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.