The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

Monard, François; Stefanov, Plamen; Uhlmann, Günther

doi:10.1007/s00220-015-2328-6

Cited by 45 publications

(98 citation statements)

References 38 publications

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“…Some of these artifacts are created near each "source bump" as well as their respective conjugate loci. This is an effect that will be discussed at length in a forthcoming work studying non-simple metrics theoretically and numerically, see [11].…”

Section: Manifolds With a Radially Symmetric Metricmentioning

confidence: 87%

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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Section: Manifolds With a Radially Symmetric Metricmentioning

confidence: 87%

Numerical Implementation of Geodesic X-Ray Transforms and Their Inversion

Monard¹

2014

SIAM J. Imaging Sci.

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“…When we have limited X-ray data it is not guaranteed that we can see all the singularities of f from the data. Singularities which are invisible in the microlocal sense are related to the instability of inverting f from its limited X-ray data [22,29,30,31]. See also [21,32] for discussion of which part of the wave front set is visible in limited data tomography.…”

Section: 2mentioning

confidence: 99%

Unique continuation of the normal operator of the x-ray transform and applications in geophysics

Ilmavirta

Mönkkönen²

2020

Inverse Problems

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We show that the normal operator of the X-ray transform in R d , d ≥ 2, has a unique continuation property in the class of compactly supported distributions. This immediately implies uniqueness for the X-ray tomography problem with partial data and generalizes earlier results to lower regularity. Our proof also gives a unique continuation property for certain Riesz potentials. We present applications to local and global seismology. These include linearized travel time tomography with half-local data and global tomography based on shear wave splitting in a weakly anisotropic elastic medium.

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“…In geometries with conjugate points, another separation between two-and higher-thanthree dimensions occurs: in two dimensions, conjugate points on surfaces unconditionally destroy stability of X-ray transforms [114,70] while the question needs to be refined in higher dimensions and exhibits a tradeoff between the order of conjugate points considered and the dimension of the manifold [44]. In fact, there is more at play in higher dimensions: the mere existence of a foliation by strictly convex hypersurfaces allows to prove global injectivity and stability [120,116,87].…”

Section: The Inverse Problems Agenda In a Geometric Contextmentioning

confidence: 99%

4. Integral geometry on manifolds with boundary and applications

Ilmavirta¹,

Monard²

2019

The Radon Transform

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We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on manifolds with boundary.Johann Radon and his contemporaries formulated several integral geometric problems, not only in linear but also in non-linear settings [42,123]. Such problems, namely travel-time tomography and boundary rigidity as later formulated in [72,62], are concerned with recovering a Riemannian metric from the shortest length between any two boundary points. Such problems and their cousins (described below), now make the field of integral geometry, or how to reconstruct geometric features of a manifold from integral functionals defined over that manifold.Nowadays this field forms the basis of several non-invasive approaches to imaging internal properties of materials: seismology [42,123], or how to reconstruct the density inside the Earth from first arrival times of seismic wavefronts; medical imaging since the development of X-ray Computerized Tomography [75,119,25]; Single-Photon Emission Computerized Tomography using the attenuated X-ray transform [76,74,78]; vector tomography in helio-seismology [51,94,52]; ocean imaging [73]; X-ray diffraction strain tomography [59,19] and tomography in elastic media [103, Ch. 7][108]; neutron imaging, as applied to the imaging of vertebrate remains [102] and shales [11]. Non-linear integral geometric problems also continue to find new applications: recently, Neutron Spin Tomography [99] as a means to measure magnetic fields in materials, has arised as a novel method which can be of use in electrical engineering, superconductivity, etc. The transform to invert in this case is a non-linear operator, the so-called "non-abelian X-ray transform" of the magnetic field, see Problem 3 below.Recent breakthroughs have fuelled the field, exploiting a combination of old and new methods. Examples of such methods are: the systematic use of analysis on the unit sphere bundle combining energy methods (also coined "Pestov identities"), initiated by Mukhometov [71] and generalized in [91,103], and harmonic analysis on the tangent fibers [15,83,86]; in dimensions three and higher, the discovery in [120] that the existence of a foliation of the domain by strictly convex hyperfsurfaces, local or global, yields a powerful and robust approach to integral geometric inversions [116,118,125, 127], via a successful use of Melrose's scattering calculus [60]; the systematic use of analytic microlocal analysis to produce 'generic' results, implying the unique identifiability of unknown parameters in an open and dense subset of all cases [112,45, 128]; finally, recent results in the context of Anosov flows, leading to positive results for certain geometries with trapped sets [33,34,38].This review article aims at giving an overview of the arsenal of these methods, and to describe to what extent they help coping with various geometric settings, whose complexity is mainly governed by two features ...

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The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

Cited by 45 publications

References 38 publications

Numerical Implementation of Geodesic X-Ray Transforms and Their Inversion

Numerical Implementation of Geodesic X-Ray Transforms and Their Inversion

Unique continuation of the normal operator of the x-ray transform and applications in geophysics

4. Integral geometry on manifolds with boundary and applications

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