Computer Modelling in Tomography and Ill Posed Problems

✍ Scribed by Michail M. Lavrent'ev; M. Mikhail Mikhailovich Lavrentev; S. M. Zerkal; Oleg Evgenʹevich Trofimov; O. E. Trofimov

• Publisher: VSP
• Year: 2001
• Tongue: English
• Leaves: 136
• Category: Library

✦ Synopsis

The last decades of the 20th century were marked by the appearance of a new field of mathematics: computerized tomography. Its theory forms the basis for the solution of many applied problems. The methods of computerized tomography make it possible study the interior structure of a body by examining the characteristics of radiation passing through the object under study (transmission tomography). Depending on the type of radiation used, X-ray, optical, seismic, and some other kinds of tomography can be distinguished. Comparatively weakly researched, untraditional tomography problems are being solved because of new achievements in calculation mathematics and the theory of ill-posed problems (3D cone-beam tomography, geo-tomography). 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 basic themes of the book are: mathematical basis of the method of computerized tomography; algorithms for 3D cone-beam tomography; and inverse kinematics problems in tomographic settings (geo-tomography). This volume in the Inverse and Ill-Posed Problems Series will be of interest to researchers, graduates and post-graduates in X-ray, optical, seismic, as well as some other kinds of tomography in both academia and industry.

✦ Table of Contents

Introduction
Chapter 1. Mathematical basis of the method of computerized tomography
1.1. Basic notions of the theory of ill-posed problems
1.2. Problem of integral geometry
1.3. The Radon transform
1.4. Radon problem as an example of an ill-posed problem
1.5. The algorithm of inversion of the two-dimensional Radon transform based on the convolution with the generalized function 1/z2
Chapter 2. Cone-beam tomography reconstruction
2.1. Reducing the inversion formulas of cone-beam tomography reconstruction to the form convenient for constructing numerical algorithms
2.2. Elements of the theory of generalized functions in application to problems of inversion of the ray transformation
2.3. The relations between the Radon, Fourier, and ray transformations
Chapter 3. Inverse kinematic problem in the tomographic setting
3.1. Direct kinematic problem and numerical solution for three-dimensional regular media
3.2. Formulation of the inverse kinematic problem with the use of a tomography system of data gathering
3.3. Deduction of the basic inversion formula and the algorithm of solving the inverse kinematic problem in three-dimensional linearized formulation
3.4. Model experiment and numerical study of the algorithm
3.5. 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