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

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".

This paper is a combined replacement for the papers "A Neural Network Anomaly Detector using the Random Cluster Model" (arXiv:1501.07227), "On the Sharp Threshold Interval Length of Partially Connected Random Geometric Graphs During K-Means Classification" (arXiv:1412.4178) and "Estimating the Mean Number of K-Means Clusters to Form" (arXiv:1503.03488), which have all been withdrawn

References in corpus (5)

• Statistical performance of support vector machines
• Continuum percolation of wireless ad hoc communication networks
• Monotone properties of random geometric graphs have sharp thresholds
• Continuum Limits of Markov Chains with Application to Network Modeling
• The spectrum of a random geometric graph is concentrated

Cited by in corpus (1)

• Image Segmentation, Compression and Reconstruction from Edge Distribution Estimation with Random Field and Random Cluster Theories