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

TL;DR

This paper develops methods to reconstruct symmetric tensor fields within a disk from V-line transforms with vertices on a circle, providing explicit kernels and inversion algorithms for practical tensor tomography.

Contribution

It introduces two approaches for tensor field recovery from combined V-line transforms and derives explicit inversion formulas and algorithms in a circular geometric setting.

Findings

Explicit kernel descriptions for V-line transforms.

Inversion algorithms for tensor field reconstruction.

Recovery of tensor fields from first moments of V-line transforms.

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.