Comparatively weakly researched untraditional tomography problems are solved because of new achievements in calculation mathematics and the theory of ill-posed problems, the regularization process of solving ill-posed problems, and the increase of stability. Experiments show possibilities and applicability of algorithms of processing tomography data. This monograph is devoted to considering these problems in connection with series of ill-posed problems in tomography settings arising from practice.The book includes chapters to the following themes:

Mathematical basis of the method of computerized tomography
Cone-beam tomography reconstruction
Inverse kinematic problem in the tomographic setting

Autorentext

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.

Inhalt

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.