Reconstruction Formulas For X-Ray Transforms In Negative Curvature

Colin Guillarmou (1) , François Monard (2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 Department of Mathematics - University of Michigan

We give reconstruction formulas inverting the geodesic X-ray transform over functions (call it I_0) and solenoidal vector fields on surfaces with negative curvature and strictly convex boundary. These formulas generalize the Pestov-Uhlmann formulas in [PeUh] (established for simple surfaces) to cases allowing geodesics with infinite length on surfaces with trapping. 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.

Keywords: X-ray transform; inversion formula; negative curvature

Domains: Spectral Theory [math.SP]; Differential Geometry [math.DG]; Dynamical Systems [math.DS]

Colin Guillarmou, François Monard. Reconstruction Formulas For X-Ray Transforms In Negative Curvature. Annales de l'Institut Fourier, 2017, 67 (4), pp.1353--1392. ⟨10.5802/aif.3112⟩. ⟨hal-01236235⟩