On classification approaches for crystallographic symmetries of noisy 2D periodic patterns

Abstract

The classifications approaches for the crystallographic symmetries of patterns that are more or less periodic in two dimensions are critically reviewed and their relative performance qualitatively evaluated. The information theory based approach of the author utilizes digital images and turns out to be the only one that allows for fully objective classifications of the crystallographic symmetries, i.e. Bravais lattice type, Laue class, and plane symmetry group, of noisy real-world images. His information theory based crystallographic symmetry classifications utilize geometric bias-corrected sums of squared residuals, i.e. pertinent first order information, and enable the most meaningful crystallographic averaging in the spatial frequency domain, which suppresses generalized noise much more effectively than traditional Fourier filtering. Taking account of the fact that it is fundamentally unsound to assign an abstract mathematical concept such as a single symmetry type, class, or group with 100 % certainty to a more or less 2D periodic record of a noisy real-world imaging experiment that involved a real-world sample, the information theory based approach to crystallographic symmetry classifications delivers probabilistic classifications. Recent applications of deep convolutional neural networks to classifications of crystallographic translation symmetries in 2D and crystals in three dimensions are discussed as these machines deliver probabilistic classifications by non-analytical means.