Weighted Radon transforms of vector fields, with applications to magnetoacoustoelectric tomography

Abstract

Currently, theory of ray transforms of vector and tensor fields is well developed, but the Radon transforms of such fields have not been fully analyzed. We thus consider linearly weighted and unweighted longitudinal and transversal Radon transforms of vector fields. As usual, we use the standard Helmholtz decomposition of smooth and fast decreasing vector fields over the whole space. We show that such a decomposition produces potential and solenoidal components decreasing at infinity fast enough to guarantee the existence of the unweighted longitudinal and transversal Radon transforms of these components.
It is known that reconstruction of an arbitrary vector field from only longitudinal or only transversal transforms is impossible. However, for the cases when both linearly weighted and unweighted transforms of either one of the types are known, we derive explicit inversion formulas for the full reconstruction of the field. Our interest in the inversion of such transforms stems from a certain inverse problem arising in magnetoacoustoelectric tomography (MAET). The connection between the weighted Radon transforms and MAET is exhibited in the paper. Finally, we demonstrate performance and noise sensitivity of the new inversion formulas in numerical simulations.

Inversion of generalized Radon transforms acting on 3D vector and symmetric tensor fields

“…The formulas allow one to restore the potential and solenoidal parts of the field independently of each other. This favorably distinguishes the methods we have developed from the approach proposed in the article [24].…”

“…In this section we demonstrate how to use the results of section 4 to solve the problem considered in [24] in the three-dimensional case.…”

“…To solve the problem, the values of all longitudinal and one weighted Radon transforms are required. The algorithm proposed in [24] is aimed at restoring the complete vector field in R n . To reconstruct its potential part it is necessary to rely on the result of preliminary restoration of the solenoidal part of the vector field under consideration.…”

“…Note that the works [16,17,24,29,34,35] did not consider the issues of describing the kernels and images of each of the longitudinal and mixed Radon transforms separately.…”

Inversion of the attenuated momenta ray transform of planar symmetric tensors

“…, mth attenuated moment ray transforms. This problem is motivated by some engineering applications: for m = 1 in Doppler tomography [7,27,35], and Magneto-acousto-electrical tomography [17,23], for m = 2 in inverse kinematic problems in isotropic elastic media [1,18] and for m = 4 in anisotropic media [2,33]. The non-attenuated case also arises in the linearization of the boundary rigidity problem [33,36,37].…”