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.


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.