Momentum ray transforms, II: range characterization in the Schwartz space

Krishnan, Venkateswaran P.; Manna, Ramesh; Sahoo, Suman Kumar; Sharafutdinov, V. A.

doi:10.1088/1361-6420/ab6a65

Cited by 20 publications

(14 citation statements)

References 8 publications

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“…Using the above definition, we generalize the well-known momentum ray transforms mapping vector or tensor fields to weighted integrals of their components along straight lines (e.g. see [1,10,28,29,30,36,38]) to the case of transforms integrating along V-line paths as follows.…”

Section: Definitions and Notationsmentioning

confidence: 99%

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Generalized V-line transforms in 2D vector tomography

Ambartsoumian¹,

Jebelli²,

Mishra³

2020

Inverse Problems

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We study the inverse problem of recovering a vector field in R 2 from a set of new generalized V-line transforms in three different ways. First, we introduce the longitudinal and transverse V-line transforms for vector fields in R 2 . We then give an explicit characterisation of their respective kernels and show that they are complements of each other. We prove invertibility of each transform modulo their kernels and combine them to reconstruct explicitly the full vector field. In the second method, we combine the longitudinal and transverse V-line transforms with their corresponding first moment transforms and recover the full vector field from either pair. We show that the available data in each of these setups can be used to derive the signed V-line transform of both scalar component of the vector field, and use the known inversion of the latter. The final major result of this paper is the derivation of an exact closed form formula for reconstruction of the full vector field in R 2 from its star transform with weights. We solve this problem by relating the star transform of the vector field to the ordinary Radon transform of the scalar components of the field.

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Section: Definitions and Notationsmentioning

confidence: 99%

“…The first four concepts were motivated by the analogous generalizations of the classical Radon transform to vector fields (e.g. see [1,11,12,23,24,26,28,29,30,31,32,33,36,37,38,41,42,48]). The vector star transform is a natural extension of the longitudinal and transverse VLTs to the case of trajectories with more than two branches.…”

Section: Introductionmentioning

confidence: 99%

Generalized V-line transforms in 2D vector tomography

Ambartsoumian¹,

Jebelli²,

Mishra³

2020

Inverse Problems

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“…The corresponding problem for tensors of higher order in non-Euclidean spaces has also been considered in [36,32,2,18], see [31] for a comprehensive review. On simple Riemannian surfaces, the range characterization of the geodesic X-ray transform of compactly supported functions has been established in terms of the scattering relation in the breakthrough work in [32].…”

Section: Introductionmentioning

confidence: 99%

On the range of the planar $X$-ray transform on the Fourier lattice of the torus

Sadiq¹,

Tamasan²

2022

Preprint

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“…The systematic study of tensor tomography in non-Euclidean spaces originated in [42]. On simple Riemannian surfaces, the range characterization of the geodesic X-ray of compactly supported 0 and 1 has been established in terms of the scattering relation in [36], and the results were extended to symmetric tensors of arbitrary order in [2], and to attenuating media in [20]; see [35] for a comprehensive survey.…”

Section: Introductionmentioning

confidence: 99%