Asymptotic inversion formulas in 3D vector field tomography for different geometries

Abstract

Abstract We study the problem of recovering scalar fields and solenoidal vector fields supported in the unit ball in from tomographic data. We consider three different measurement settings: the case of full data, where the ray sources cover the entire unit sphere and the rays spread in all directions, the case of parallel geometry which is to be understood slice-by-slice as well as the case of cone beam data where the sources are located on a trajectory surrounding the object and the rays are emitted in directions that are bounded by a cone. We formulate an asymptotic inversion formula for the case of full data using an expansion of the searched object in orthogonal polynomials and show how this inversion formula remains valid even for the parallel and cone beam geometry, where for the latter one the source trajectory has to satisfy a certain Tuy condition for vector fields.