Towards more reasonable identifications of the symmetries in noisy digital images from periodic and aperiodic crystals

Abstract

A geometric form of information theory allows for reasonable, i.e. probabilistic, evidence-ranking based, and generalized noise-level dependent, classifications of the crystallographic and quasicrystallographic symmetries in noisy digital images. Such classifications are based solely on the image pixel intensity values, justifiable assumptions about the aggregate distribution of generalized noise in the images, asymptotic extrapolations to zero-noise images, and rational symmetry model selections with maximized predictive accuracy in the presence of both symmetry-inclusion relations and pseudo-symmetries. Preferring a well developed geometric form of information theory over a theoretically possible geometric-Bayesian approach for these classifications is the only subjective choice made. Using digital data planes and assuming approximately Gaussian distributed generalized noise, reasonable crystallographic and quasicrystallographic symmetry classifications can be made for noisy images from both scanning probe and transmission electron microscopes. A binary type classification of structurally very similar mate-rials into either a quasicrystal or one of its rational/crystalline approximants based on the approximate point symmetries in their noisy digital images is proposed here for the first time.