The 3D Doppler transform: elementary properties and computation of reconstruction kernels

The 3D Doppler transform maps a vector field to its line integrals over tha t component of the field which is parallel to the line. In this paper we co nsider only lines aligned to the coordinate axes. Since the Doppler transfo rm describes the mathematical model for the vector tomography, efficient in version formulae are necessary in order to solve the reconstruction problem . The approximate inverse represents a numerical inversion scheme based on scalar products of the data with so-called reconstruction kernels. We chara cterize these reconstruction kernels as solutions of a normal equation conn ected with the Doppler transform and a mollifier. To solve this equation el ementary properties of the underlying operator are investigated and a smoot hing property is proved. We succeed in computing a reconstruction kernel fo r one special mollifier and give a representation with the help of the sing ular value decomposition of the 2D Radon transform.