## Mathematical morphology and statistical methods for the analysis of directional data with applications in civil engineering

• We encounter directional data in numerous application areas such as astronomy, biology or engineering. Examples include the direction of arrival of cosmic rays, the direction of flight of migratory birds or the orientation of steel fibres in fibre-reinforced concrete. In part I, we define and apply morphological operators, quantiles and depths for directional data. The morphological operators are defined for $$\mathcal{S}^{d−1}$$-valued images with $$\mathcal{S}^{d−1} = \{x \in \mathbb{R}^d :\sqrt{x^T x} = 1\}$$ , $$d \geq 2$$. Since an ordered structure is necessary for a definition of these operators, which is not naturally given between vectors, an order is determined with the help of the theory of statistical depth functionals. This allows for defining the basic operators erosion and dilation as well as morphological (multi-scale) operators for $$\mathcal{S}^{d−1}$$-valued images based on them. The operators introduced are related to their grey value counterparts. Furthermore, quantiles and the "angular Mahalanobis" depth for directional data introduced by Ley et al. (2014) are extended. The concept of Ley et al. (2014) provides useful geometric properties of the depth contours (such as convexity and rotational equivariance) and a Bahadur-type representation of the quantiles. Their concept is canonical for rotationally symmetric depth contours. However, it also produces rotationally symmetric depth contours when the underlying distribution is not rotationally symmetric. We solve this lack of flexibility for distributions with elliptical depth contours. The basic idea is to deform the elliptic contours by a diffeomorphic mapping to rotationally symmetric contours, thus reverting to the canonical case in Ley et al. (2014). Our results are confirmed by a Monte Carlo simulation study and applied to the analysis of fibre directions in fibre-reinforced concrete. In Part II, we elaborate interdisciplinary results of statistical analysis and stochastic modelling in civil engineering. Our statistical analysis of the correlation between production parameters (fibre length, fibre diameter, fibre volume fraction as well as casting method, superplasticiser content and specimen size) of ultra-high performance fibre reinforced concrete and the fibre system (spatial arrangement and orientation of the fibres) provides users with a better understanding of this relatively new composite material. The fibre system is modelled by a Boolean model and the fibre orientation by a one-parameter distribution. In addition, the behaviour under tensile loading is modelled.

Author: Konstantin Hauch urn:nbn:de:hbz:386-kluedo-69614 https://doi.org/10.26204/KLUEDO/6961 Claudia Redenbach Doctoral Thesis English 2022/10/07 2022 Technische Universität Kaiserslautern Technische Universität Kaiserslautern 2022/09/08 2022/10/07 152 Kaiserslautern - Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik Creative Commons 4.0 - Namensnennung (CC BY 4.0)