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Tensor tomography using V-line transforms with vertices restricted to a circle

Rohit Kumar Mishra, Anamika Purohit, Indrani Zamindar

Abstract

In this article, we study the problem of recovering symmetric $m$-tensor fields (including vector fields) supported in a unit disk $\mathbb{D}$ from a set of generalized V-line transforms, namely longitudinal, transverse, and mixed V-line transforms, and their integral moments. We work in a circular geometric setup, where the V-lines have vertices on a circle, and the axis of symmetry is orthogonal to the circle. We present two approaches to recover a symmetric $m$-tensor field from the combination of longitudinal, transverse, and mixed V-line transforms. With the help of these inversion results, we are able to give an explicit kernel description for these transforms. We also derive inversion algorithms to reconstruct a symmetric $m$-tensor field from its first $(m+1)$ moment longitudinal/transverse V-line transforms.

Tensor tomography using V-line transforms with vertices restricted to a circle

Abstract

In this article, we study the problem of recovering symmetric -tensor fields (including vector fields) supported in a unit disk from a set of generalized V-line transforms, namely longitudinal, transverse, and mixed V-line transforms, and their integral moments. We work in a circular geometric setup, where the V-lines have vertices on a circle, and the axis of symmetry is orthogonal to the circle. We present two approaches to recover a symmetric -tensor field from the combination of longitudinal, transverse, and mixed V-line transforms. With the help of these inversion results, we are able to give an explicit kernel description for these transforms. We also derive inversion algorithms to reconstruct a symmetric -tensor field from its first moment longitudinal/transverse V-line transforms.

Paper Structure

This paper contains 13 sections, 9 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Main results
  4. Proof of Theorem \ref{['Recovery from M_lf']} (Approach 1)
  5. Vector fields ($m=1$)
  6. Symmetric m-tensor fields
  7. Proof of Theorem \ref{['Recovery from M_lf']} (Approach 2)
  8. Vector fields
  9. Symmetric 2-tensor fields
  10. Proof of Theorem \ref{['Recovery from longi/trans moments']}
  11. Vector fields
  12. Symmetric m-tensor fields
  13. Acknowledgements

Key Result

Theorem 2.3

Let $h\in C^{\infty}_c(\mathbb{D})$, then $h$ can be recovered from $\mathcal{V}_{w}h$ from any of the following inversion formulas: where $h_n$ and $(\mathcal{V}_{w}h)_{n}$ are the $n^{th}$ Fourier coefficients of the $h$ and $\mathcal{V}_{w}h$, respectively.

Figures (1)

  • Figure 1: A V-line with vertex $\Phi(\phi)=(\cos{\phi},\sin{\phi})$ and directions $\textbf{u} = \Phi(\pi + \phi - \psi)$ and $\textbf{v} = \Phi(\pi + \phi + \psi)$. Both the rays are at a distance $s = \sin \psi$ from the origin and are orthogonal to unit vectors $\Phi(\phi - \psi+ \pi/2)$ and $\Phi(\phi + \psi+ 3\pi/2) = \Phi(\phi + \psi -\pi/2)$, respectively.

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Remark 1
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 17 more