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The mathematics of computerized tomography. (English) Zbl 0617.92001
Stuttgart: B. G. Teubner; Chichester etc.: John Wiley & Sons. X, 222 p. DM 72.00 (1986).
Computerized tomography deals with the reconstruction of a function on \(R^ n\) from its line, plane or hyperplane integrals. The best known applications include diagnostic radiology, where a cross-section of a part of a body is scanned by X-ray beams, whose intensity loss is recorded and processed by a computer to produce a two-dimensional image. Other applications stem from electron microscopy, positron emission tomography and NMR-spectroscopy, to mention only some.
In this context the Radon transform plays a key role, since it maps functions on \(R^ n\) into the set of their integrals over hyperplanes of \(R^ n\). The reconstruction problem simply calls for the inversion of the Radon transform (or related transforms). In principle, this inversion problem was solved by Radon in 1917. For practical applications, however, Radon’s inversion formula is not very useful, since in practice the integrals can be measured only for a finite number of lines, whose arrangement is determined by the scanning geometry. This causes severe complications, since one has to deal with ill posed and incomplete data problems.
The present book gives a well written account on the underlying mathematics. Chapter I contains the basic practical examples. Chapter II treats the Radon transform and related transforms. In particular, inversion formulas, theorems on uniqueness and Sobolev space estimates are derived.
In Chapter III sampling and resolution is discussed, that is, it is investigated for which directions the transformed function has to be known in order that the original function can be recovered reliably. It turns out that the (essentially) band-limited functions play an important role.
Chapter IV is on ill-posed problems and accuracy. In particular, it contains error estimates and a singular value decomposition theorem of the Radon transform. Chapter V deals with reconstruction algorithms and Chapter VI is dedicated to incomplete data problems. Finally, Chapter VII contains some mathematical tools that are used throughout, including Fourier analysis, discrete Fourier transform, special functions and Sobolev spaces.
The book will not only be of interest to mathematicians working in this field, but it could also serve as an introductory text for readers having a background in the topics that are reviewed in chapter VII.
Reviewer: R.Bürger

MSC:
92-02 Research exposition (monographs, survey articles) pertaining to biology
92C50 Medical applications (general)
44-02 Research exposition (monographs, survey articles) pertaining to integral transforms
78A70 Biological applications of optics and electromagnetic theory
78-02 Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory
92F05 Other natural sciences (mathematical treatment)