Morphological processing of univariate Gaussian distribution-valued images based on Poincaré upper-half plane representation

Mathematical morphology is a nonlinear image processing methodology based on the application of complete lattice theory to spatial structures. Let us consider an image model where at each pixel is given a univariate Gaussian distribution. This model is interesting to represent for each pixel the measured mean intensity as well as the variance (or uncertainty) for such measurement. The aim of this paper is to formulate morphological operators for these images by embedding Gaussian distribution pixel values on the Poincaré upper-half plane. More precisely, it is explored how to endow this classical hyperbolic space with various families of partial orderings which lead to a complete lattice structure. Properties of order invariance are explored and application to morphological processing of univariate Gaussian distribution-valued images is illustrated.