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Pestov, Leonid; Uhlmann, Günther

doi:10.1155/s1073792804142116

Cited by 65 publications

(23 citation statements)

References 6 publications

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“…Moving on to the description of the range of I 0 , the scattering relation ( 9) plays again a fundamental role on simple surfaces, as is shown by the same authors in Pestov and Uhlmann (2004). To state their result, we let A * ± be the adjoint of (10) with expression…”

Section: Motivationmentioning

confidence: 92%

“…The transform (5) has been thoroughly studied for the past few decades, and is quite wellunderstood in many respects. Its injectivity holds in various flexible contexts: in two dimensions, simple 1 geometries Mukhometov (1975); Paternain et al (2013), including further range characterizations and inversion formulas Krishnan (2010); Pestov and Uhlmann (2004); in higher dimensions, geometries admitting convex foliations Stefanov et al (2018); Uhlmann and Vasy (2016). On the smoothing properties of (5) and stability estimates, it is generally understood that, in the absence of conjugate points, the operator I ♯ 0 I 0 is an elliptic pseudodifferential operator of order −1 on M int .…”

Section: Introduction 1main Setting and Questionsmentioning

confidence: 99%

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Functional Relations, Sharp Mapping Properties, and Regularization of the X-Ray Transform on Disks of Constant Curvature

Monard¹

2020

SIAM J. Math. Anal.

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This article surveys recent results aiming at obtaining refined mapping estimates for the X-ray transform on a Riemannian manifold with boundary, which leverage the condition that the boundary be strictly geodesically convex. These questions are motivated by classical inverse problems questions (e.g. range characterization, stability estimates, mapping properties on Hilbert scales), and more recently by uncertainty quantification and operator learning questions.2 A family of Banach (or Hilbert) spaces {(Eα, • α)}α 0 <α<β 0 is a Banach scale in the sense of Krein and Petunin (1966) if (1) E β is densely embedded in Eα when β > α with x α ≤ α,β x β for all x ∈ E β , and (2) interpolation inequalities hold in the sense that there is a function C(α, β, γ) finite at all points of the domain α0 ≤ α < β < γ ≤ β0 such that x β ≤ C(α, β, γ) x

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Section: Motivationmentioning

confidence: 92%

Section: Introduction 1main Setting and Questionsmentioning

confidence: 99%

Functional Relations, Sharp Mapping Properties, and Regularization of the X-Ray Transform on Disks of Constant Curvature

Monard¹

2020

SIAM J. Math. Anal.

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“…Combining Theorem 1.4 with the constructive invertibility of the geodesic ray transform on a simple two-dimensional Riemannian manifold, see [50], [31], [54], see also [43], [44], we obtain the following unconditional result.…”

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confidence: 86%

Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds

Yan¹

2023

IPI

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<p style='text-indent:20px;'>We prove that a continuous potential <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula> can be constructively determined from the knowledge of the Dirichlet–to–Neumann map for the perturbed biharmonic operator <inline-formula><tex-math id="M2">\begin{document}$ \Delta_g^2+q $\end{document}</tex-math></inline-formula> on a conformally transversally anisotropic Riemannian manifold of dimension <inline-formula><tex-math id="M3">\begin{document}$ \ge 3 $\end{document}</tex-math></inline-formula> with boundary, assuming that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [<xref ref-type="bibr" rid="b56">56</xref>]. In particular, our result is applicable and new in the case of smooth bounded domains in the <inline-formula><tex-math id="M4">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>–dimensional Euclidean space as well as in the case of <inline-formula><tex-math id="M5">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>–dimensional admissible manifolds.</p>

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“…This paper studies the inversion of the ray transform on the hyperbolic plane, which arises as a particular linearization of travel time tomography: when the background speed of propagation increases linearly with depth. The relevance of this simplified model is that it captures the effect of diving waves curving back to the surface, due to varying speed of propagation, and it can be explicitly solved [5,20,26,32].…”

Section: Introductionmentioning

confidence: 99%

Semi-local inversion of the geodesic ray transform in the hyperbolic plane

Sáez

2013

Inverse Problems

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The inversion of the ray transform on the hyperbolic plane has applications in geophysical exploration and in medical imaging techniques (such as electrical impedance tomography). The geodesic ray transform has been studied in more general geometries and including attenuation, but all of the available inversion formulas require knowledge of the ray transform for all the geodesics. In this paper we present a different inversion formula for the ray transform on the hyperbolic plane, which has the advantage of only requiring knowledge of the ray transform in a reduced family of geodesics. The required family of geodesics is directly related to the set where the original function is to be recovered.

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Cited by 65 publications

References 6 publications

Functional Relations, Sharp Mapping Properties, and Regularization of the X-Ray Transform on Disks of Constant Curvature

Functional Relations, Sharp Mapping Properties, and Regularization of the X-Ray Transform on Disks of Constant Curvature

Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds

Semi-local inversion of the geodesic ray transform in the hyperbolic plane

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