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.

The Radon transform generates many other integral transforms in integral geometry and tensor tomography. Together with complicated geometrical objects, the weighted ray and Radon transforms arise. Usually symmetric tensor fields are considered as the object should be reconstructed. We consider complete, symmetric and difference non-symmetric tensor fields of small ranks as the objects for applications in integral geometry and tensor tomography. The structure and differential properties of such fields are investigated. We establish a decomposition theorem for a complete tensor field, properties and attributes of solenoidal and potential fields.

Within the framework of inverse problems of photometry, we study questions on reconstruction of the spatial location and luminosity of a Lambertian curve from a stereo pair of its images. We study sources of ambiguity in reconstruction of the location of such a curve and suggest criteria for existence of a unique solution of the problem for general weight functions. We apply the results to specific classes of weight functions that model the degree of transparency of the medium (including its absorption or scattering).

В рамках постановок обратных задач фотометрии изучаются вопросы определения пространственного расположения и светимости ламбертовой оптической поверхности по ее изображениям, полученным с помощью малого числа оптических систем. Изучены причины возникновения неоднозначности определения расположения таких поверхностей. Установлены критерии единственности решения обратной задачи восстановления светящихся поверхностей по трем изображениям для весовых функций общего вида. Результаты распространены на конкретные классы весовых функций, моделирующих степень прозрачности среды, в том числе ее поглощение или рассеяние.

В рамках постановок обратных задач фотометрии изучается вопрос единственности определения расположения и светимости кривой, излучающей в соответствии с законом Ламберта, по стереопаре ее изображений. Изучены причины расположения таких кривых. Установлены критерии единственности решения задачи восстановления ламбертовой кривой по стереопаре для произвольных весовых функций. Результаты распространены на конкретные семейства весовых функций, моделирующих степень прозрачности среды, ее поглощение или рассеяние.

Within the framework of inverse problems of photometry, we study questions on reconstruction of the spatial location and luminosity of a Lambertian optical surface from its images obtained with the use of a small number of optical systems. We study causes of ambiguity in reconstruction of the location of such a surface. We suggest criteria for existence of a unique solution of the inverse problem on reconstruction of a luminous surface from three images for general weight functions and apply the results to specific classes of weight functions that model the degree of transparency of the medium (including its absorption or scattering).

Within the framework of geometric tomography, inverse problems of photometry, wave optics, and discrete tomography, we study questions on reconstruction of the spatial location and luminosity of a discrete distribution of radiant sources from its images obtained with the use of a small number of optical systems. We analyze the problem on finding geometric parameters of such a distribution and describe sources of ambiguity. We consider the inverse problem on reconstruction of a discrete distribution that consists of incoherent and monochromatic sources and suggest uniqueness criteria for its solution. We also suggest a constructive approach to numerical solution of the inverse problem on reconstruction of the coordinates and luminosity of a family of radiant pinpoint sources from their images.

The paper considers ray transforms of the moments of symmetric tensor fields of arbitrary rank given in the unit disk. The basic geometric and differential properties of mixed ray transforms of tensor fields and mixed ray transforms of the moments of tensor fields are established. A simple algorithm for reconstructing a low-rank tensor field from known mixed ray transforms of its moments is proposed and justified.

Some features of mathematical modeling of tensor fields in tomography are con-sidered. We describe them with respect to the problem of recovery discontinuous objects from tomographic data. In a brief overview we pay attention also to the results of application of the method of tensor fields representation by potentials in tensor tomography. The mentioned approach allowed to establish the images and kernels of the ray transforms for symmetric tensor fields, the connections between these transforms and the Radon transform for their potentials. An application of tools of the Riemannian geometry for modeling the refraction phenomenon in tomography is described. A problem of representation of solenoidal tensor fields given in 2D and 3D Riemannian domains with a special metric in terms of scalar po-tentials is considered. The properties of tensor fields of small ranks given in the Riemannian domains are discussed. Connections between the fields and metric characteristics in a form of systems of equations are established. Along with the solenoidal fields of general type we consider a case of the toroidal vector field.

We consider the inverse kinematic problem of seismics (IKPS) with internal sources.It consists in determining the velocities of the longitudinal and transverse waves by the traveltimes from earthquake sources in the focal zone to a group of seismic stations. We propose analgorithm for the numerical solution of the problem which bases on the eikonal equation and thetechnology of smoothing multidimensional splines, which give an approximation of the velocitystructure of the focal zone. The paper presents some theoretical results that substantiate thealgorithm for solving the problem by approximation methods on using smoothing withmultidimensional splines from data on irregular grids. We describe the results of the numericalsolution of the problem, the calculations with real data on earthquakes in the focal zone, and givethe estimates of the velocity and elastic parameters of a medium.

In this article generalized attenuated ray transforms (ART) and integral angular moments are investigated. Starting from the Radon transform, the attenuated ray transform and the longitudinal ray transform, we derive the concept of ART-operators of order k over functions defined on the phase space and depending on time. The ART-operators are generalized for complex-valued absorption coefficient as well as weight functions of polynomial and exponential type. Connections between ART operators of various orders are established by means of the application of the linear part of a transport equation. These connections lead to inhomogeneous differential equations of order (k+1) for the ART of order k. Uniqueness theorems for the corresponding boundary-value and initial boundary-value problems are proved. Properties of integral angular moments of order p are considered and connections between the moments of different orders are deduced. A close connection of the considered operators with mathematical models for tomography, physical optics and integral geometry allows to treat the inversion of ART of order k as an inverse problem of determining the right-hand side of a corresponding differential equation.

We present an approach for solving the inverse kinematic problem of seismic with internal sources, based on the method of multidimensional data approximation on irregular grids. The times of arrival of elastic waves to the seismic stations are considered as known. The hodographs from earthquake to the stations are approximated for further determining the velocities of longitudinal and transverse waves using the eikonal equation. The ratio of these velocities determines the Poisson’s ratio, and the other elastic parameters of the medium can be found in units of the density. The results of implementation of the approach, based on the real data, are presented.

The Helmholtz decomposition of a vector field on potential and solenoidal parts is much more natural from physical and geometric points of view then representations through the components of the vector in the Cartesian coordinate system of Euclidean space. The structure, representation through potentials and detailed decomposition for 2D symmetric m-tensor fields in a case of the Euclidean metric is known. For the Riemannian metrics similar results are known for vector fields. We investigate the properties of the solenoidal vector and 2-tensor two-dimensional fields given in the Riemannian domain with the conformal metric and establish the connections between the fields and metrics.

A problem of the refraction tomography consists in reconstructing a function or tensor field by their attenuated ray transforms. Usually an absorption coefficient and refraction in the medium are assumed to be known. A solution to the refraction tomography problems can be obtained by approximate methods within the corresponding mathematical models, and the modeling of refraction is one of important elements of the model. We adduce summary data on the Riemannian metrics, suitable for the implementation in numerical tests, methods of their construction and main characteristics.

In the paper we consider a problem of recovering a 3D vector eld given in cylinder by means of jointly known nuclear magnetic resonance (NMR) images and ray transforms.The NRM images and 2D longitudinal and transverse ray transforms are known in every plane orthogonal to the cylinder axis.The 3D ray transforms of new type connected with a family of the parallel planes are dened.Simulation conrms the legitimacy and further perspective of the proposed approach.

Under consideration are the operators of angular moments which map the values of generalized attenuated ray transforms (ART) into the set of symmetric p -tensor fields. The differential relations between the values of ARTs of various orders, acting on stationary or dynamic source distributions f , serve as the basis for establishing the differential connections between the tensor fields of angular moments of various orders k and ranks p . The particular cases are indicated allowing us to obtain some previously known results. Connections of the ARTs of order k with the problems of tomography and integral geometry as well as the established properties and connections between ARTs and angular moments can be useful as additional information when constructing the iterative algorithms for solving the problems of dynamic refraction tensor tomography.

Properties of operators of generalized attenuated ray transforms (ART) are investigated. Starting with Radon transform in the mathematical model of computer tomography, attenuated ray transform in emission tomography and longitudinal ray transform in tensor tomography, we come to the operators of ART of order k over symmetric m-tensor fields, depending on spatial and temporal variables. The operators of ART of order k over tensor fields contain complex-valued absorption, different weights, and depend on time. Connections between ART of various orders are established by means of application of linear part of transport equation. This connections lead to the inhomogeneous k-th order differential equations for the ART of order k over symmetric m-tensor field. The right hand parts of such equations are m-homogeneous polynomials containing the components of the tensor field as the coefficients. The polynomial variables are the components ξ ^j of direction vector ξ participating in differential part of transport equation. Uniqueness theorems of boundary-value and initial boundary-value problems for the obtained equations are proved, with significant application of Gauss-Ostrogradsky theorem. The connections of specified operators with integral geometry of tensor fields, emission tomography, photometry and wave optics allow to treat the problem of inversion of the ART of order k as the inverse problem of determining the right hand part of certain differential equation.

Some results of numerical investigations are presented for the problem of determination of discontinuities of an unknown function that has the meaning of the internal source distribution and is given in a domain with absorption and refraction, on using the attenuated ray transform of the function. The refraction and the absorption coefficient are assumed to be given. The behavior of the available and newly constructed discontinuity indicator operators is investigated in some numerical tests. Some modification of discontinuity indicators was carried out for the purpose of applying them in the model of refractive tomography with absorption. Numerical methods are applied to investigate the possibility of using the operators of this kind for solving the problem of finding the discontinuities of a function from its attenuated ray transform; the degree is investigated of the influence on the recovery quality of such factors as the level of the noise introduced into the generated data, the parameters of metrics, the magnitude and variation of the absorption coefficient.