A Critical Connectivity Radius for Segmenting Randomly-Generated, High Dimensional Data Points

TL;DR

This paper introduces a probabilistic framework to identify a critical connectivity radius in high-dimensional data, revealing when data transitions from short-range to long-range correlations, with applications in image segmentation.

Contribution

It develops a mathematical model to determine a critical radius for connectivity in high-dimensional data projections, generalizing image segmentation techniques.

Findings

Calculated critical thresholds for data correlation

Defined a neighbor concept for data structures

Estimated the radius interval for correlation transition

Abstract

Motivated by a $2$-dimensional (unsupervised) image segmentation task whereby local regions of pixels are clustered via edge detection methods, a more general probabilistic mathematical framework is devised. Critical thresholds are calculated that indicate strong correlation between randomly-generated, high dimensional data points that have been projected into structures in a partition of a bounded, $2$-dimensional area, of which, an image is a special case. A neighbor concept for structures in the partition is defined and a critical radius is uncovered. Measured from a central structure in localized regions of the partition, the radius indicates strong, long and short range correlation in the count of occupied structures. The size of a short interval of radii is estimated upon which the transition from short-to-long range correlation is virtually assured, which defines a demarcation of when an image ceases to be "interesting".