The constructive methods and algorithms of reconstruction of a solenoidal part of the tensor field, given in domains with Euclidean or Riemannian metrics, by its ray transform are developed (some part of the work has been done together with V. A. Sharafutdinov, I. G. Kashina, M. A. Bezuglova, S. B. Sorokin). The finite difference approximations of covariant derivative and some other geometrical objects and operators, with conservation of main geometrical properties, are constructed in the domains with a given Riemannian metric. The polynomial approximations of solenoidal and potential tensor fields are investigated. For a refracted medium with an arbitrary absorption the numerical solution to the emission tomography problem is suggested (with V. A. Sharafutdinov, A. G. Kleshchev). The influence of refraction to the accuracy of the solution for the problem is investigated (with R. Dietz, A. K. Louis, T. Schuster). In the spaces of distributions with compact support the examples of distributions, that have vanishing images in the arbitrary given finite set of points of $R^{n}$, are constructed. For the problem of reconstruction of the optical surfaces of some types the questions of uniqueness are considered, and the constructive methods and algorithms are created, if two or three images of the surface are given.
Main publications:
Derevtsov E. Yu. Ghost distributions in the cone-beam tomography // J. Inverse Ill-posed Problems, 1997, 5(5), 411–426.
Derevtsov E. Yu., Kleshchev A. G., Sharafutdinov V. A. Numerical solution of the emission 2D-tomography problem for a medium with absorption and refraction // J. Inverse Ill-posed Problems, 1999, 7(1), 83–103.
Derevtsov E. Yu., Dietz R., Louis A. K., Schuster Th. Influence of refraction to the accuracy of a solution for the 2D-emission tomography problem // J. Inverse Ill-posed Problems, 2000, 8(2), 161–191.
