Voronoi-Markov chain and spatial entropy for point pattern analysis

2026-09-28

AI Summary

This study addresses the lack of computable and interpretable spatial entropy metrics in pathological point pattern analysis by proposing Dense Basin Entropy and a simplified Brownian motion Markov chain. Methodologically, novel spatial entropy metrics are defined based on Voronoi/Delaunay tessellations, and a Markov chain approximating Brownian motion is constructed. Combined with spectral gap analysis, this framework enables the generalization of multi-set hypothesis testing and feature extraction. In histopathological experiments, the proposed metric demonstrates high sensitivity to spatial clustering, providing analytical insights complementary to conventional methods. Ultimately, this work establishes a new paradigm for quantifying the spatial structure of complex point patterns.

Abstract

In this paper, we revisit the concept of entropy in the spatial context with the aim of deriving computable and interpretable metrics for point pattern analysis in domains such as histopathology. We discuss stochastic point process assumptions like Poisson homogeneity and Simple Sequential Inhibition (SSI) and review established results for Voronoi and Delaunay tilings of point sets for their potential to provide null distributions for rigorous empirical hypothesis testing. We present (1) a novel "dense basin entropy" defined in terms of the Ord distribution supported on Voronoi basins, which is shown to be sensitive to clustering, and (2) a related Markov chain designed as a simplification or approximation of Brownian motion in the underlying planar domain. We investigate bounds on the dense basin entropy in the SSI regime and in empirical data, and show that the entropy rate and spectral gap of the Markov chain lead to sharp discrimination metrics. Finally, we demonstrate a generalization to multiple-set analysis via aggregation of one set over the Markov invariant measure for another. Simulations and experiments in histopathology show that our proposed metrics offer insights complementary to other instantiations of entropy in spatial analysis. The code can be found on our SMProfiler GitHub: https://github.com/nadeemlab/SMProfiler.

Problem

• point pattern analysis
• spatial entropy
• histopathology
• Voronoi tessellation
• Markov chain

Innovation

• Voronoi-Markov chain
• spatial entropy
• dense basin entropy
• point pattern analysis
• histopathology