  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".
