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Reconstruction of a vector field and a symmetric $2$-tensor field from the moment ray transforms in $\mathbb{R}^2$

Rahul Bhardwaj, Karishman B. Solanki

TL;DR

This work addresses recovering a real-valued vector field $\mathbf{f}$ and a symmetric tensor field $\mathbf{F}$ supported in a strictly convex domain in $\mathbb{R}^{2}$ from the $\mathfrak{a}$-attenuated moment ray transforms of orders $0$ to $2$. The authors convert the inverse problem into a boundary-value problem for a coupled transport system on $\overline{\Omega}\times\mathbb{S}^{1}$ and solve it using the Bukhgeim A-analytic theory, employing the Bukhgeim-Cauchy and Pompeiu operators to handle Fourier components. They prove unique recovery of $\mathbf{f}$ and $\mathbf{F}$ from the data set $\mathcal{M}_{\mathfrak{a}}(\mathcal{F})$, and establish a stability estimate that bounds $\|\mathbf{f}\|_{L^{2}(\Omega)}+\|\mathbf{F}\|_{L^{2}(\Omega)}$ by norms of the measured transforms. The attenuated case is treated via an integrating factor, reducing it to the non-attenuated scenario and enabling the same reconstruction and stability framework. This provides a complete reconstruction procedure for full tensor fields from moment ray data with quantitative stability.

Abstract

We present a technique for recovering a vector field and a symmetric $2$-tensor field, both real-valued and compactly supported in some strictly convex bounded domain with smooth boundary in the Euclidean plane, from the sum of their attenuated moment ray transforms. In addition, we provide a stability estimate for recovering both the vector field and the symmetric $2$-tensor field from the aforementioned ray transform.

Reconstruction of a vector field and a symmetric $2$-tensor field from the moment ray transforms in $\mathbb{R}^2$

TL;DR

This work addresses recovering a real-valued vector field and a symmetric tensor field supported in a strictly convex domain in from the -attenuated moment ray transforms of orders to . The authors convert the inverse problem into a boundary-value problem for a coupled transport system on and solve it using the Bukhgeim A-analytic theory, employing the Bukhgeim-Cauchy and Pompeiu operators to handle Fourier components. They prove unique recovery of and from the data set , and establish a stability estimate that bounds by norms of the measured transforms. The attenuated case is treated via an integrating factor, reducing it to the non-attenuated scenario and enabling the same reconstruction and stability framework. This provides a complete reconstruction procedure for full tensor fields from moment ray data with quantitative stability.

Abstract

We present a technique for recovering a vector field and a symmetric -tensor field, both real-valued and compactly supported in some strictly convex bounded domain with smooth boundary in the Euclidean plane, from the sum of their attenuated moment ray transforms. In addition, we provide a stability estimate for recovering both the vector field and the symmetric -tensor field from the aforementioned ray transform.
Paper Structure (8 sections, 4 theorems, 73 equations)

This paper contains 8 sections, 4 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and statement of the main result
  3. Transport model and Bukhgeim A-analytic theory
  4. Proofs
  5. The non-attenuated case:
  6. The attenuated case:
  7. Proof of the proposition \ref{['proposition; transport model']}
  8. Acknowledgements

Key Result

Theorem 2.3

Let $\Omega$ be a strictly convex bounded domain in $\mathbb{R}^2$ with the smooth boundary $\Gamma$. If ${\textbf{f}} \in H_0^{s}(\Omega; S^{1})$, ${\textbf{F}} \in H_0^{s}(\Omega; S^{2})$ and $\mathfrak{a} \in C^{1,\alpha}(\overline{\Omega})$, for $s\geq0$ and $1/2<\alpha\leq1$, then $\textbf{f}$

Theorems & Definitions (6)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Main Theorem
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1