Regensburg 2016 – wissenschaftliches Programm
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DY: Fachverband Dynamik und Statistische Physik
DY 14: Complex Systems
DY 14.4: Vortrag
Montag, 7. März 2016, 17:00–17:15, H47
Anisotropic Gaussian random fields characterized by Minkowski tensors — •Michael A. Klatt1,2, Max Hörmann2, and Klaus Mecke2 — 1KIT, Institut für Stochastik, Englerstraße 2, 76131 Karlsruhe — 2FAU, Institut für Theoretische Physik, Staudtstraße 7, 91058 Erlangen
Anisotropic Gaussian random fields are important models of anisotropic disordered structures that appear in very different physical and biological systems. Often physical insight is best achieved by a rigorous structure characterization, for which comprehensive shape descriptors are needed. We here show how the Minkowski tensors as sensitive and robust measures of anisotropy, which extend the notion of volume and surface area to scalar and tensorial morphometric measures [1], comprehensively characterize the shape of level sets of Gaussian random fields.
We give explicit expressions for the mean values and compare them to simulation results. We also provide explicit integral expressions for the second moments of the Minkowski functionals. Which additional information is contained in higher rank Minkowski tensors? We find that tensors of higher rank indeed contain additional anisotropy information as compared to the tensor of rank two. Surprisingly, we can show that the latter is nevertheless sufficient to estimate the model parameters which are necessary to determine all Minkowski tensors of arbitrary rank. Using this relation a null hypothesis test for non-Gaussianities in anisotropic random fields can be defined.
[1] G. E. Schröder-Turk et al., Adv. Mater. 23:2535, 2011.