Rotational Integral Geometry and its Applications

This self-contained book offers an extensive state-of-the-art exposition of rotational integral geometry, a field that has reached significant maturity over the past four decades. Through a unified description of key results previously scattered across various scientific journals, this book provides a cohesive and thorough account of the subject. Initially, rotational integral geometry was driven by applications in fields such as optical microscopy. Rotational integral geometry has now evolved into an independent mathematical discipline. It contains a wealth of theorems paralleling those in classical kinematic integral geometry for Euclidean spaces, such as the rotational Crofton formulae, rotational slice formulae, and principal rotational formulae. The present book presents these for very general tensor valuations in a convex geometric setting. It also discusses various applications in the biosciences, explained with a mathematical audience in mind. This book is intended for a diverse readership, including specialists in integral geometry, and researchers and graduate students working in integral, convex, and stochastic geometry, as well as geometric measure theory.

Eva B. Vedel Jensen obtained a Doctor of Science degree in 1987 from Aarhus University (AU). She has been a staff member at the Department of Mathematics, AU, since 1976, as full professor since 1998. Since 2020, she has been affiliated to AU as professor emerita. Member of the Danish Natural Science Research Council, 2001-2007, and Royal Danish Academy of Sciences and Letters, 2010-. In 2010, she received The Order of Dannebrog and in 2013 she became honorary doctor at University of Bern. In 2010-2020, she was the scientific director of Centre for Stochastic Geometry and Advanced Bioimaging with rotational integral geometry as one of the focus points. She is the author of two research monographs and approximately 90 scientific papers (stereology, integral geometry, stochastic geometry, geometric measure theory, probability theory, statistics).

Markus Kiderlen obtained an Erasmus Diploma in mathematics from the Université Grenoble Alpes in 1990 and a Doctor of Science degree in 1999 from the Karlsruhe Institute of Technology (KIT). After a postdoc period at KIT, he has been a staff member at the Department of Mathematics, AU, since 2004, as associate professor since 2007. In 2011-2020 he was co-leader of the project Image analysis and spatial statistics within the DFG research unit Geometry and Physics of Random Spatial Systems. He was leader of the Stochastic Geometry Group within the Centre for Stochastic Geometry and Advanced Bioimaging in 2016-2020. He has published an edited volume on Tensor Valuations and Their Applications in Stochastic Geometry and Imaging together with Eva B. Vedel Jensen and approximately 45 peer-reviewed papers on stereology, convex and integral geometry, stochastic geometry, geometric tomography and inverse problems.

- 1. Introduction.- 2. Convex Bodies and their Classical Integral Geometry.- 3. Integral Geometric Transformations.- 4. Rotational Crofton Formulae for Intrinsic Volumes.- 5. Rotational Crofton Formulae for Minkowski Tensors.- 6. Rotational Slice Formulae.- 7. Further Rotational Integral Geometric Formulae.- 8. Applications to Particle Populations.- 9. Implementation in Optical Microscopy.