Vector- and tensor-valued descriptors for spatial patterns

TL;DR

This paper extends scalar Minkowski valuations to vector- and tensor-valued measures, enabling detailed analysis of spatial patterns in various scientific fields, including astrophysics and molecular geometry.

Contribution

It introduces a comprehensive framework for vector- and tensor-valued Minkowski valuations and demonstrates their application to real and toy data across multiple disciplines.

Findings

Effective description of galaxy cluster morphology

Insights into spiral galaxy structures

Application to molecular geometry

Abstract

Higher-rank Minkowski valuations are efficient means for describing the geometry and connectivity of spatial patterns. We show how to extend the framework of the scalar Minkowski valuations to vector- and tensor-valued measures. The properties of these extensions are described in detail. We show the versatility of these measures by using simple toy models as well as real data. Our applications cover the morphology of galaxy clusters, the structure of spiral galaxies, and the geometry of molecules. Furthermore, we consider a physical ansatz closely related to higher-rank Minkowski valuations, the Rosenfeld functional known from density functional theory.