3-D 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 eld is the inner product of the Radon transform of a vector eld with a unit vector, called the probe, which may be a function of the projection orientation. Previous work has given reconstruction formulas for arbitrary elds and for those known to be divergence-free or curl-free in the case that the eld is zero on its boundary. This paper considers the possibility that the eld may not be zero on its boundary, and it 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.