A vector field \(F\) in \(\mathbb{R}^d\) is split into its potential and solenoidal component. The authors first prove that if the components of \(F\) are in the Schwartz space \({\mathcal S}(\mathbb{R}^d)\) of rapidly decreasing functions, the longitudinal and transversal Radon transforms of both components exist.
In practice, one of the transform types often cannot be measured, which makes reconstruction impossible. However, in certain measurement schemes such as magnetoacoustoelectric tomography (MAET), the longitudinal Radon transform as well as linearly weighted longitudinal transforms can be measured.
The authors present explicit inversion formulas for two distinct cases: First, reconstruction from one transversal and \(d-1\) linearly weighted transversal transforms. Second, reconstruction from \(d-1\) longitudinal and one linearly weighted longitudinal transform, which applies to MAET.
The formulas are illustrated by numerical simulations.