The attenuated geodesic x-ray transform

Holman, Sean; Monard, François; Stefanov, Plamen

doi:10.1088/1361-6420/aab8bc

Cited by 21 publications

(40 citation statements)

References 39 publications

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“…Picking f 1 ∈ L 2 (U 1 ) arbitrary and f 2 = 0, and writing the Landweber iteration at leading order, the iterations converge to Landweber solution = (Id − P )f 1 + F 21 P f 1 , P := (Id + F * 21 F 21 ), see [43,Sec. 3.2].…”

Section: Two Dimensionsmentioning

confidence: 99%

“…The work in [70,43] also covers the case of attenuations, namely in constructing corresponding artifact-generating operators in the unstable case, and studying the long-term behavior of the Landweber iteration when applied to the attenuated X-ray transform. A characteristic example of the locality of the discussion regarding the interplay between the geometry and the presence of a weight can be found Fig.…”

Section: Two Dimensionsmentioning

confidence: 99%

See 1 more Smart Citation

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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Section: Two Dimensionsmentioning

confidence: 99%

Section: Two Dimensionsmentioning

confidence: 99%

4. Integral geometry on manifolds with boundary and applications

Ilmavirta¹,

Monard²

2019

The Radon Transform

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“…Non-detectability and invisibility results have been extensively studied for inverse problems, see [5,6,7,8] and references therein. For the Riemannian geodesic ray transform, it was shown in [24], see also [14], that in presence of conjugate points, singularities cannot be resolved locally, at least, i.e., knowing the ray transform near a single (directed) geodesic. We will prove an analogous result in the Lorentzian case in 1 + 2 dimensions.…”

Section: Cancellation Of Singularities In Two Dimensionsmentioning

confidence: 99%

The Light Ray Transform on Lorentzian Manifolds

Lassas

Oksanen

Stefanov

et al. 2020

Commun. Math. Phys.

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“…One would expect and we confirm that recovery of singularities are affected by the existence of conjugate points on ν. Much work has been done for the class of X-ray type transform with conjugate points [32,31,22,11]. In the case of transform for a generic family of smooth curves [8], if there are no conjugate points, the Partly supported by NSF Grant DMS-1600327. localized normal operator is an elliptic pseudodifferential operator of order −1.…”

Section: Introductionmentioning

confidence: 99%

“…It is also proved that the attenuated geodesic ray transform is well posed under certain conditions. Most recently, [11] provides a thorough analysis of the stability of attenuated geodesic ray transform and shows what artifacts we can expect when using the Landweber iterative reconstruction for unstable problems.…”

Section: Introductionmentioning

confidence: 99%

Artifacts in the inversion of the broken ray transform in the plane

Zhang¹

2020

Inverse Problems &Amp; Imaging

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We study the integral transform over a general family of broken rays in R 2 . It is natural for broken rays to have conjugate points, for example, when they are reflected from a curved boundary. If there are conjugate points, we show that the singularities cannot be recovered from local data and therefore artifacts arise in the reconstruction. We apply these conclusions to two examples, the V-line Radon transform and the parallel ray transform. In each example, a detailed discussion of the local and global recovery of singularities is given and we perform numerical experiments to illustrate the results.

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The attenuated geodesic x-ray transform

Cited by 21 publications

References 39 publications

4. Integral geometry on manifolds with boundary and applications

4. Integral geometry on manifolds with boundary and applications

The Light Ray Transform on Lorentzian Manifolds

Artifacts in the inversion of the broken ray transform in the plane

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