TY - JOUR

T1 - Inversion of generalized Radon transforms acting on 3D vector and symmetric tensor fields

AU - Svetov, Ivan E.

AU - Polyakova, Anna P.

N1 - The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme ‘Rich and Nonlinear Tomography—a multidisciplinary approach’ when work on this paper was undertaken. This programme was supported by EPSRC Grant Number EP/R014604/1. While in Cambridge, authors received support from the Simons Foundation.

PY - 2024/1

Y1 - 2024/1

N2 - Currently, theory of the ray transforms of vector and tensor fields is well developed, but the generalized Radon transforms of such fields have not been fully studied. We consider the normal, longitudinal and mixed Radon transforms (with integration over planes) acting on three-dimensional vector and symmetric tensor fields. We prove that these operators are continuous. In case when values of all generalized Radon transforms are known, inversion formulas are derived for componentwise reconstruction of vector and symmetric m-tensor fields, m ⩾ 2 . Novel detailed decompositions of 3D vector and symmetric 2-tensor fields as a sum of pairwise orthogonal terms are obtained. For construction of each term in the sum only one function is required. With usage of these decompositions we have described the kernels and images of the generalized Radon transforms. In addition, we have obtained inversion formulas for each of the generalized Radon transforms acting on 3D vector and symmetric 2-tensor fields and have formulated theorems similar to the projection theorem for the Radon transform. For the cases m ⩾ 3 similar statements are formulated as hypotheses. In addition, we consider the weighted longitudinal Radon transforms of 3D vector fields. Formulas are obtained for reconstructing the potential part of a 3D vector field from the known values of the longitudinal Radon transforms and one weighted Radon transform. Finally, we discuss the problem of vector fields reconstruction in R n , n ⩾ 4 .

AB - Currently, theory of the ray transforms of vector and tensor fields is well developed, but the generalized Radon transforms of such fields have not been fully studied. We consider the normal, longitudinal and mixed Radon transforms (with integration over planes) acting on three-dimensional vector and symmetric tensor fields. We prove that these operators are continuous. In case when values of all generalized Radon transforms are known, inversion formulas are derived for componentwise reconstruction of vector and symmetric m-tensor fields, m ⩾ 2 . Novel detailed decompositions of 3D vector and symmetric 2-tensor fields as a sum of pairwise orthogonal terms are obtained. For construction of each term in the sum only one function is required. With usage of these decompositions we have described the kernels and images of the generalized Radon transforms. In addition, we have obtained inversion formulas for each of the generalized Radon transforms acting on 3D vector and symmetric 2-tensor fields and have formulated theorems similar to the projection theorem for the Radon transform. For the cases m ⩾ 3 similar statements are formulated as hypotheses. In addition, we consider the weighted longitudinal Radon transforms of 3D vector fields. Formulas are obtained for reconstructing the potential part of a 3D vector field from the known values of the longitudinal Radon transforms and one weighted Radon transform. Finally, we discuss the problem of vector fields reconstruction in R n , n ⩾ 4 .

KW - decomposition of tensor fields

KW - integral geometry of tensor fields

KW - inversion formula

KW - longitudinal Radon transform

KW - mixed Radon transform

KW - normal Radon transform

KW - tensor tomography

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85179890018&origin=inward&txGid=c9c2a1eb52d159f59085e86d65b6a98d

UR - https://www.mendeley.com/catalogue/c954ed21-3f2b-37f2-97f1-fe7b433355dd/

U2 - 10.1088/1361-6420/ad0fac

DO - 10.1088/1361-6420/ad0fac

M3 - Article

VL - 40

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 1

M1 - 015009

ER -