Reconstruction of scalar functions and vector fields from weighted V-line transforms with swinging branches

Abstract

Weighted V-line transforms map a symmetric tensor field of order m\ge0 to a linear combination of certain integrals of those fields along two rays emanating from the same vertex. A significant focus of current research in integral geometry centers on the inversion of V-line transforms in formally determined setups. Of particular interest are the restrictions of these operators in which the vertices of integration trajectories can be anywhere inside the support of the field, while the directions of the pair of rays, often called branches of the V-line, are determined by the vertex location. Such transforms have been thoroughly investigated under at least one of the following simplifying assumptions: the weights of integration along each branch are the same, while the branch directions are either constant or radial. In this paper we lift the first restriction and substitute the second one by a much weaker requirement in the case of transforms defined on scalar functions and vector fields. We extend multiple previously known results on the kernel description, injectivity, and inversion of the transforms with simplifying assumptions and prove pertinent statements for more general setups not studied before.