Ivan E. Svetov

The algorithms for reconstructing a vector field from its known longitudinal or transverse ray transforms of its moment are proposed and justified. The properties of several algorithms are studied depending on the degree of data discretization, the level and type of noise introduced into the data, the smoothness of the vector field, and the degree of connectivity of its support. Numerical simulations show good results of reconstructing vector fields from their momentum ray transforms.

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 .

The paper considers the vector tomography problem of reconstructing a three-dimensional vector field based on the values of unweighted (normal and longitudinal) and weighted Radon transforms. Using the detailed decomposition of vector fields obtained in the paper, connections are established between the unweighted and weighted Radon transforms acting on vector fields and the Radon transform acting on functions. In particular, the kernels of tomographic integral operators acting on vector fields are described. Some statements of tomography problems for the reconstruction of vector fields are considered, and inversion formulas for their solution are obtained.

The operators of longitudinal, transverse and mixed ray transforms acting on two-dimensional symmetric tensor fields of arbitrary degree m in an unit disk are considered in the article. The singular value decompositions of the operators for a parallel scheme of data acquisition are constructed. Orthogonal bases in original spaces and image spaces are constructed using harmonic, Jacobi and Gegenbauer polynomials. Based on the obtained decompositions the polynomial expressions for the (pseudo)inverse and adjoint operators are obtained.

We consider a problem of dynamic 2D vector tomography, i.e. the object under investigation changes during the data acquisition. More precisely, we consider the case when the object motion is a combination of rotation and shifting. The task is then to reconstruct the searched-for vector field by known values of the dynamic ray transforms. In order to solve this dynamic inverse problem, we first study properties of the dynamic ray transforms operators. In particular, the singular value decompositions of the operators are constructed using classic orthogonal polynomials. Following from this study, a numerical algorithm for solving the dynamic problem is proposed based on the truncated singular value decomposition method.

We propose an approach for reconstructing a three-dimensional function from the known values of the Radon transform. The approach is based on the method of approximate inverse. The result obtained is the basis of two approaches for reconstructing a potential part of the vector and symmetric 2-tensor fields from the known values of the normal Radon transform. The first approach allows the recovery of components of the potential part of fields, and the second one reconstructs a potential of the potential part of fields.

We consider the problem of reconstructing the potential part of a symmetric 2-tensor field defined in a unit ball by its known values of the normal Radon transform. To solve the problem, a singular value decomposition of the normal Radon transform operator is constructed. The basis functions and fields using the Jacobi, Gegenbauer polynomials and spherical harmonics are constructed.

We consider the problem of the dynamic two-dimensional 2-tensor tomography. An object motion is a combination of rotation and shifting. Properties of the dynamic longitudinal, mixed and transverse ray transform operators are investigated. The singular value decompositions of the operators with usage of the classic orthogonal polynomials are constructed.