Microlocal Analysis of a Restricted Ray Transform on Symmetric $m$-Tensor Fields in $\mathbb{R}^{n}$

Krishnan, Venkateswaran P.; Mishra, Rohit Kumar

doi:10.1137/18m1178530

Cited by 9 publications

(2 citation statements)

References 37 publications

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“…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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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: Introductionmentioning

confidence: 99%

Generalized V-line transforms in 2D vector tomography

Ambartsoumian¹,

Jebelli²,

Mishra³

2020

Inverse Problems

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“…The reconstruction problem for both the longitudinal and transverse ray transforms in n-dimensions (n ⩾ 3) is overdetermined, therefore one would like to have a reconstruction algorithm using only an n-dimensional restriction of the data. Such questions related to restricted data are also wellstudied in different settings; for instance, see [11,12,23,25,35] and references therein. These problems, sometimes collectively called tomography of tensor fields, have applications in plasma physics, polarization imaging, prediction of earthquakes, and many other areas related to the propagation of radiation or waves through anisotropic media.…”

Section: Introductionmentioning

confidence: 99%

V-line 2-tensor tomography in the plane

Ambartsoumian,

Mishra,

Zamindar

2024

Inverse Problems

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In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-tensor fields in R2. The operators of interest include the longitudinal, transverse, and mixed VLTs, their integral moments, and the star transform. With the exception of the star transform, all these operators are natural generalizations to the broken-ray trajectories of the corresponding well-studied concepts defined for straight-line paths of integration. We characterize the kernels of the VLTs and derive exact formulas for reconstruction of tensor fields from various combinations of these transforms. The star transform on tensor fields is an extension of the corresponding concepts that have been previously studied on vector fields and scalar fields (functions). We describe all injective configurations of the star transform on symmetric 2-tensor fields and derive an exact, closed-form inversion formula for that operator.

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Unique Continuation Results for Certain Generalized Ray Transforms of Symmetric Tensor Fields

2022

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Let Im denote the Euclidean ray transform acting on compactly supported symmetric m-tensor field distributions f , and I * m be its formal L 2 adjoint. We study a unique continuation result for the normal operator Nm = I * m Im. More precisely, we show that if Nm vanishes to infinite order at a point x 0 and if the Saint-Venant operator W acting on f vanishes on an open set containing x 0 , then f is a potential tensor field. This generalizes two recent works of Ilmavirta and Mönkkönen who proved such unique continuation results for the ray transform of functions and vector fields/1-forms. One of the main contributions of this work is identifying the Saint-Venant operator acting on higher order tensor fields as the right generalization of the exterior derivative operator acting on 1-forms, which makes unique continuation results for ray transforms of higher order tensor fields possible. In the second half of the paper, we prove analogous unique continuation results for momentum ray and transverse ray transforms.

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Microlocal Analysis of a Restricted Ray Transform on Symmetric $m$-Tensor Fields in $\mathbb{R}^{n}$

Cited by 9 publications

References 37 publications

Generalized V-line transforms in 2D vector tomography

Generalized V-line transforms in 2D vector tomography

V-line 2-tensor tomography in the plane

Unique Continuation Results for Certain Generalized Ray Transforms of Symmetric Tensor Fields

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