Generalized V-line transforms in 2D vector tomography

TL;DR

This paper introduces new generalized V-line transforms for 2D vector fields, characterizes their kernels, and provides explicit inversion formulas, enabling the reconstruction of vector fields from various transform data.

Contribution

It presents novel longitudinal and transverse V-line transforms, their kernel characterizations, and explicit inversion formulas, including a closed-form reconstruction from star transform data.

Findings

Explicit kernel characterizations of the transforms.

Invertibility of each transform modulo kernels.

Closed-form formula for vector field reconstruction from star transform.

Abstract

We study the inverse problem of recovering a vector field in $\mathbb{R}^2$ from a set of new generalized $V$-line transforms in three different ways. First, we introduce the longitudinal and transverse $V$-line transforms for vector fields in $\mathbb{R}^2$. We then give an explicit characterization of their respective kernels and show that they are complements of each other. We prove invertibility of each transform modulo their kernels and combine them to reconstruct explicitly the full vector field. In the second method, we combine the longitudinal and transverse V-line transforms with their corresponding first moment transforms and recover the full vector field from either pair. We show that the available data in each of these setups can be used to derive the signed V-line transform of both scalar component of the vector field, and use the known inversion of the latter. The final major result of this paper is the derivation of an exact closed form formula for reconstruction of the full vector field in $\mathbb{R}^2$ from its star transform with weights. We solve this problem by relating the star transform of the vector field to the ordinary Radon transform of the scalar components of the field.