Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve

The x-ray transform I associates with a covariant symmetric tensor field f (x) of order m, x is an element of R-n, the line function If (l) by integration over the line l. In this paper, new reconstruction formulae for the solenoidal part of f are obtained. The case when If is known only for the family of lines intersecting a given curve in R 3 is considered as well. The formulae obtained contain only a finite number of differentiations, integrations, solving of linear algebraic equations and application the Laplacian powers. The formulae are obtained under the assumption that every plane intersecting the support of f intersects the curve m + 1 times. It is shown that the number of intersection points cannot be decreased. It is shown that the similar considerations are valid for two other admissible families of lines in R 3 : lines parallel to a given curve of directions and lines tangential to a given surface.