Rotational integral geometry and local stereology – with a view to image analysis. (English) Zbl 1343.53079
Schmidt, Volker (ed.), Stochastic geometry, spatial statistics and random fields. Models and algorithms. Selected papers based on the presentations at the summer academy on stochastic analysis, modelling and simulation of complex structures, Söllerhaus, Hirschegg, Germany, September 11–24, 2011. Cham: Springer (ISBN 978-3-319-10063-0/pbk; 978-3-319-10064-7/ebook). Lecture Notes in Mathematics 2120, 233-255 (2015).
This article is a short report on Chapter 8 of the book [Volker Schmidt (ed.), Stochastic geometry, spatial statistics and random fields. Models and algorithms].
Authors’ abstract: “This chapter contains an introduction to rotational integral geometry that is the key tool in local stereological procedures for estimating quantitative properties of spatial structures. In rotational integral geometry, focus is on integrals of geometric functionals with respect to rotation invariant measures. Rotational integrals of intrinsic volumes are studied. The opposite problem of expressing intrinsic volumes as rotational integrals is also considered. It is shown how to express intrinsic volumes as integrals with respect to geometric functionals defined on lower dimensional linear subspaces. Rotational integral geometry of Minkowski tensors is shortly discussed as well as a principal rotational formula. These tools are then applied in local stereology leading to unbiased stereological estimators of mean intrinsic volumes for isotropic random sets. At the end of the chapter, emphasis is put on how these procedures can be implemented when automatic image analysis is available. Computational procedures play an increasingly important role in the stereological analysis of spatial structures and a new sub-discipline, computational stereology, is emerging.”
The chapter under review provides an introduction to various aspects of stochastic geometry, spatial statistics and random fields, with special emphasis on fundamental classes of models and algorithms as well as on their applications. It provides also a thorough introduction to rotational integral geometry and its application in local stereological methods. Stereology is a set of methods that allow unbiased and efficient estimation of geometric quantities defined in arbitrary structures.
In the first section the main concepts and results of Rotational Integral Geometry are summarized. Section 2 consists of the applications of the concepts developed in the previous section to four interesting examples of local stereology. In Section 3, the authors deal with some techniques for reducing the variance of the estimators defined in the previous section. Section 4 discusses without much details a few proposals of computational procedures applicable to the estimates of the previous sections that the authors define as “computational stereology”. The Chapter under review is clearly written and the proofs are rigorously established. Probably, for a non-experienced reader it would be a little bit hard to read. However for those researchers working in affine fields the whole book is an excellent asset.
For the entire collection see [Zbl 1301.60005].
Authors’ abstract: “This chapter contains an introduction to rotational integral geometry that is the key tool in local stereological procedures for estimating quantitative properties of spatial structures. In rotational integral geometry, focus is on integrals of geometric functionals with respect to rotation invariant measures. Rotational integrals of intrinsic volumes are studied. The opposite problem of expressing intrinsic volumes as rotational integrals is also considered. It is shown how to express intrinsic volumes as integrals with respect to geometric functionals defined on lower dimensional linear subspaces. Rotational integral geometry of Minkowski tensors is shortly discussed as well as a principal rotational formula. These tools are then applied in local stereology leading to unbiased stereological estimators of mean intrinsic volumes for isotropic random sets. At the end of the chapter, emphasis is put on how these procedures can be implemented when automatic image analysis is available. Computational procedures play an increasingly important role in the stereological analysis of spatial structures and a new sub-discipline, computational stereology, is emerging.”
The chapter under review provides an introduction to various aspects of stochastic geometry, spatial statistics and random fields, with special emphasis on fundamental classes of models and algorithms as well as on their applications. It provides also a thorough introduction to rotational integral geometry and its application in local stereological methods. Stereology is a set of methods that allow unbiased and efficient estimation of geometric quantities defined in arbitrary structures.
In the first section the main concepts and results of Rotational Integral Geometry are summarized. Section 2 consists of the applications of the concepts developed in the previous section to four interesting examples of local stereology. In Section 3, the authors deal with some techniques for reducing the variance of the estimators defined in the previous section. Section 4 discusses without much details a few proposals of computational procedures applicable to the estimates of the previous sections that the authors define as “computational stereology”. The Chapter under review is clearly written and the proofs are rigorously established. Probably, for a non-experienced reader it would be a little bit hard to read. However for those researchers working in affine fields the whole book is an excellent asset.
For the entire collection see [Zbl 1301.60005].
MSC:
| 53C65 | Integral geometry |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |
| 62M40 | Random fields; image analysis |
| 60D05 | Geometric probability and stochastic geometry |
