Abstract. In this paper we suggest the method of 3D vector tomography problem solving. The problem consists in determination of potential part of 3D vector field by its known the normal Radon transform. The singular value decomposition of the normal Radon transform operator is obtained. Based on obtained decomposition inversion formula is derived. The decomposition can be the basis for numerical solution of given problem.
IntroductionThe development both of mathematical approaches and of systems of data measurements and processing induced new mathematical models in tomography such as thermotomography, diffusion tomography, vector and tensor tomography. The models appear due to the necessity of the reconstruction of properties of media with different degree of complication. Thus the tomography of vector fields arises for the description of vector characteristics of currents of fluids, vectors of electromagnetic fields inside the conductor in inhomogeneous media, and many others. [1] We consider here a method of solving the 3D vector tomography problem in the case of parallel scheme of observation. As the problem of scalar tomography consists in the inversion of the Radon transform for a function, the vector tomography problem is the problem of inversion of normal Radon transform operator applying to potential part of 3D vector fields. In other words, one has to solve operator equations Af = g of the first kind. Here A is a linear, bounded operator. In the operator equation g is a known right hand-side (data of tomographic measurements), and f is an unknown vector field to be determined.The method of singular value decomposition (SVD) is well known and often used for inversion of compact linear operators. The idea of the approach consists in representing the operator in a form of series of singular numbers and basic elements in the image space. Then the inverse operator is a similar series with the reciprocal of the singular numbers and pre-images of the bases elements. The solution of 3D vector tomography problem, which is proposed in the paper, is based on the possibility of 3D vector field potential part representation by using of potentials, which are constructed as a product of harmonic functions and classical orthogonal polynomials.