On Classification Approaches for Crystallographic Symmetries of Noisy 2D Periodic Patterns

The analytical classification approaches for the crystallographic symmetries of patterns that are more or less periodic in two dimensions (2D) are reviewed. Pseudo-symmetries of different types, which are not rare as far as 2D projections from three-dimensional (3D) crystals are concerned, are briefly mentioned as they present severe challenges to most classification approaches when noise levels are moderate to high. The author's information theory based approaches utilize digital images and geometric Akaike Information Criteria. They perform well in the presence of pseudo-symmetries and turn out to be the only ones that allow for researcher independent (but generalized noise level dependent), classifications of the full range of crystallographic symmetries, i.e., Bravais lattice type, Laue class, and plane symmetry group, of noisy real-world images.