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

The 3D Doppler transform maps a vector field to its line integrals over that component of the field which is parallel to the line. In this paper we consider only lines aligned to the coordinate axes. Since the Doppler transform describes the mathematical model for the vector tomography, efficient inversion 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 characterize these reconstruction kernels as solutions of a normal equation connected with the Doppler transform and a mollifier. To solve this equation elementary properties of the underlying operator are investigated and a smoothing property is proved. We succeed in computing a reconstruction kernel for one special mollifier and give a representation with the help of the singular value decomposition of the 2D Radon transform.