An improved exact inversion formula for cone beam vector tomography

TL;DR

This paper introduces an improved exact inversion formula for 3D cone beam vector tomography that simplifies implementation and reduces computational complexity by eliminating an inefficient integration step, while maintaining accuracy.

Contribution

It presents a simplified and more efficient inversion formula for cone beam vector tomography, improving upon previous methods by reducing integration complexity.

Findings

The new formula is easier to implement.

It reduces the integration from 4D to 3D.

Numerical tests confirm the theoretical accuracy.

Abstract

In this article we present an improved exact inversion formula for the 3D cone beam transform of vector fields. It is well known that only the solenoidal part of a vector field can be determined by the longitudinal ray transform of a vector field in cone beam geometry. The exact inversion formula, as it was developed in A. Katsevich and T. Schuster, 'An exact inversion formula for cone beam vector tomography', Inverse Problems 29 (2013), consists of two parts. the first part is of filtered backprojection type, whereas the second part is a costly 4D integration and very inefficient. In this article we tackle this second term and achieve an improvement which is easily to implement and saves one order of integration. The theory says that the first part contains all information about the curl of the field, whereas the second part presumably has information about the boundary values. This suggestion is supported by the fact that the second part vanishes if the exact field is divergence free and tangential at the boundary. A number of numerical tests, that are also subject of this article, confirm the theoretical results and the exactness of the formula.