Two approaches to the problem of defect correction in vector field tomography solving boundary value problems

Abstract

The problem of vector field tomography consists of reconstructing a vector field from integrals of certain projections of the field over geodesic lines. In case of an homogeneous medium the underlying Riemannian metric is the Euclidean one and the geodesics are straight lines. Modeling inhomogeneous media we take also non-Euclidean metrics into account. It is well known that the corresponding integral transform has a nontrivial kernel consisting of potential fields with potentials being equal to a constant on the boundary of the domain. Thus, the reconstruction from these data may contain a potential part of this type. Since we consider only solenoidal fields, e.g.velocity fields of incompressible fluids, the accuracy of the solution can be improved detecting the undesirable potential part of the reconstruction and subtracting it. Depending on the boundary condition the sought potential is the solution of either a Dirichlet or a Neumann boundary value problem. In this article two approaches to recover the potential are suggested: One deals with the solution of the Dirichlet problem with a finite difference scheme, the other solves the Neumann problem with the help of boundary element methods.