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Random field and random cluster theory are used to describe certainmathematical results concerning the probability distribution of image pixelintensities characterized as generic $2D$ integer arrays. The size of thesmallest bounded region within an image is estimated for segmenting an image,from which, the equilibrium distribution of intensities can be recovered. Fromthe estimated bounded regions, properties of the sub-optimal and equilibriumdistributions of intensities are derived, which leads to an image compressionmethodology whereby only slightly more than half of all pixels are required fora worst-case reconstruction of the original image. A custom deep belief networkand heuristic allows for the unsupervised segmentation, detection andlocalization of objects in an image. An example illustrates the mathematicalresults.

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