We study the microlocal properties of the geodesic X-ray transform X on a manifold with boundary allowing the presence of conjugate points. Assuming that there are no self-intersecting geodesics and all conjugate pairs are nonsingular we show that the normal operator N = X t • X can be decomposed as the sum of a pseudodifferential operator of order −1 and a sum of Fourier integral operators. We also apply this decomposition to prove inversion of X is only mildly ill-posed in dimension three or higher.

This article deals with stability issues related to geodesic X-ray transforms, where an interplay between the (attenuation type) weight in the transform and the underlying geometry strongly impact whether the problem is stable or unstable. In the unstable case, we also explain what types of artifacts are expected in terms of the underlying conjugate points and the microlocal weights at those points. We show in particular that the well-known iterative reconstruction Landweber algorithm cannot provide accurate reconstruction when the problem is unstable, though the artifacts generated, specific for the reconstruction algorithm, can be properly described.

We consider the tomography problem of recovering a covector field on a simple Riemannian manifold based on its weighted Doppler transformation over a family of curves Γ. This is a generalization of the attenuated Doppler transform. Uniqueness is proven for a generic set of weights and families of curves under a condition on the weight function. This condition is satisfied in particular if the weight function is never zero, and its derivatives along the curves in Γ is never zero.

MSC: 35R30 86A15 53C21Keywords: Geometric inverse problems Riemannian manifold Shape operatorWe analyze the inverse problem, if a manifold and a Riemannian metric on it can be reconstructed from the sphere data. The sphere data consist of an open set U ⊂M and the pairs (t, Σ) where Σ ⊂ U is a smooth subset of a generalized metric sphere of radius t. This problem is an idealization of a seismic inverse problem, originally formulated by Dix [8], of reconstructing the wave speed inside a domain from boundary measurements associated with the single scattering of waves. In this problem, one considers a domain M with a varying and possibly anisotropic wave speed which we model as a Riemannian metric g. For our data, we assume that M contains a dense set of point diffractors and that in a subset U ⊂M , we can measure the wave fronts of the waves generated by these. The inverse problem we study is to recover the metric g in local coordinates anywhere on a set M ⊂M up to an isometry (i.e. we recover the isometry type of M ). To do this we show that the shape operators related to wave fronts produced by the point diffractors within M satisfy a certain system of differential equations which may be solved along geodesics of the metric. In this way, assuming that we know g as well as the shape operator of the wave fronts in the region U , we may recover g in certain coordinate systems (e.g. Riemannian normal coordinates centered at point diffractors). This generalizes the method of Dix to metrics which may depend on all spatial variables and be anisotropic. In particular, the novelty of this solution lies in the fact that it can be used to reconstruct the metric also in the presence of the caustics. (M.V. de Hoop), sean.holman@manchester.ac.uk (S.F. Holman), Einar.Iversen@norsar.com (E. Iversen), matti.lassas@helsinki.fi (M. Lassas), bjorn.ursin@ntnu.no (B. Ursin). http://dx.doi.org/10.1016/j.matpur.2014.09.003 0021-7824/© 2014 Elsevier Masson SAS. All rights reserved. M.V. de Hoop et al. / J. Math. Pures Appl. 103 (2015) 830-848 831 r é s u m éOn analyse un problème inverse, si une variété riemannienne peut être reconstruite à partir des données sphère. Les données sphère sont constituées d'un ensemble ouvert U ⊂M et les paires (t, Σ), où Σ ⊂ U set un sous-ensemble lisse d'une sphère métrique généralisée. Ce problème est une idéalisation d'un problème sismique inverse, à l'origine formulé par Dix [8], consistant à reconstruire la vitesse d'onde dans un domaine à partir des mesures aux frontières associées à la dispersion simple des ondes sismiques. On considère un domaine M avec une vitesse d'onde variable et éventuellement anisotrope modélisée par une métrique riemannienne g. On suppose que M contient une densité élevée de points diffractants et que dans un sous-ensemble U ⊂M , correspondant à un domaine contenant les instruments de mesure, on peut mesurer les fronts d'onde de la diffusion simple des ondes diffractées depuis les points diffractants. Le problème inverse étudié consiste à reconstruire la métrique g en coordon...

Abstract-Inspired by the recent advances on minimizing nonsmooth or bound-constrained convex functions on models using varying degrees of fidelity, we propose a line search multigrid (MG) method for full-wave iterative image reconstruction in photoacoustic tomography (PAT) in heterogeneous media. To compute the search direction at each iteration, we decide between the gradient at the target level, or alternatively an approximate error correction at a coarser level, relying on some predefined criteria. To incorporate absorption and dispersion, we derive the analytical adjoint directly from the first-order acoustic wave system. The effectiveness of the proposed method is tested on a total-variation penalized Iterative Shrinkage Thresholding algorithm (ISTA) and its accelerated variant (FISTA), which have been used in many studies of image reconstruction in PAT. The results show the great potential of the proposed method in improving speed of iterative image reconstruction.