Reconstruction formulas for X-ray transforms in negative curvature

Guillarmou, Colin; Monard, François

Annales de l'Institut Fourier, Tome 67 (2017) no. 4, pp. 1353-1392.

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 $\Pi =I_0^{*}{I}_0$. Des exemples numériques de reconstructions sont fournis en dernière section.

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 ${\Pi }_0=I_0^{*}{I}_0$. Numerical examples are provided at the end.

Classification : 44A12, 35R30, 58J32, 58J47
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

[1] Anantharaman, Nalini; Zelditch, Steve Patterson-Sullivan distributions and quantum ergodicity, Ann. Henri Poincaré, Volume 8 (2007) no. 2, pp. 361-426 | DOI

[2] Anikonov, Yurii E. Some methods for the study of multidimensional inverse problems for differential equations, Nauka Sibirsk. Otdel, Novosibirsk, 1978, 118 pages

[3] Berenstein, Carlos A.; Tarabusi, Enrico Casadio Inversion formulas for the k-dimensional Radon transform in real hyperbolic spaces, Duke Math. J., Volume 62 (1991) no. 3, pp. 613-631 | DOI

[4] Borthwick, David Spectral theory of infinite-area hyperbolic surfaces, Progress in Mathematics, 256, Birkhäuser, 2007

[5] Bowen, Rufus; Ruelle, David The ergodic theory of Axiom A flows, Invent. Math., Volume 29 (1975) no. 3, pp. 181-202 | DOI

[6] Dyatlov, Semyon; Guillarmou, Colin Microlocal limits of plane waves and Eisenstein functions, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014), pp. 371-448 | DOI

[7] Fridman, Buma; Kuchment, Peter; Lancaster, Kirk; Lissianoi, Serguei; Mogilevsky, Mila Numerical harmonic analysis on the hyperbolic plane, Appl. Anal., Volume 76 (2000) no. 3-4, pp. 351-362 | DOI

[8] Guillarmou, Colin Invariant distributions and X-Ray transform for Anosov flows, J. Differ. Geom., Volume 105 (2017), pp. 177-208 | DOI

[9] Guillarmou, Colin Lens rigidity for manifolds with hyperbolic trapped set, J. Amer. Math. Soc., Volume 30 (2017), pp. 561-599 | DOI

[10] Guillarmou, Colin; Naud, Frédéric Wave decay on convex co-compact hyperbolic manifolds, Commun. Math. Phys., Volume 287 (2009) no. 2, pp. 489-511 | DOI

[11] Guillemin, Victor; Kazhdan, David Some inverse Spectral results for negatively curved 2-manifolds, Topology, Volume 19 (1980), pp. 301-312 | DOI

[12] Guillopé, Laurent; Zworski, Maciej Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity, Asymptotic Anal., Volume 11 (1995) no. 1, pp. 1-22

[13] Harish-Chandra Plancherel formula for the $2\times 2$ real unimodular group, Proc. Nat. Acad. Sci. U. S. A., Volume 38 (1952), pp. 337-342 | DOI

[14] Helgason, Sigurdur The totally-geodesic Radon transform on constant curvature spaces, Integral geometry and tomography (Arcata, CA, 1989) (Contemp. Math.), Volume 113, Amer. Math. Soc. (1990), pp. 141-149

[15] Hörmander, Lars The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis, Grundlehren der Mathematischen Wissenschaften, 256, Springer, 1983, ix+391 pages

[16] Iwaniec, Henryk Spectral methods of automorphic forms, Graduate Studies in Mathematics, 53, Amer. Math. Soc., 2002, xi+220 pages

[17] Krishnan, Venkateswaran P. On the inversion formulas of Pestov and Uhlmann for the geodesic ray transform, J. Inverse Ill-Posed Probl., Volume 18 (2010) no. 4, pp. 401-408 | DOI

[18] Lang, Serge ${SL}_2\left(\mathbf{R}\right)$, Graduate Texts in Mathematics, 105, Springer, 1985, xiv+428 pages

[19] Mazzeo, Rafe Elliptic theory of differential edge operators. I, Commun. Partial Differ. Equations, Volume 16 (1991) no. 10, pp. 1615-1664 | DOI

[20] Mazzeo, Rafe; Melrose, Richard B. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., Volume 75 (1987) no. 2, pp. 260-310 | DOI

[21] Monard, François Numerical implementation of geodesic X-ray transforms and their inversion, SIAM J. Imaging Sci., Volume 7 (2014) no. 2, pp. 1335-1357 (electronic only) | DOI

[22] Monard, François On reconstruction formulas for the X-ray transform acting on symmetric differentials on surfaces, Inverse Problems, Volume 30 (2014), 065001 pages (electronic) | DOI

[23] Monard, François Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces, SIAM J. Math. Anal., Volume 48 (2016) no. 2, pp. 1155-1177 | DOI

[24] Mukhometov, R. G. The reconstruction problem of a two-dimensional Riemannian metric, and integral Geometry, Dokl. Akad. Nauk. SSSR, Volume 232 (1977) no. 1, pp. 32-35 (Russian)

[25] Natterer, Frank The Mathematics of Computerized Tomography, Classics in Applied Mathematics, 32, SIAM, 2001, xvii+222 pages

[26] Patterson, Samuel James The limit set of a Fuchsian group, Acta Math., Volume 136 (1976), pp. 241-273 | DOI

[27] Patterson, Samuel James On a lattice-point problem in hyperbolic space and related questions in spectral theory, Ark. Mat., Volume 26 (1988) no. 1, pp. 167-172 | DOI

[28] Pestov, Leonid; Uhlmann, Gunther On the Characterization of the Range and Inversion Formulas for the Geodesic X-Ray Transform, Int. Math. Res. Not., Volume 80 (2004), pp. 4331-4347 | DOI

[29] Pestov, Leonid; Uhlmann, Gunther Two-dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, Volume 161 (2005) no. 2, pp. 1093-1110 | DOI

[30] Radon, Johann Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Leipz. Ber., Volume 69 (1917), pp. 262-277

[31] Sharafutdinov, Vladimir Integral geometry of tensor fields, Inverse and Ill-Posed Problems Series, 1, Walter de Gruyter, 1994, 271 pages

[32] Sullivan, Dennis The density at infinity of a discrete group of hyperbolic motions, Publ. Math., Inst. Hautes Étud. Sci., Volume 50 (1979), pp. 171-202 | DOI