3D vector tomography on bounded domains

This paper considers the problem of vector tomography on an arbitrary bounded domain in three dimensions. The probe transform of a vector field is the inner product of the Radon transform of a vector field with a unit vector, called the probe, which may be a function of the projection orientation. Previous work has given reconstruction formulae for arbitrary fields and for those known to be divergence-free or curl-free in the case that the field is zero on its boundary. This paper considers the possibility that the field may not be zero on its boundary and may, therefore, have a harmonic component, which is both divergence-free and curl-free. It is shown that the curl-free component can be reconstructed using only one probe measurement and the divergence-free component can be reconstructed using only two probe measurements. No boundary measurements are necessary.