The inverse problem for the local geodesic ray transform

Uhlmann, Günther; Vasy, András

doi:10.1007/s00222-015-0631-7

Cited by 154 publications

(412 citation statements)

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“…The latter allows the use of the clean intersection calculus of Duistermaat and Guillemin [9] to show that X * X is a pseudodifferential operator (ΨDO), elliptic when κ = 0. In the geodesic case under consideration, a microlocal study of Xf has been done in [31,32,10,35,37,39], and in some of those works, f can even be a tensor field.…”

Section: 1)mentioning

confidence: 99%

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

Monard

Stefanov

Uhlmann

2015

Commun. Math. Phys.

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Abstract. We study the geodesic X-ray transform X on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of X and relate it to the conjugate locus. We present numerical examples illustrating the cancellation of singularities. We also show that the attenuated X-ray transform is well posed if the attenuation is positive and there are no more than two conjugate points along each geodesic; but still ill-posed, if there are three or more conjugate points. Those results follow from our analysis of the weighted X-ray transform.

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Section: 1)mentioning

confidence: 99%

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

Monard

Stefanov

Uhlmann

2015

Commun. Math. Phys.

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“…Sharafutdinov established in [24] s-injectivity over tensors of any order on spherically symmetric layers satisfying the Herglotz non-trapping condition. In dimension three or higher, Stefanov and Uhlmann proved in [27] s-injectivity for realanalytic metrics satisfying some additional conditions, and Uhlmann and Vasy proved in [29] local injectivity of the ray transform on manifolds satisfying a foliation condition, including a reconstruction algorithm. While the question of s-injectivity remains open for general domains and metrics, it is shown in [28] using microlocal analysis that when the metric has caustics of fold type and the manifold is two-dimensional, the singularities of the unknown function that are conormal to a fold can no longer be resolved by the ray transform, thus showing that caustic sets have detrimental effects on the stability of the ray transform.…”

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Numerical Implementation of Geodesic X-Ray Transforms and Their Inversion

Monard¹

2014

SIAM J. Imaging Sci.

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Abstract. We present a numerical implementation of the geodesic ray transform and its inversion over functions and solenoidal vector fields on two-dimensional Riemannian manifolds. For each problem, inversion formulas previously derived by Pestov and Uhlmann in [Int. Math Research Notices 80 (2004)] then extended by Krishnan in [J. Inv. Ill-Posed Problems 18 (2010)] are implemented in the case of simple and some non-simple metrics. These numerical tools are also used to better understand and gain intuition about non-simple manifolds, for which injectivity and stability of the corresponding integral geometric problems are still under active study.Key words. geodesic ray transform, Radon transform, tensor tomography problem, inverse problems, Riemann surfaces AMS subject classifications. 65R10, 65R32, 53D25, 44A121. Introduction. The present article discusses a numerical implementation in MatLab of geodesic X-Ray transforms of functions and solenoidal (i.e. divergence-free) vector fields and their inversion on two dimensional Riemannian manifolds with boundary.Geodesic X-ray transforms appear in problems of mathematical physics where particles travel along some curves and "gather information" along their path. Probably the best known example of geodesic ray transform is that of the Radon transform in two dimensions (also known as the X-Ray transform, as these two transforms coincide in two dimensions), which is the collection of integrals of a given function over all straight lines in the plane, corresponding to the case of a Euclidean metric. Reconstructing a function from its integrals along lines was first considered and solved in [21] and is now used every day in medical imaging. A thorough account of theoretical and numerical aspects for this transform may be found in [13]. In the Euclidean case, solenoidal tensors of any order were also explicitely reconstructed in [23].When optical rays propagate in a medium with variable index of refraction, their trajectories, no longer straight lines, come as geodesics of some Riemannian metric. In this framework, the same questions (injectivity, stability, range characterization, reconstruction algorithms, inverse problems with partial data) as for the straight line case are still under active theoretical study. To the author's knowledge, numerical simulations for these transforms remain to be documented.When the metric is simple, injectivity over functions was proved in [12] and injectivity over solenoidal vector fields was established in [1,2]. Under the same simplicity assumption, the problem was recently proved in [17] to be injective over solenoidal tensors ("s-injective") of any order, and previously in [4] under assumptions on the curvature. Independently, stability estimates were given in [26] via a microlocal study of the normal operator.

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“…Experts are taking the claim seriously, in part because it builds on a technical step from a linear form of the problem that the community has accepted as a breakthrough 4 , adds UCL mathematician Yaroslav Kurylev. So far, says Paternain, the impression is "excellent".…”

Section: From Theory To Realitymentioning

confidence: 99%

World’s largest wind-mapping project spins up in Portugal

Witze¹

2017

Nature

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The inverse problem for the local geodesic ray transform

Cited by 154 publications

References 19 publications

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

Numerical Implementation of Geodesic X-Ray Transforms and Their Inversion

World’s largest wind-mapping project spins up in Portugal

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