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Crystallography, Symmetry groups, Mathematical crystallography, Information theory


The existing types of classification approaches for the crystallographic symmetries of patterns that are more or less periodic in two dimensions (2D) are reviewed. Their relative performance is evaluated in a qualitative manner. Pseudo-symmetries of different kinds are discussed 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 fully objective (completely researcher independent) and 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. His method’s identification of the plane symmetry group that can with the highest likelihood be assigned to a noisy 2D periodic image enables the most meaningful crystallographic averaging in the spatial frequency domain. This kind of averaging 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 author’s information theory based approaches to crystallographic symmetry classifications deliver probabilistic (rather than definitive) classifications. Recent applications of deep convolutional neural networks (DCNNs) to classifications of crystallographic translation symmetry types in 2D and crystals in three dimensions (3D) are discussed as these “correlation detection and optimization” machines deliver probabilistic classifications by other – non-analytical – means. The discussed DCNN classifications ignore the fact that many crystallographic symmetries are hierarchic, i.e. that the classification classes are often non-disjoint. They are currently also incapable of dealing with pseudo-symmetries. DCNNs for classifications of crystals in 3D are discussed separately in an appendix.


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