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**Numerik des Radonschen Problems.**
*(German)*
Zbl 0575.65130

This is a brilliant review of the state of the art in numerical inversion of the Radon transform, especially in connection with computer tomography (CT). In § 1 meaning and development of computer tomography is represented. In § 2 the relevant integral transforms and related inversion formulae are discussed. The continuity of Rf in certain Sobolev spaces is shown and it is to be seen, that the inversion is mildly ill- posed. In § 3 numerical reconstruction algorithms are reviewed. There are iterative and direct algorithms for the solution of the discretized problem and on the other hand numerical realizations of the inversion formulae of § 2, which are looked at in more detail.

In § 4 problems with incomplete data are discussed, which appear in most practical cases. There are (a) outer problems, \(| x| \geq a\), (b) inner problems, \(| x| \leq a\), (c) problems with restricted angle, (d) the divergent beam X-ray transform, (e) only few data, f.i. some directions only. If one has an infinite number of directions or sources, one gets uniqueness for (a), (c) and (d). One can get uniqueness for (e), too, if one can assume convex, homogeneous objects. Problems with incomplete data are badly ill-posed, but an example shows, that the reconstruction can become quite good, if the incompleteness of the data is geometrically insignificant. In 3D-reconstruction one either has got a very ill-posed problem or the sensors must not be rotationally symmetric, i.e.: 3D-CT will remain a dream for at least some time.

In § 5 the practical problem of having only a finite number of sources is considered. There is the possibility of frequency-range-limitation, which filters highly oscillating phantom objects. Today no reconstruction algorithm is known, which delivers the possible resolution. In § 6 developments and generalizations are discussed. These are important for applications in seismology and nuclear magnetic resonance, f.i. The Radon transform is a limit case of the inverse scattering problem for the wave equation, but it is not yet known, if results can be carried over. For any of these problems a lot of numerical work is to be done.

In § 4 problems with incomplete data are discussed, which appear in most practical cases. There are (a) outer problems, \(| x| \geq a\), (b) inner problems, \(| x| \leq a\), (c) problems with restricted angle, (d) the divergent beam X-ray transform, (e) only few data, f.i. some directions only. If one has an infinite number of directions or sources, one gets uniqueness for (a), (c) and (d). One can get uniqueness for (e), too, if one can assume convex, homogeneous objects. Problems with incomplete data are badly ill-posed, but an example shows, that the reconstruction can become quite good, if the incompleteness of the data is geometrically insignificant. In 3D-reconstruction one either has got a very ill-posed problem or the sensors must not be rotationally symmetric, i.e.: 3D-CT will remain a dream for at least some time.

In § 5 the practical problem of having only a finite number of sources is considered. There is the possibility of frequency-range-limitation, which filters highly oscillating phantom objects. Today no reconstruction algorithm is known, which delivers the possible resolution. In § 6 developments and generalizations are discussed. These are important for applications in seismology and nuclear magnetic resonance, f.i. The Radon transform is a limit case of the inverse scattering problem for the wave equation, but it is not yet known, if results can be carried over. For any of these problems a lot of numerical work is to be done.

Reviewer: N.Köckler

### MSC:

65R10 | Numerical methods for integral transforms |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

44A15 | Special integral transforms (Legendre, Hilbert, etc.) |

43A85 | Harmonic analysis on homogeneous spaces |

45H05 | Integral equations with miscellaneous special kernels |

53C30 | Differential geometry of homogeneous manifolds |

92F05 | Other natural sciences (mathematical treatment) |