Generalized V-line transforms in 2D vector tomography

Abstract

We study the inverse problem of recovering a vector field in R 2 from a set of new generalized V-line transforms in three different ways. First, we introduce the longitudinal and transverse V-line transforms for vector fields in R 2 . We then give an explicit characterisation of their respective kernels and show that they are complements of each other. We prove invertibility of each transform modulo their kernels and combine them to reconstruct explicitly the full vector field. In the second method, we combine the longitudinal and transverse V-line transforms with their corresponding first moment transforms and recover the full vector field from either pair. We show that the available data in each of these setups can be used to derive the signed V-line transform of both scalar component of the vector field, and use the known inversion of the latter. The final major result of this paper is the derivation of an exact closed form formula for reconstruction of the full vector field in R 2 from its star transform with weights. We solve this problem by relating the star transform of the vector field to the ordinary Radon transform of the scalar components of the field.

V-line 2-tensor tomography in the plane

“…• In the case of ray transforms (integrating along straight lines), it is known that a symmetric m-tensor field f can be recovered from the knowledge of its first (m + 1)-integral moments. In the V-line setup, the results are identical for vector fields (m = 1) as reported in an earlier work [7], but fail for symmetric 2-tensor fields (see lemma 1 and the discussion below it). Moreover, in the case of transforms integrating symmetric 2-tensor fields along V-lines, we can write the kth integral moments for k ⩾ 2 in terms of 0th and 1th order moments.…”

“…The V-line and star transforms of vector fields in R 2 have been analyzed in a pair of recent works [7,8]. The authors extended the notions of the longitudinal, transverse, and momentum ray transforms from operators integrating vector fields along straight lines to those integrating along broken-ray trajectories.…”