The mathematics of computerized tomography (Q2714092)

This book is a reprint of the 1986 edition, see the review Zbl 0617.92001. Computerized Tomography (CT) means reconstruction of a function from its line or plane integrals, irrespective of the field of application. Chapter~1 describes a few typical applications of CT including diagnostic radiology, giving an idea of the scope, limitations and potential of CT. Chapter~2 studies various integral transforms from a theoretical viewpoint. The n-dimensional Radon transform is detailed. Inversion formulas are derived and the problem of uniqueness of the restored function is tackled. Establishing the ranges of the operators introduced, the attenuated Radon transform concludes the chapter.NEWLINENEWLINENEWLINESampling and resolution are the topics of Chapter~3. The fundamental Shannon sampling theorem is given. The resolution issue and some two-dimensional sampling schemes are discussed next. To enable a serious mathematical study of CT, Chapter~4 brings in notions from the theory of ill-posed problems. The accuracy that can be expected for CT problems is discussed, given the accuracy of the data and the number of projections. The singular value decomposition of the Radon transform is derived.NEWLINENEWLINENEWLINESome well-known reconstruction algorithms are detailed in Chapter~5. The chapter starts with the filtered backprojection algorithm. Given next are an error analysis of the Fourier algorithm and an analysis of the convergence properties of the Kaczmarz method for the iterative solution of under- and overdetermined linear systems. Several versions of the algebraic reconstruction technique, the direct algebraic method and a survey of other reconstruction methods conclude the chapter.NEWLINENEWLINENEWLINEChapter~6 deals with incomplete data problems. The features shared by such problems are described: the nature of the artefacts, the degree of ill-posedness, and the data completion. Four specific problems are analyzed: the limited angle problem, the exterior problem, the interior problem, and the restricted source problem. It is demonstrated that homogeneous objects can be restored from very few data. The mathematical prerequisites for understanding the text are given as an auxiliary Chapter~8. These include: Fourier analysis, integration over spheres, special functions, Sobolev spaces, and the discrete Fourier transform.