Slice-by-slice reconstruction algorithm for vector tomography with incomplete data

Abstract

For a finite collection of planes, we consider the problem of recovering the solenoidal part of a vector (symmetric second rank tensor) field on from ray integrals known over all lines parallel to one of the planes. Two different planes are sufficient for the uniqueness in the case of vector fields, but three planes in the general position are needed for the stable reconstruction. In the case of symmetric second rank tensor fields, three (six) planes in the general position are needed for the unique (stable) reconstruction. The reconstruction algorithm is presented for each of these cases. The main ingredients of the algorithm are the 3D Fourier transform and multiple application of the 2D back-projection operator.