<title>Algebra of image transformations</title>

Abstract

This paper consists of two distinct contributions. The first contribution formulates the theory of translational invariant image transformations in the context of symbolic transformation groups and presents results describing the decomposability of certain image transformations. The ultimate objective of such studies is to decompose image transformations efficiently by decomposing them into short sequences of simpler transformations. Our methods are used to show particular indecomposable transformations, so that where previously only existence results were shown (Nasu's theorem), we can now explicitly search for indecomposable transformations. The second contribution discusses related stochasticconcepts. It formulates several Markovian approximations to images with potential application to image segmentation, image generation, and image coding. One family of models is closely related to the renormalizationgroup concepts that were originally formulated to address problems in statistical physics and quantum field theory, and have more recently been used to provide hierarchical methods for image processing. We apply these approximations to a specific binary valued random field- the two dimensionalIsing model- and demonstrate that segmentation can be achieved, and that a hierarchical model haspromise in this regard.