Reconstruction of vector field which given in ball by its known the normal Radon transform

The problem of recovering of the vector field, which is defined in the ball, by its known normal Radon transform, which is an integral along the planes of the projection of the vector field on the normal to the plane. It is shown that solenoidal fields, which are tangential on the boundary of the ball, are formed the core of the normal Radon transform. It is therefore possible to recover only potential part of the vector field. In this paper, for the subspace of potential fields with the potentials, which are equal to zero at the boundary, an orthogonal basis is constructed and normal Radon transform of these basic vector functions is calculated. The result is a singular value decomposition of the normal Radon transform in this space. The resulting decomposition can be used as a basis for the numerical solution of the problem of recovery of potential part of vector field on the assumption that the harmonic part of the original vector field is absent.