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Inversion of a restricted transverse ray transform with sources on a curve

Rohit Kumar Mishra, Chandni Thakkar

Abstract

In this paper, a restricted transverse ray transform acting on vector and symmetric $m$-tensor fields is studied. We developed inversion algorithms using restricted transverse ray transform data to recover symmetric $m$-tensor fields in $\mathbb{R}^3$ and vector fields in $\mathbb{R}^n$. We restrict the transverse ray transform to all lines going through a fixed curve $γ$ that satisfies the Kirillov-Tuy condition. We show that the known restricted data can be used to reconstruct a specific weighted Radon transform of the unknown vector/tensor field's components, which we then use to explicitly recover the unknown field.

Inversion of a restricted transverse ray transform with sources on a curve

Abstract

In this paper, a restricted transverse ray transform acting on vector and symmetric -tensor fields is studied. We developed inversion algorithms using restricted transverse ray transform data to recover symmetric -tensor fields in and vector fields in . We restrict the transverse ray transform to all lines going through a fixed curve that satisfies the Kirillov-Tuy condition. We show that the known restricted data can be used to reconstruct a specific weighted Radon transform of the unknown vector/tensor field's components, which we then use to explicitly recover the unknown field.
Paper Structure (6 sections, 9 theorems, 66 equations)

This paper contains 6 sections, 9 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Some notations and statement of main theorems
  3. Proof of Theorem \ref{['th:Main Theorem']}
  4. Proof of Theorem \ref{['th:Main Theorem 2']}
  5. Appendix
  6. Acknowledgements

Key Result

Theorem 1

Let $f$ be a symmetric $m$-tensor field in $\mathbb{R}^3$, which is supported in $B \subset \mathbb{R}^3$. Assume that a curve $\gamma \subset \mathbb{R}^3$, satisfying the Kirillov-Tuy condition of order $m$, encompasses $B$ and the transverse ray transform $\mathcal{T}_i f$, for $i = 0, \dots, m$

Theorems & Definitions (28)

  • Definition 1: Microlocal_2021Sharafutdinov_1994
  • Definition 2: Microlocal_2021
  • Remark 1
  • Definition 3
  • Definition 4: Generic vectors Denisjuk_2006
  • Definition 5: Kirillov-Tuy condition
  • Example 1
  • Definition 6: Modified Kirillov-Tuy condition
  • Definition 7: Vertgeim_2000
  • Remark 2
  • ...and 18 more