Currents and K-functions for Fiber Point Processes

TL;DR

This paper introduces a novel K-function for analyzing fiber point processes by embedding fiber shapes as currents, allowing for the assessment of spatial and shape distributions in fiber data such as myelin sheaths.

Contribution

It develops a shape-valued K-function based on currents and extends Ripley's K-function to fiber data, providing a new statistical tool for fiber shape and spatial analysis.

Findings

The K-function effectively detects deviations from spatial homogeneity.

Application to real myelin data reveals spatial and shape patterns.

Simulation studies validate the method's accuracy.

Abstract

Analysis of images of sets of fibers such as myelin sheaths or skeletal muscles must account for both the spatial distribution of fibers and differences in fiber shape. This necessitates a combination of point process and shape analysis methodology. In this paper, we develop a K-function for shape-valued point processes by embedding shapes as currents, thus equipping the point process domain with metric structure inherited from a reproducing kernel Hilbert space. We extend Ripley's K-function which measures deviations from spatial homogeneity of point processes to fiber data. The paper provides a theoretical account of the statistical foundation of the K-function and its extension to fiber data, and we test the developed K-function on simulated as well as real data sets. This includes a fiber data set consisting of myelin sheaths, visualizing the spatial and fiber shape behavior of myelin configurations at different debts.