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

• Main Authors: Kunyansky, L. (Author), McDugald, E. (Author), Shearer, B. (Author)
• Format: Article
• Language: English
• Published: Institute of Physics 2023
• Subjects: explicit inversion formula; Explicit inversion formula; Inverse problems; Inversion formulae; longitudinal Radon transform; Longitudinal radon transform; Mathematical transformations; Radon; Radon Transform; Tomography; transversal Radon transform; Transversal radon transform; Vector fields; Vector spaces; vector tomography; Vector tomography; Vectors; weighted Radon transform; Weighted radon transform
• Online Access: View Fulltext in Publisher; View in Scopus

 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. © 2023 The Author(s). Published by IOP Publishing Ltd.