V-line 2-tensor tomography in the plane

TL;DR

This paper introduces and analyzes various V-line transforms for symmetric 2-tensor fields in the plane, providing kernel characterizations and explicit inversion formulas, extending known concepts from scalar and vector fields.

Contribution

It generalizes V-line transforms to tensor fields, characterizes their kernels, and derives explicit inversion formulas, including for the star transform on tensor fields.

Findings

Kernel characterizations of V-line transforms

Explicit inversion formulas for tensor field reconstruction

Injectivity conditions for the star transform

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.