Computer Modelling in Tomography and Ill-Posed Problems

Mikhail M. Lavrent'ev †; Sergei M. Zerkal, Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk, Russia; Oleg E. Trofimov, Institute of Automation and Electrometry, Russian Academy of Sciences, Novosibirsk, Russia.

Introduction
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 transform 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
Bibliography