Morphometry on the sphere: Cartesian and irreducible Minkowski tensors explained and implemented

Abstract

Minkowski tensors are comprehensive shape descriptors that robustly capture n-point information in complex random geometries and that have already been extensively applied in the Euclidean plane. Here, we devise a framework for Minkowski tensors on the sphere. We first advance the theory by introducing irreducible Minkowski tensors, which avoid the redundancies of previous representations. We, moreover, generalize Minkowski sky maps to the sphere. These maps are a concept of local anisotropy, which easily adjusts to masked data. We demonstrate the power of our new procedure by applying it to simulations and real data of the Cosmic Microwave Background, finding an anomalous region close to the well-known Cold Spot. The accompanying open-source software, litchi, used to generate these maps from data in the HEALPix-format is made publicly available to facilitate broader integration of Minkowski maps in other fields, such as fluid demixing, porous structures, or geosciences more generally.