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V-line 2-tensor tomography in the plane

Gaik Ambartsoumian, Rohit Kumar Mishra, Indrani Zamindar

Abstract

In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-tensor fields in $\mathbb{R}^2$. 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.

V-line 2-tensor tomography in the plane

Abstract

In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-tensor fields in . 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.
Paper Structure (9 sections, 19 theorems, 110 equations, 1 figure)

This paper contains 9 sections, 19 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions, notations, and preliminary results
  3. Kernel descriptions of transforms $\mathcal{L}$, $\mathcal{T}$, and $\mathcal{M}$
  4. Reconstruction results for special kinds of tensor fields
  5. Recovery of a tensor field $\textbf{f}$ from $\mathcal{L} \textbf{f}$, $\mathcal{T} \textbf{f}$, and $\mathcal{M}\textbf{f}$
  6. Reconstruction of a tensor field using integral moments
  7. Full recovery of a tensor field from its star transform
  8. Additional remarks
  9. Acknowledgements

Key Result

Theorem 1

Let $\textbf{f}\in C_c^2\left(S^2;D_1\right)$. Then the knowledge of $\mathcal{L} \textbf{f}$ and $\mathcal{T} \textbf{f}$ determines the trace of $\textbf{f}$ as follows: where $\textbf{e}_2=(0,1)$.

Figures (1)

  • Figure 1: From Gaik_Mohammad_Rohit.

Theorems & Definitions (53)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • ...and 43 more