Reconstruction formulas for X-ray transforms in negative curvature
Annales de l'Institut Fourier, Volume 67 (2017) no. 4, p. 1353-1392
We give reconstruction formulas inverting the geodesic X-ray transform over functions (call it I 0 ) and divergence-free 1-forms on surfaces with negative curvature and strictly convex boundary. These formulas generalize formulas by L. Pestov and G. Uhlmann previously established for simple surfaces, to cases allowing geodesics with infinite length. Such formulas take the form of Fredholm equations, where the analysis of error operators requires deriving new estimates for the normal operator Π 0 =I 0 * I 0 . Numerical examples are provided at the end.
Nous dérivons des formules d’inversion pour la transformée Rayons X géodésique de fonctions (appellée I 0 ) et 1-formes à divergence nulle, définie sur des surfaces à courbure negative et bord convexe. Ces formules généralisent celles de L. Pestov et G. Uhlmann dans [28] (valides pour le cas de surfaces dites « simples ») à des cas autorisant des géodésiques captées (i.e., de longueur infinie). Les formules prennent la forme d’équations de Fredholm, dans lesquelles l’analyse des opérateurs d’erreur requiert la dérivation de nouvelles estimées de continuité pour l’operateur normal Π=I 0 * I 0 . Des exemples numériques de reconstructions sont fournis en dernière section.
Received : 2015-12-24
Revised : 2016-06-06
Accepted : 2016-09-16
Published online : 2017-09-26
Classification:  44A12,  35R30,  58J32,  58J47
Keywords: geodesic X-ray transform, Radon transform, inversion formulas, surfaces of negative curvature, integral geometry
     author = {Guillarmou, Colin and Monard, Fran\c cois},
     title = {Reconstruction formulas for X-ray transforms in negative curvature},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {4},
     year = {2017},
     pages = {1353-1392},
     doi = {10.5802/aif.3112},
     language = {en},
     url = {}
Guillarmou, Colin; Monard, François. Reconstruction formulas for X-ray transforms in negative curvature. Annales de l'Institut Fourier, Volume 67 (2017) no. 4, pp. 1353-1392. doi : 10.5802/aif.3112.

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