Geometric Functionals of Fractal Percolation

Fractal percolation exhibits a dramatic topological phase transition, changing abruptly
from a dust-like set to a system-spanning cluster. Where this transition occurs is unknown
and notoriously difficult to estimate. In continuum percolation, the percolation thresholds have been approximated well using Minkowski functionals. Motivated by the question of whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process.

M. A. Klatt and S. Winter. Geometric functionals of fractal percolation. Adv. Appl. Probab. 52:1085–1126 (2020)

We establish the existence of the limit functionals and obtain explicit formulas for them as well as for their finite approximations (in excellent agreement with simulation result). They provide new geometric insights into fractal percolation. For example, we provide a mathematically rigorous notion of fractal subdimensions that have been previously defined in physics as an approach to refined information about fractal sets beyond fractal dimension.

My code to simulate fractal percolation and to analyze our geometric functionals is freely available at GitHub.


Detecting structured sources in noisy images via Minkowski maps

Distinguishing a feature from background noise is often the key to discovering physical phenomena in spatial data for a wide range of applications. We show how the sensitivity of a null hypothesis test can be dramatically increased by additional geometric information without the need for any prior knowledge about potential sources.

M. A. Klatt and K. Mecke. Detecting structured sources in noisy images via Minkowski maps. EPL (Europhysics Letters) 128:60001-p1–8 (2019)

We present a morphometric analysis technique, based on Minkowski functionals, to quantify the shape of structural deviations in greyscale images. It identifies important features in noisy spatial data, especially for short observation times and low statistics.
Without assuming any prior knowledge about potential sources, the additional shape information can increase the sensitivity by 14 orders of magnitude compared to previous methods. Rejection rates can increase by an order of magnitude.

As a key ingredient to such a dramatic increase, we accurately describe the distribution of the homogeneous background noise in terms of the density of states for the area, perimeter, and Euler characteristic of random black-and-white images. Remarkably, the density of states can vary by 64 orders of magnitude. We apply our technique to data of the H.E.S.S. experiment.

The density of states and my code for Minkowski maps are freely available at GitHub.

All simulated data and parameters underlying our study are available at Zenodo.


Strength of bacterial adhesion on nanostructured surfaces quantified by substrate morphometry

In this paper, we combine physics, microbiology and mathematics to study how bacteria attach to nanostructured surfaces, and how topography of nanostructured substrates can directly quantify the adhesion.

C. Spengler, F. Nolle, J. Mischo, T. Faidt, S. Grandthyll, N. Thewes, P. Jung, M.
Koch, F. Müller, M. Bischoff, M. A. Klatt, K. Jacobs. Strength of bacterial adhesion on nanostructured surfaces quantified by substrate morphometry. Nanoscale 11:19713–19722 (2019)

Aiming for 3D-structured materials for or against bio-adhesion, we studied mechanisms by which the bacteria adhere to different types of materials. Our insights help, for example, to prevent adhesion of pathogens (like Staphylococcus aureus) before the bacteria can begin to grow a biofilm, which protects individual bacteria from attack by antibiotics.

We found that the measured forces with which the individual bacterial cells adhered to nanostructured surfaces decrease with an increasing size of these nanostructures. In fact, with our morphometric analysis based on Minkowski functionals, we found that the strength of the adhesion force can be predicted from measurements of the surface area that is accessible to the tethering proteins with which the bacteria adhere to the surface. More precisely, we found that the ratio of the experimentally measured adhesion forces at the rough surface compared to a smooth surface agrees with the fraction of surface area that is accessible to the tethering proteins, which are about 50 nm long.

There are press releases in English and German.


Goodness-of-fit tests for complete spatial randomness based on Minkowski functionals of binary images

In this paper we use the Minkowski functionals for a rigorous null hypothesis test for complete spatial randomness.

Excursion set of a binned Poisson point process

B. Ebner, N. Henze, M. A. Klatt, K. Mecke. Goodness-of-fit tests for complete spatial randomness based on Minkowski functionals of binary images. Electron. J. Statist 12:2873–2904 (2018)

We propose a class of goodness-of-fit tests for complete spatial randomness (CSR). In contrast to standard tests, our procedure utilizes a transformation of the data to a binary image, which is then characterized by geometric functionals. Under a suitable limiting regime, we derive the asymptotic distribution of the test statistics under the null hypothesis and almost sure limits under certain alternatives. The new tests are computationally efficient, and simulations show that they are strong competitors to other tests of CSR. The tests are applied to a real data set in gamma-ray astronomy, and immediate extensions are presented to encourage further work.


Characterization of maximally random jammed sphere packings. I. Voronoi correlation functions

In this paper, we characterize the local and global structure of maximally random jammed sphere packings.

MRJ sphere packing: (left) only the spheres, (right) spheres and Voronoi cells

M. A. Klatt and S. Torquato. Characterization of Maximally Random Jammed Sphere Packings: Voronoi Correlation Functions. Phys. Rev. E, 90:052120-1–12 (2014)

We characterize the structure of the maximally disordered packing among the set of all packings of monodisperse frictionless hard spheres, the so-called maximally random jammed (MRJ) sphere packing. Therefore, we compute the Minkowski functionals of the associated Voronoi cells and compare the structure to that of the Poisson point process (ideal gas) and of an equilibrium hard-sphere liquid.

In particular, we consider correlation functions or probability density functions of these Voronoi characteristics. Here we introduce and compute correlation functions and probability density functions of Minkowski functionals to quantify the global structure of the Voronoi diagram.

The local analysis using the distribution of the Voronoi volumes finds no qualitative difference for the structure of liquid or random jammed hard-sphere packings. In contrast to this, the higher-order statistical descriptors introduced here qualitatively distinguish the Voronoi structure of the MRJ sphere packings (prototypical glasses) from that of a hard-sphere liquid. We find strong anti-correlations in the MRJ sphere packings that arise because the MRJ state is “hyperuniform”.



Morphometric analysis in gamma-ray astronomy using Minkowski functionals. I. Source detection via structure quantification

In this paper, we introduce a morphometric data analysis of gamma-ray sky maps.

How to compute a Minkowski sky map

D. Göring, M. A. Klatt, C. Stegmann, and K. Mecke. Morphometric Analysis in Gamma-Ray Astronomy using Minkowski Functionals. Astron. Astrophys., 555:A38-1–7 (2013)

We introduce a novel approach to source detection via structural deviations from typical features of a random homogeneous background. Minkowski functionals are powerful tools from integral geometry; in 2D, they are the area, the perimeter and the Euler characteristics (a topological quantity, which is given by the sum of all components minus the sum of all holes).

Via a combination of these different geometric measures that quantify the shape of level sets of a counts map, more information can be taken out of the same data without the need to assume prior knowledge about potential sources.

We introduce Minkowski sky maps that quantify local structural deviations. Moreover, they localize and visualize potential sources.