Computer Modelling in Tomography and Ill-Posed Problems

Mathematical basis of the method of computerized tomography; basic notions of the theory of ill-posed problems; problem of integral geometry; the Radon transfer; Radon problem as an example of an ill-posed problem; the algorithm of inversion of the two-dimensional Radon trans form based on the convolution with the generalized function 1/z2 cone-beam tomography reconstruction; reducing the inversion formulas of cone-beam tomography reconstruction to the form convenient for constructing numerical algorithms; elements of the theory of generalized functions in application to problems of inversion of the ray transformation; the relations between the Radon, Fourier and ray transformations; inverse kinematic problem in the tomographic setting; direct kinematic problem and numerical solution for three-dimensional regular media; formulation of the inverse kinematic problem with the use of a tomography system of data gathering; deduction on the basic inversion formula and the algorithm of solving the inverse kinematic problem in three-dimensional linearized formulation; model experiment and numerical study of the algorithm; solution of the inverse kinematic problem by the method of computerized tomography for media with opaque inclusions. Appendix: reconstruction with the use of the standard model.