing Riemannian metrics. Sibirsk. Mat. Zh., 22(3), 119–135, 237.Google Scholar
Nakamura, G., and Uhlmann, G. 2002. Complex geometrical optics solutions and pseudoanalytic matrices. Pages 305–338 of: Ill-posed and Inverse Problems. VSP, Zeist.Google Scholar
Natterer, F. 2001. The Mathematics of Computerized Tomography. Classics in Applied Mathematics, vol. 32. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Reprint of the 1986 original.Google Scholar
Novikov, R. G. 2002a. On determination of a gauge field on ℝd from its non-abelian Radon transform along oriented straight lines. J. Inst. Math. Jussieu, 1(4), 559–629.Google Scholar
Novikov, R. G. 2014. Weighted Radon transforms and first order differential systems on the plane. Mosc. Math. J., 14(4), 807–823, 828.Google Scholar
Novikov, Roman. 2019. Non-abelian Radon transform and its applications. Pages 115–128, of: The Radon Transform: The First 100 Years and Beyond.Google Scholar
Novikov, Roman, and Sharafutdinov, Vladimir. 2007. On the problem of polarization tomography. I. Inverse Problems, 23(3), 1229–1257.Google Scholar
Novikov, Roman G. 2002b. An inversion formula for the attenuated X-ray transformation. Ark. Mat., 40(1), 145–167.Google Scholar
Oksanen, Lauri, Salo, Mikko, Stefanov, Plamen, and Uhlmann, Gunther. 2020. Inverse Problems for Real Principal Type Operators. To appear in Amer. J. Math.Google Scholar
Otal, Jean-Pierre. 1990. Le spectre marqué des longueurs des surfaces à courbure négative. Ann. Math. (2), 131(1), 151–162.Google Scholar
Palais, Richard S. 1959. Natural operations on differential forms. Trans. Am. Math. Soc., 92, 125–141.Google Scholar
Paternain, Gabriel P. 1999. Geodesic Flows. Progress in Mathematics, vol. 180. Birkhäuser Boston, Inc., Boston, MA.Google Scholar
Paternain, Gabriel P. 2009. Transparent connections over negatively curved surfaces. J. Mod. Dyn., 3(2), 311–333.Google Scholar
Paternain, Gabriel P. 2013. Inverse problems for connections. Pages 369–409 of: Inverse Problems and Applications: Inside Out. II. Math. Sci. Res. Inst. Publ., vol. 60. Cambridge Univ. Press, Cambridge.Google Scholar
Paternain, Gabriel P., and Salo, Mikko. 2018. Carleman Estimates for Geodesic X-ray Transforms. To appear in Ann. Sci. École Norm. Sup.Google Scholar
Paternain, Gabriel P., and Salo, Mikko. 2020. The Non-Abelian X-ray Transform on Surfaces. To appear in J. Differ. Geom..Google Scholar
Paternain, Gabriel P., and Salo, Mikko. 2021. A sharp stability estimate for tensor tomography in non-positive curvature. Math. Z., 298(3-4), 1323–1344.Google Scholar
Paternain, Gabriel P., Salo, Mikko, and Uhlmann, Gunther. 2012. The attenuated ray transform for connections and Higgs fields. Geom. Funct. Anal., 22(5), 1460–1489.Google Scholar
Paternain, Gabriel P., Salo, Mikko, and Uhlmann, Gunther. 2013. Tensor tomography on surfaces. Invent. Math., 193(1), 229–247.Google Scholar
Paternain, Gabriel P., Salo, Mikko, and Uhlmann, Gunther. 2014a. Spectral rigidity and invariant distributions on Anosov surfaces. J. Differ. Geom., 98(1), 147–181.Google Scholar
Paternain, Gabriel P., Salo, Mikko, and Uhlmann, Gunther. 2014b. Tensor tomography: progress and challenges. Chin. Ann. Math. Ser. B, 35(3), 399–428.Google Scholar
Paternain, Gabriel P., Salo, Mikko, and Uhlmann, Gunther. 2015a. Invariant distributions, Beurling transforms and tensor tomography in higher dimensions. Math. Ann., 363(1-2), 305–362.Google Scholar
Paternain, Gabriel P., Salo, Mikko, and Uhlmann, Gunther. 2015b. On the range of the attenuated ray transform for unitary connections. Int. Math. Res. Not. IMRN, 4, 873–897.Google Scholar
Paternain, Gabriel P., Salo, Mikko, Uhlmann, Gunther, and Zhou, Hanming. 2019. The geodesic X-ray transform with matrix weights. Am. J. Math., 141(6), 1707–1750.Google Scholar
Pestov, L. N., and Sharafutdinov, V. A. 1987. Integral geometry of tensor fields on a manifold of negative curvature. Dokl. Akad. Nauk SSSR, 295(6), 1318–1320.Google Scholar
Pestov, Leonid, and Uhlmann, Gunther. 2004. On characterization of the range and inversion formulas for the geodesic X-ray transform. Int. Math. Res. Not., 80, 4331–4347.Google Scholar
Pestov, Leonid, and Uhlmann, Gunther. 2005. Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. Math. (2), 161(2), 1093–1110.Google Scholar
Pressley, Andrew, and Segal, Graeme. 1986. Loop Groups. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York. Oxford Science Publications.Google Scholar
Quinto, Eric Todd. 2006. An introduction to X-ray tomography and Radon transforms. Pages 1–23 of: The Radon Transform, Inverse Problems, and Tomography. Proc. Sympos. Appl. Math., vol. 63. American Mathematical Society, Providence, RI.Google Scholar
Romanov, V. G. 1967. Reconstructing a function by means of integrals along a family of curves. Sibirsk. Mat. Ž., 8, 1206–1208.Google Scholar
Romanov, V. G. 1987. Inverse Problems of Mathematical Physics. VNU Science Press, b.v., Utrecht. With a foreword by V. G. Yakhno, Translated from the Russian by L. Ya. Yuzina.Google Scholar
Royden, H. L. 1956. Rings of analytic and meromorphic functions. Trans. Amer. Math. Soc., 83, 269–276.Google Scholar
Sakai, Takashi. 1996. Riemannian Geometry. Translations of Mathematical Monographs, vol. 149. American Mathematical Society, Providence, RI. Translated from the 1992 Japanese original by the author.Google Scholar
Salo, Mikko, and Uhlmann, Gunther. 2011. The attenuated ray transform on simple surfaces. J. Differ. Geom., 88(1), 161–187.Google Scholar
Sepanski, Mark R. 2007. Compact Lie Groups. Graduate Texts in Mathematics, vol. 235. Springer, New York.Google Scholar
Serre, Jean-Pierre. 1951. Homologie singulière des espaces fibrés. Applications. Ann. Math. (2), 54, 425–505.Google Scholar
Sharafutdinov, V. A. 1994. Integral Geometry of Tensor Fields. Inverse and Ill-posed Problems Series. VSP, Utrecht.Google Scholar
Sharafutdinov, V. A. 1997. Integral geometry of a tensor field on a surface of revolution. Sibirsk. Mat. Zh., 38(3), 697–714, iv.Google Scholar
Sharafutdinov, V. A. 2000. On the inverse problem of determining a connection on a vector bundle. J. Inverse Ill-Posed Probl., 8(1), 51–88.Google Scholar
Sharafutdinov, V. A. 2011. The geometric problem of electrical impedance tomography in the disk. Sibirsk. Mat. Zh., 52(1), 223–238.Google Scholar
Sharafutdinov, Vladimir. 2007. Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds. J. Geom. Anal., 17(1), 147–187.Google Scholar
Sharafutdinov, Vladimir, Skokan, Michal, and Uhlmann, Gunther. 2005. Regularity of ghosts in tensor tomography. J. Geom. Anal., 15(3), 499–542.Google Scholar
Stefanov, Plamen. 2008. A sharp stability estimate in tensor tomography. Page 012007 of: Journal of Physics: Conference Series, (Vol 124: Proceeding of the First International Congress of the IPIA).Google Scholar
Stefanov, Plamen, and Uhlmann, Gunther. 2004. Stability estimates for the X-ray transform of tensor fields and boundary rigidity. Duke Math. J., 123(3), 445–467.Google Scholar
Stefanov, Plamen, and Uhlmann, Gunther. 2005. Boundary rigidity and stability for generic simple metrics. J. Am. Math. Soc., 18(4), 975–1003.Google Scholar
Stefanov, Plamen, and Uhlmann, Gunther. 2009. Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds. J. Differ. Geom., 82(2), 383–409.Google Scholar
Stefanov, Plamen, Uhlmann, Gunther, and Vasy, Andras. 2016. Boundary rigidity with partial data. J. Am. Math. Soc., 29(2), 299–332.Google Scholar
Stefanov, Plamen, Uhlmann, Gunther, Vasy, Andras, and Zhou, Hanming. 2019. Travel time tomography. Acta Math. Sin. (Engl. Ser.), 35(6), 1085–1114.Google Scholar
Stefanov, Plamen, Uhlmann, Gunther, and Vasy, András. 2021. Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge. Ann. Math. (2), 194(1), 1–95.Google Scholar
Stein, Elias M. 1993. Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton, NJ. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.Google Scholar
Stein, Elias M., and Weiss, Guido. 1971. Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, NJ.Google Scholar
Strichartz, Robert S. 1982. Radon inversion—variations on a theme. Am. Math. Monthly, 89(6), 377–384, 420–423.Google Scholar
Struwe, Michael. 1996. Variational Methods. Second edn. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 34. Springer-Verlag, Berlin. Applications to nonlinear partial differential equations and Hamiltonian systems.Google Scholar
Taylor, Michael E. 2011. Partial Differential Equations I. Basic Theory. Second edn. Applied Mathematical Sciences, vol. 115. Springer, New York.Google Scholar
Thorbergsson, Gudlaugur. 1978. Closed geodesics on non-compact Riemannian manifolds. Math. Z., 159(3), 249–258.Google Scholar
Uhlmann, Gunther. 2004. The Cauchy data and the scattering relation. Pages 263–287 of: Geometric Methods in Inverse Problems and PDE Control. IMA Vol. Math. Appl., vol. 137. Springer, New York.Google Scholar
Uhlmann, Gunther. 2014. Inverse problems: Seeing the unseen. Bull. Math. Sci., 4(2), 209–279.Google Scholar
Uhlmann, Gunther, and Vasy, András. 2016. The inverse problem for the local geodesic ray transform. Invent. Math., 205(1), 83–120.Google Scholar
Vertgeĭm, L. B. 1991. Integral geometry with a matrix weight and a nonlinear problem of the reconstruction of matrices. Dokl. Akad. Nauk SSSR, 319(3), 531–534.Google Scholar
Vertgeĭm, L. B. 2000. Weighted integral geometry of matrices. Sibirsk. Mat. Zh., 41(6), 1325–1337, ii.Google Scholar
Ward, R. S. 1988. Soliton solutions in an integrable chiral model in 2 + 1 dimensions. J. Math. Phys., 29(2), 386–389.Google Scholar
Warner, Frank W. 1983. Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, vol. 94. Springer-Verlag, New York-Berlin. Corrected reprint of the 1971 edition.Google Scholar
Wen, Haomin. 2015. Simple Riemannian surfaces are scattering rigid. Geom. Topol., 19(4), 2329–2357.Google Scholar
Wiechert, E., and Geiger, L. 1910. Bestimmung des Weges der Erdbebenwellen im Erdinnern. I. Theoretisches. Physik. Zeitschr., 11, 294–311.Google Scholar
Zhou, Hanming. 2017. Generic injectivity and stability of inverse problems for connections. Comm. Partial Differ. Equ., 42(5), 780–801.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×