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 when the branch directions are either constant or radial. In addition to that, in most of the prior research on this subject, it was assumed that the weights of integration along each branch are the same. In this paper we analyze the transforms defined on scalar functions and vector fields, satisfying a much weaker assumption on the branch directions. The weights restriction is also lifted in all but one setup. Consequently, 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.