On a quantitative partial imaging problem in vector tomography

TL;DR

This paper addresses the challenging problem of reconstructing a vector field in 2D from partial ray transform data intersecting a specific arc, proposing a new method that stabilizes numerical solutions despite the problem's ill-posedness.

Contribution

It introduces a novel reconstruction method for partial vector tomography data on a subset of lines intersecting an arc, with implementation and numerical validation.

Findings

Reconstruction is feasible in the convex hull of the arc.

Discretization stabilizes the ill-posed problem.

Numerical experiments demonstrate the method's effectiveness.

Abstract

In two dimensions, we consider the problem of reconstructing a vector field from partial knowledge of its zeroth and first moment ray transforms. Different from existing works the data is known on a subset of lines, namely the ones intersecting a given arc. The problem is non-local and, for partial data, severely ill-posed. We present a reconstruction method which recovers the vector field in the convex hull of the arc. An algorithm based on this method is implemented on some numerical experiments. While still ill-posed the discretization stabilizes the numerical reconstruction.