The mathematics of computerized tomography
Author(s)
Bibliographic Information
The mathematics of computerized tomography
B.G. Teubner , Wiley, c1986
- : us
- : gw
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Note
Bibliography: P. [213]-220
Includes index
Description and Table of Contents
- Volume
-
: us ISBN 9780471909590
Description
The central subject of this book is the reconstruction of a function from line or plane integrals, with special emphasis on applications in science, radiology and engineering. It not only covers the relevant mathematical theory of the Radon transform and related transforms, but also studies more practical questions such as sampling, resolution, stability and accuracy. Much of the book is devoted to the derivation, analysis and practical examination of reconstruction algorithms, both for standard problems and problems with incomplete data. An appendix gives a brief review of the mathematical background needed, making the book self-contained. Also included is a four- page section of colour plates showing the results of image reconstruction by various methods, using both complete and incomplete data.
- Volume
-
: gw ISBN 9783519021032
Table of Contents
I. Computerized Tomography.- I.1 The basic example: transmission computerized tomography.- I.2 Other applications.- I.3 Bibliographical notes.- II. The Radon Transform and Related Transforms.- II.1 Definition and elementary properties of some integral operators.- II.2 Inversion formulas.- II.3 Uniqueness.- II.4 The ranges.- II.5 Sobolev space estimates.- II.6 The attenuated Radon transform.- II.7 Bibliographical notes.- III. Sampling and Resolution.- III.1 The sampling theorem.- III.2 Resolution.- III.3 Some two-dimensional sampling schemes.- III.4 Bibliographical notes.- IV. Ill-posedness and Accuracy.- IV.1 Ill-posed problems.- IV.2 Error estimates.- IV.3 The singular value decomposition of the Radon transform.- IV.4 Bibliographical notes.- V. Reconstruction Algorithms.- V.1 Filtered backprojection.- V.2 Fourier reconstruction.- V.3 Kaczmarz's method.- V.4 Algebraic reconstruction technique (ART).- V.5 Direct algebraic methods.- V.6 Other reconstruction methods.- V.7 Bibliographical notes.- VI. Incomplete Data.- VI.1 General remarks.- VI.2 The limited angle problem.- VI.3 The exterior problem.- VI.4 The interior problem.- VI.5 The restricted source problem.- VI.6 Reconstruction of homogeneous objects.- VI.7 Bibliographical notes.- VII. Mathematical Tools.- VII.1 Fourier analysis.- VII.2 Integration over spheres.- VII.3 Special functions.- VII.4 Sobolev spaces.- VII.5 The discrete Fourier transform.- References.
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