The Radon transform and local tomography. (English) Zbl 0863.44001

Ramm, A. G.; Katsevich, A. I.

Boca Raton, FL: CRC Press. xviii, 485 p. (1996).

The authors start by introductory chapters where they explain the contents of the book, give several physical motivations to the problems of tomography, and call again the basic definitions and properties about the Radon transform and related transforms. The main aim of the book is the theory developed by the authors which deal with the study of singularities of the Radon transform and its applications to imaging in tomography.
They give methods for finding discontinuities of functions and the sizes of the jumps of these functions across the discontinuities from local tomographic data. The methods proposed are generalized local tomography, pseudolocal tomography and geometrical tomography. They study also incomplete tomographic data and the inversion of the cone-beam data. Moreover, they discuss a multidimensional algorithm for finding discontinuities of signals from noisy discrete data, and a general test of randomness and its applications.
In this book, the authors present and discuss numerical examples which show that the algorithms for tomographic imaging proposed are of practical values.

Reviewer: K.Trimeche (Tunis)

MSC:

44A12	Radon transform
92C55	Biomedical imaging and signal processing
65R10	Numerical methods for integral transforms
65R30	Numerical methods for ill-posed problems for integral equations
44-02	Research exposition (monographs, survey articles) pertaining to integral transforms

Keywords:

medical diagnostics; X-ray transforms; backprojection operator; Legendre transform; image processing; Radon transform; local tomography; pseudolocal tomograph; geometrical tomography; incomplete tomographic data; cone-beam data; multidimensional algorithm; discontinuities of signals; test of randomness; numerical examples