On the Self-Similarity of Natural Stochastic Textures

Self-similarity is the essence of fractal images and, as such, characterizes natural stochastic textures. This paper is concerned with the property of self-similarity in the statistical sense in the case of fully-textured images that contain both stochastic texture and structural (mostly deterministic) information. We firstly decompose a textured image into two layers corresponding to its texture and structure, and show that the layer representing the stochastic texture is characterized by random phase of uniform distribution, unlike the phase of the structured information which is coherent. The uniform distribution of the the random phase is verified by using a suitable hypothesis testing framework. We proceed by proposing two approaches to assessment of self-similarity. The first is based on patch-wise calculation of the mutual information, while the second measures the mutual information that exists across scales. Quantifying the extent of self-similarity by means of mutual information is of paramount importance in the analysis of natural stochastic textures that are encountered in medical imaging, geology, agriculture and in computer vision algorithms that are designed for application on fully-textures images.