Problem Of Integral Geometry Research Articles

Weekly Ill-posed integral geometry problems of Volterra type in three-dimensional space

In this paper considers the problem of recovering a function from families of spheres in space. The uniqueness of the solution of the problem is proved by reducing it to the Volterra integral equation of the first and then the second kind. Fourier transform methods are also used. Uniqueness theorems are proved for some new classes of operator equations of Volterra type in three-dimensional space.

Inversion formula for an integral geometry problem over surfaces of revolution

AbstractAn integral geometry problem is considered on a family of ‐dimensional surfaces of revolution whose vertices lie on a hyperplane and directions of symmetry axes are fixed and orthogonal to this plane, in . More precisely, the reconstruction of a function , , , from the integrals of the form extended over a chosen side of all surfaces of revolution of a given family is investigated. Unlike the usual Radon transform, the integrals considered here are not taken with respect to the surface area element. A Fourier slice identity and a backprojection‐type inversion formula are obtained with a method based on the Fourier and Hankel transforms. The reconstruction procedure and some analytical and numerical implementations of the obtained inversion formulas in the cases of and are provided.

ON ONE PROBLEM OF INTEGRAL GEOMETRY

The paper is devoted to a problem of integral geometry on a two-dimensional plane. In the band of the two-dimensional space , a two-parameter family of curves is given. Each of these curves consists of two arcs , which are adjacent to each other at the vertex of the curve at an angle , and the other ends rest on the axis. With respect to the branches of the curve , certain smoothness conditions are assumed. The family of curves can be parametrized using the coordinates of the vertices of the curves, that is, each curve from this family can be put in an unambiguous correspondence with the coordinates of its vertex. For each point , there is only one curve belonging to the specified family and having the point as its vertex. It is required to determine the function from the following equation for a given function where – region bounded curves and the axis . The theorem (uniqueness of the solution) is proved. If , then the solution of equation (*) is unique in the class of continuous finite functions with a carrier in . In proving the theorem, the invariance of the family of curves L(ξ,n) and weight functions to shifts along the abscissa axis is significantly used. Further, using the Fourier transform from the function with respect to the variable and the permissible transformations, a family of Volterra integral equations of the second kind is obtained with respect to the Fourier image from the function with respect to the first variable. The paper also provides an estimate of the conditional stability of the considered integral geometry problem

Ray Statement of the Acoustic Tomography Problem

The ray statement of the inverse problem of determining three unknown variable coefficients in the linear acoustic equation is studied. These coefficients are assumed to differ from given constants only inside some bounded domain. There are point pulse sources and acoustic receivers on the boundary of this domain. Acoustic signals are measured by a receiver near the moment of time at which the signal from a source arrives at the receiver. It is shown that this information makes it possible to uniquely determine all the three desired coefficients. Algorithmically, the original inverse problem splits into three subproblems solved successively. One of them is a well-known inverse kinematic problem (of determining the speed of sound), while the other two lead to the same integral geometry problem for a family of geodesic lines determined by the speed of sound.

Some integral geometry problems for wave equations

We consider the Cauchy problem and the source problem for normally hyperbolic operators on the Minkowski spacetime, and study the determination of solutions from their integrals along light-like geodesics. For the Cauchy problem, we give a new proof of the stable determination result obtained by Vasy and Wang (2021 Commun. Math. Phys. 384 503–32). For the source problem, we obtain stable determination for sources with space-like singularities. Our proof is based on the microlocal analysis of the normal operator of the light ray transform composed with the parametrix for strictly hyperbolic operators.

INTEGRAL GEOMETRY PROBLEMS WITH PERTURBATION IN THE FAMILY OF PARABOLAS WITH A WEIGHT FUNCTION OF A SPECIAL FORM

In this article, we present a statement of the integral geometry problem with perturbation over a family of parabolas in a strip. The weight function of which has a fairly general form. A uniqueness theorem for its solution in the class of twice continuously differentiable and compactly supported functions is proved, and stability estimates are obtained in spaces of finite smoothness.

On the stability of the difference analogue of the boundary value problem for a mixed type equation

This paper considers a difference problem for a mixed-type equation, to which a problem of integral geometry for a family of curves satisfying certain regularity conditions is reduced. These problems are related to numerous applications, including interpretation problem of seismic data, problem of interpretation of Xray images, problems of computed tomography and technical diagnostics. The study of difference analogues of integral geometry problems has specific difficulties associated with the fact that for finite-difference analogues of partial derivatives, basic relations are performed with a certain shift in the discrete variable. In this regard, many relations obtained in a continuous formulation, when transitioned to a discrete analogue, have a more complex and cumbersome form, which requires additional studies of the resulting terms with a shift. Another important feature of the integral geometry problem is the absence of a theorem for existence of a solution in general case. Consequently, the paper uses the concept of correctness according to A.N.Tikhonov, particularly, it is assumed that there is a solution to the problem of integral geometry and its differential-difference analogue. The stability estimate of the difference analogue of the boundary value problem for a mixed-type equation obtained in this work is vital for understanding the effectiveness of numerical methods for solving problems of geotomography, medical tomography, flaw detection, etc. It also has a great practical significance in solving multidimensional inverse problems of acoustics, seismic exploration.

ИНТЕГРАЛДЫҚ ГЕОМЕТРИЯНЫҢ САЛМАҚ ФУНКЦИЯСЫ БАР ЕСЕБІНІҢ ДИФФЕРЕНЦИАЛДЫҚ-АЙЫРЫМДЫҚ АНАЛОГЫНЫҢ ОРНЫҚТЫЛЫҒЫ

In this paper, we consider the problem of Integral geometry, which is brought to the problem of difference for a mixed-type equation for a bunch of curves that satisfy some regularity conditions. The study of distinctive analogues of Integral geometry problems has its own complex points, due to the fact that for limited-distinctive analogues of independent derivatives, the main relations are carried out with a certain shift over a discrete variable. Therefore, many relationships obtained in continuous representation take on a more complex form when switching to a discrete analog, and require further research on the connectors that occur duringthe shift. Since these problems do not have a theorem on the existence of a solution in the general case, the concept of conditional correctness is used, that is, it is assumed that the problem of Integral geometry and its differential-differential analoghave a solution. The stability assessment of the differential analog of the boundary problem for the mixed-type equation obtained in the work is carried out by geotomography, medical tomography, defectoscopy, etc. it is used to justify the compactness of numerical problem solving methods.

An inverse problem for a generalized kinetic equation in semi-geodesic coordinates

The aim of this article is to investigate the uniqueness of solution of an inverse source problem for a generalized kinetic equation on a Riemannian manifold. The problem is related with an integral geometry problem in semi-geodesic coordinates. We prove the uniqueness in a convex domain by the help of Riemannian coordinates.

THE PROBLEM OF RESTORING A FUNCTION BY FAMILIES OF SPHERES IN SPACE

In this chapter, we study the question of reducing the problem of integral geometry for a special class of surfaces to the Cauchy problem for some equation of evolutionary type. Fourier transform methods are also used. Methods are obtained that allow reducing the problem of integral geometry for special families of curves and surfaces to the Cauchy problem for equations of evolutionary type, and classes of such problems are distinguished. Uniqueness theorems are proved for some new classes of operator equations of Voltaire type in three-dimensional space.

A phaseless inverse problem for electrodynamic equations in the dispersible medium

ABSTRACT For electrodynamic equations related to non-conducting dispersible medium we consider the inverse problem of recovering two variable coefficients from a given phaseless information of solutions to the equations. One of these coefficients is the permittivity while the second one characterizes the time dispersion of the medium. We suppose that unknown coefficients differ from given constants inside of a compact domain Ω. A plane electromagnetic wave going in the direction ν from infinity fall down on this domain and modulus of the electric strength is measured on a part of the boundary of Ω for all . The inverse problem consists in determining unknown functions from this information. We reduce the inverse problem to two problems: (1) the inverse kinematic problem for recovering the refractive index and (2) the integral geometry problem for recovering the second coefficient related to the dispersion. An uniqueness theorem for the first problem is stated on the base of known results. The second problem differs from have studied by the more general weight function and it is still open. Then we demonstrate that under some natural assumption the weight function uniformly close to 1. Replacing the weight function by 1, we obtain the integral geometry problem for which the uniqueness theorem and stability estimate are established and some numerical algorithms are proposed.

The geodesic X-ray transform with matrix weights

Author(s): Paternain, GP; Salo, M; Uhlmann, G; Zhou, H | Abstract: Consider a compact Riemannian manifold of dimension ≥ 3 with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray transform in the presence of an arbitrary connection and Higgs field is injective modulo the natural obstruction for functions and one-forms. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo gauge transformations. The proofs involve a reduction to a local result showing that the geodesic X-ray transform with a matrix weight can be inverted locally near a point of strict convexity at the boundary, and a detailed analysis of layer stripping arguments based on strictly convex exhaustion functions. As a somewhat striking corollary, we show that these integral geometry problems can be solved on strictly convex manifolds of dimension ≥ 3 having nonnegative sectional curvature (similar results were known earlier in negative sectional curvature). We also apply our methods to solve some inverse problems in quantum state tomography and polarization tomography.

On the stability of a differential-difference analogue of a two-dimensional problem of integral geometry

A differential-difference analogue of a two-dimensional problem of integral geometry with a weight function is studied. A stability estimate for the problem considered is obtained.

ON THE GINZBURG-FEINBERG PROBLEM OF FREQUENCY ELECTROMAGNETIC SOUNDING FOR UNAMBIGUOUS DETERMINATION OF THE ELECTRON DENSITY IN THE IONOSPHERE

In the present work, we investigate an inverse problem of frequency electromagnetic sounding for unambiguous determination of the electron density in the ionosphere. Direct statement of this problem is known as the Ginzburg-Feinberg problem that has, in general case, an essential nonlinearity. Inverse statement of the Ginzburg-Feinberg problem has the boundary-value formulation relative to two functions: the sought-for electric-field strength and the distribution of the electron density (or rather two-argument function appearing in the additive decomposition formula for distribution of the electron density) in the ionosphere. In the present work, we prove the existence and uniqueness of the solution of the Ginzburg-Feinberg problem as well as we propose the analytical method, permitting: first, to reduce it to the problem of integral geometry, and, thereupon, having applied the adjusted variant of the Lavrentiev's theorem, to reduce the obtained problem of integral geometry to the first kind matrix integral equation of Volterra type with a weak singularity

The X-ray Transform on a General Family of Curves on Finsler Surfaces

For a compact oriented Finsler surface with smooth boundary, we consider the scalar and vector integral geometry problems over a general family of curves running between boundary points and parametrized by arc length. We impose a natural condition which results in the no conjugate points condition in the case when the curves in question are geodesic lines. Our main theorem generalizes Mukhometov’s theorem in several directions. We also consider these problems on a closed oriented Finsler surface. In this case the integral geometry problems make sense provided that sufficiently many curves in the family are periodic. To this end, we assume that the induced flow on the unit circle bundle of the surface is Anosov. Also, we study the cohomological equation of thermostats without conjugate points.

Uniqueness in an integral geometry problem and an inverse problem for the kinetic equation

In this paper, we discuss the uniqueness in an integral geometry problem along the straight lines in a strongly convex domain. Our problem is related with the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation and then the uniqueness theorem is proved using the tools of Fourier analysis.

An underdetermined problem of integral geometry for the generalized radon transform

Under study is a new problem of integral geometry. All kinds of planes are considered in the three-dimensional Euclidean space. The available data are given by the integrals over all these planes of an unknown piecewise-smooth function depending both on the spatial variables and the variables characterizing the planes. The sought object is a first kind discontinuity surface of the integrand. The uniqueness theorem of a desired surface is proved. The above results are related to one of the aspects of the theory of testing unknown media by various physical signals.

The Problem of Integral Geometry of Volterra Type with a Weight Function of a Special Type

We study new problem of reconstruction of a function in a strip from their given integrals with known weight function along polygonal lines. We obtained two simply inversion formulas for the solution to the problem. We prove uniqueness and existence theorems for solutions and obtain stability estimates of a solution to the problem in Sobolev's spaces and thus show their weak ill-posedness. Then we consider integral geometry problems with perturbation. The uniqueness theorems are proved and stability estimates of solutions in Sobolev spaces are obtained.

A problem of integral geometry for a family of curves with incomplete data

The unknown boundary problem of integral geometry is considered. Known data are integrals of unknown functions over an unknown family of curves in Euclidean space of any dimension. It is mentioned that, under the traditional approach, the integral geometry problem under consideration would be underdetermined. The sought object is the discontinuity surfaces of the integrands. A uniqueness theorem for the solution of the problem specified above is proved.

Stability and uniqueness for an integral geometry problem with a weight function

We consider the integral geometry problems including a weight function along the family of plane curves of given curvatures and prove a stability estimate and the uniqueness of the solution of these problems on the space of sufficiently smooth functions. Then, these stability and uniqueness results are investigated for the problems along the families of some special plane curves and some consequences are presented.