The Mathematics of Computerized Tomography

F. Natterer

ISBN 10: 351902103X / ISBN 13: 9783519021032

Editore: Vieweg+Teubner Verlag, 1986

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Druck auf Anfrage Neuware - Printed after ordering - By computerized tomography (CT) we mean the reconstruction of a function from its line or plane integrals, irrespective of the field where this technique is applied. In the early 1970s CT was introduced in diagnostic radiology and since then, many other applications of CT have become known, so me of them preceding the application in radiology by many years. In this book I have made an attempt to collect so me mathematics which is of possible interest both to the research mathematician who wants to und erstand the theory and algorithms of CT and to the practitioner who wants to apply CT in his special field of interest. I also want to present the state of the art of the mathematical theory of CT as it has developed from 1970 on. It seems that essential parts of the theory are now weIl understood. In the selection of the material I restricted myself - with very few exceptions-to the original problem of CT, even though extensions to other problems of integral geometry, such as reconstruction from integrals over arbitrary manifolds are possible in so me cases. This is because the field is presently developing rapidly and its final shape is not yet visible. Another glaring omission is the statistical side of CT which is very important in practice and which we touch on only occasionally.

Codice articolo 9783519021032

Dati bibliografici

• Titolo: The Mathematics of Computerized Tomography
• Editore: Vieweg+Teubner Verlag
• Data di pubblicazione: 1986
• Lingua: Tedesco
• ISBN 10: 351902103X
• ISBN 13: 9783519021032
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 229 L x 152 A x 14 P (millimetri) 229 millimetri di larghezza per 152 millimetri di altezza per 14 millimetri di profondità
• Peso: 354 grammi

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Riassunto

By computerized tomography (CT) we mean the reconstruction of a function from its line or plane integrals, irrespective of the field where this technique is applied. In the early 1970s CT was introduced in diagnostic radiology and since then, many other applications of CT have become known, so me of them preceding the application in radiology by many years. In this book I have made an attempt to collect so me mathematics which is of possible interest both to the research mathematician who wants to und erstand the theory and algorithms of CT and to the practitioner who wants to apply CT in his special field of interest. I also want to present the state of the art of the mathematical theory of CT as it has developed from 1970 on. It seems that essential parts of the theory are now weIl understood. In the selection of the material I restricted myself - with very few exceptions-to the original problem of CT, even though extensions to other problems of integral geometry, such as reconstruction from integrals over arbitrary manifolds are possible in so me cases. This is because the field is presently developing rapidly and its final shape is not yet visible. Another glaring omission is the statistical side of CT which is very important in practice and which we touch on only occasionally.

Contenuti

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.