Tensor tomography for a set of generalized V-line transforms in $\mathbb{R}^2$

TL;DR

This work studies the inverse problem for generalized V-line transforms of symmetric $m$-tensor fields in the plane, using broken-ray data that emanate from the boundary of a disk. It introduces longitudinal, transverse, and mixed V-line transforms $\mathcal{L}$, $\mathcal{T}$, and $\mathcal{M}^{(k)}$, and proves two inversion results: a full-data recovery for a tensor $\mathbf{f}$ supported in $\mathbb{D}_{R\sin\theta}$ from $\mathcal{L}\mathbf{f}$, $\mathcal{T}\mathbf{f}$, and $\mathcal{M}^{(k)}\mathbf{f}$ with $d\in[0,2R]$, and a stronger partial-data recovery for $m=2$ with $d\in[0,R]$ without the same support restriction. The full-data proof reduces the V-line data to straight-line (Radon) transforms of the tensor components and then applies standard Radon inversion; the partial-data result employs a Mellin-transform and Fourier-series framework to derive explicit series formulas for the tensor components. Together, these results advance tensor tomography under broken-ray measurements and have potential impact on single-scattering tomography and related imaging modalities.

Abstract

We study a set of generalized V-line transforms, namely longitudinal, mixed, and transverse V-line transforms, of a symmetric $m$-tensor field in $\mathbb{R}^2$. The goal of this article is to recover a symmetric $m$-tensor field $\textbf{f}$ supported in a disk $\mathbb{D}_R$, with radius $R$ and centered at the origin, by a combination of the aforementioned generalized V-line transforms, using two different techniques for different sets of data.

Figures (2)

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Theorems & Definitions (17)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Theorem 2.9
• Theorem 2.10