Tensorial Minkowski Functionals of Anisotropic Spatial Structures
Klaus Mecke, Gerd Schroeder-Turk
Institut für Theoretische Physik, Universität Erlangen-Nürnberg, Erlangen, Germany

We describe a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals. Tensorial physical properties such as elasticity, permeability and conductance of microstructured heterogeneous materials require quantitative measures for anisotropic characteristics of spatial structure. Tensor-valued Minkowski functionals, defined in the framework of integral geometry, provide a concise set of descriptors of anisotropic morphology. The talk provides an overview of the theory of Minkowski tensors and functionals. We describe the robust computation of these measures for microscopy images and polygonal shapes by linear-time algorithms. We demonstrate their relevance for shape description, their versatility and their robustness by applying them to experimental datasets, specifically microscopy datasets. Applications are shown in two dimensions on Turing patterns and on sections of ice grains from Antarctic cores. In three dimensions Minkowski tensors have been used to quantify the anisotropy of granular matter, of confocal microscopy images of sheared biopolymers and of triply-periodic minimal surface models for amphiphilic self-assembly.

Keywords: Integral geometry; Anisotropic spatial structures; Statistical physics

Biography: Klaus Mecke received his doctoral degree in 1993 from the University of Munich with his thesis on integral geometry in statistical physics. In 2001 he joined the Max-Planck-Institute for Metal Research in Stuttgart. Since 2004 he is full professor for theoretical physics at the Friedrich-Alexander Universität Erlangen-Nürnberg in Germany. His research focuses on statistical physics, density functional theory for complex fluids, and the morphometric description of spatial structures.