[Inversion de la transformée Rayons X géodésique sur des surfaces à courbure négative]
Nous dérivons des formules d’inversion pour la transformée Rayons X géodésique de fonctions (appellée ) et -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 . Des exemples numériques de reconstructions sont fournis en dernière section.
We give reconstruction formulas inverting the geodesic X-ray transform over functions (call it ) and divergence-free -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 . Numerical examples are provided at the end.
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Keywords: geodesic X-ray transform, Radon transform, inversion formulas, surfaces of negative curvature, integral geometry
Mot clés : transformées Rayons X géodésiques, transformée de Radon, formules d’inversion, surfaces à courbure négative, géométrie intégrale
@article{AIF_2017__67_4_1353_0,
author = {Guillarmou, Colin and Monard, Fran\c{c}ois},
title = {Reconstruction formulas for {X-ray} transforms in negative curvature},
journal = {Annales de l'Institut Fourier},
pages = {1353--1392},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {67},
number = {4},
year = {2017},
doi = {10.5802/aif.3112},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.3112/}
}
TY - JOUR AU - Guillarmou, Colin AU - Monard, François TI - Reconstruction formulas for X-ray transforms in negative curvature JO - Annales de l'Institut Fourier PY - 2017 SP - 1353 EP - 1392 VL - 67 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3112/ DO - 10.5802/aif.3112 LA - en ID - AIF_2017__67_4_1353_0 ER -
%0 Journal Article %A Guillarmou, Colin %A Monard, François %T Reconstruction formulas for X-ray transforms in negative curvature %J Annales de l'Institut Fourier %D 2017 %P 1353-1392 %V 67 %N 4 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3112/ %R 10.5802/aif.3112 %G en %F AIF_2017__67_4_1353_0
Guillarmou, Colin; Monard, François. Reconstruction formulas for X-ray transforms in negative curvature. Annales de l'Institut Fourier, Tome 67 (2017) no. 4, pp. 1353-1392. doi : 10.5802/aif.3112. http://www.numdam.org/articles/10.5802/aif.3112/
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