In this article we study the fan-beam Radon transform Dm of symmetrical solenoidal 2D tensor fields of arbitrary rank m in a unit disc D as the operator, acting from the object space L2(D ; Sm) to the data space L2([0, 2π) × [0, 2π)). The orthogonal polynomial basis s (±m) n,k of solenoidal tensor fields on the disc D was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator Dm was obtained. The inversion formula for the fan-beam tensor transform Dm 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.

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.

In the introduction we show that the inverse problems for transport equations are naturally reduced to the Cauchy problem for the so called A-analytic functions, and hence the solution is given in terms of operator analog of the Cauchy transform. In Section 2 we develop elements of the theory of A-analytic functions and obtain stability estimates for our Cauchy transform. In Section 3 we discuss numerical aspects of this transformation. In Section 4 we apply this algorithm to the 3-dimensional inverse kinematic problem with local data on the Earth surface, using modified Newton method and discuss numerical examples.

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The approach to the study of vector and 2-tensor tomography problems on the plane is presented. The new orthogonal polynomial bases of vector and symmetrical 2-tensor solenoidal fields were built with the help of bivariate Chebyshev ridge polynomials. These bases are useful not only in tomography, but also have potential applications in fluid mechanics, electromagnetism and image processing problems. This approach can be generalized for the m-tensor tomography of arbitrary rank m. The numerical results of novel inversion algorithm for vector and tensor Radon transforms are presented.