A generalization of the inhomogeneity measure for point distributions to the case of finite size objects

TL;DR

This paper extends a spatial inhomogeneity measure to finite-sized objects in binary patterns, using a sliding cell approach to analyze patterns and compare statistical and entropic measures for detecting periodicity and quasi-crystal features.

Contribution

It introduces a generalized inhomogeneity measure for finite objects and a sliding cell sampling method to analyze patterns at various scales.

Findings

The generalized measure relates to the smallest realizable value at each scale.

Comparison shows differences in peak detection between statistical and entropic measures.

Both measures can reveal periodicity and quasi-crystal features.

Abstract

The statistical measure of spatial inhomogeneity for n points placed in chi cells each of size kxk is generalized to incorporate finite size objects like black pixels for binary patterns of size LxL. As a function of length scale k, the measure is modified in such a way that it relates to the smallest realizable value for each considered scale. To overcome the limitation of pattern partitions to scales with k being integer divisors of L we use a sliding cell-sampling approach. For given patterns, particularly in the case of clusters polydispersed in size, the comparison between the statistical measure and the entropic one reveals differences in detection of the first peak while at other scales they well correlate. The universality of the two measures allows both a hidden periodicity traces and attributes of planar quasi-crystals to be explored.