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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Jensen, E.B.V., Rasmusson, A. (2015). Rotational Integral Geometry and Local Stereology - with a View to Image Analysis. In: Schmidt, V. (eds) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, vol 2120. Springer, Cham. https://doi.org/10.1007/978-3-319-10064-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-10064-7_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10063-0
Online ISBN: 978-3-319-10064-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)