This paper summarizes some of the old results obtained for the problems of inverting the two-dimensional attenuated X-ray transform and the attenuated vectorial X-ray transform, both using the fan-beam geometry. These inverse problems are considered on the language of the transport equation and two approaches are described for solving them. The first one, dating back to 1996, gives the inversion formulae on the basis of the theory of so-called A-analytic functions. And the second method (developed by the authors in 2002) yields the inversion formulae for the scalar and vector attenuated X-ray transforms without using the theory of A-analytic functions, but merely by reducing the arising inverse problems to the unattenuated case by the change of variables. Numerical implementation details are also provided.

The X-ray computerized tomography is concerned to the reconstruction of a function f from line (ray) integrals of f. In general, we may have the parallel ray or divergent ray geometry. In this paper we develop an analytical singular value decomposition method for the cone-beam transform in divergent ray geometry. For the parallel ray transform a singular value decomposition has been derived by P. Maass in 1987.

Under study is the polynomial orthogonal basis system of vector fields in the ball which corresponds to the Helmholtz decomposition and is divided into the three parts: potential, harmonic, and...

In this article we study the fan-beam Radon transform of symmetrical solenoidal 2D tensor fields of arbitrary rank m in a unit disc as the operator, acting from the object space L 2 (; S m ) to the data space L 2 ([0, 2π) × [0, 2π)). The orthogonal polynomial basis of solenoidal tensor fields on the disc was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator was obtained. The inversion formula for the fan-beam tensor transform follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.

We study some questions of the qualitative theory of differential equations. A Cauchy problem is considered for a hyperbolic system of two first-order differential equations whose right-hand sides contain some discontinuous functions. A generalized solution is defined as a continuous solution to the corresponding system of integral equations. We prove the existence and uniqueness of a generalized solution and study the differential properties of the obtained solution. In particular, its first-order partial derivatives are unbounded near certain parts of the characteristic lines. We observe that this property contradicts the common approach which uses the reduction of a system of two first-order equations to a single second-order equation.