Objective crystallographic symmetry deviation quantifiers that properly account for crystallographic supergroup-subgroup relationships and pseudo-symmetries have recently been derived for noisy 2D periodic images on the basis of geometric Akaike Information Criteria (G-AICs) [2] and associated Akaike weights. (Akaike weights represent the probability that a certain crystallographic symmetry model within a disjoint or non-disjoint model set is the one that minimizes Kullback-Leibler information loss when it is used to represent full reality.) These quantifiers are demonstrated in this contribution on examples for the first time.

Openly accessible synthetic 2D periodic images (https://nmevenkamp.github.io/UnitCellExtraction/) have been utilized for our objective crystallographic classifications with respect to their Bravais lattice types, Laue classes, and plane symmetry groups. The example images possess per design both genuine pseudo-symmetries and added Gaussian noise, which turned genuine symmetries into pseudo-symmetries of the second kind. Note that genuine symmetries constitute the symmetry group structure of the hypothetical noise-free version of an image, but are unavoidably disturbed by noise in any real world imaging process.

Genuine pseudo-symmetries and pseudo-symmetries of the second kind also cause problems in mainstream single crystal X-ray crystallography [2] so that the approach of this contribution should be generalized to the 3D case in order to achieve a larger impact. A few percent, i.e. tens of thousands, of the molecule and crystal structures in the major 3D crystallography databases have been mis-classified with respect to their crystallographic symmetry [2]. This is due to the inherent subjectivity of the currently practiced approach where pure distance measures are utilized. When generalized to 3D, the above mentioned G-AIC approach [2] combined with Akaike weights will lead to superior noise-level dependent crystallographic symmetry classifications and subsequent re-assignments of mis-classified 3D structures to a range of symmetry types, classes, and groups where the probabilities of belonging to certain classifications is in each case quantified in an objective way.

[1] Zou, X. Hovmöller, S., Oleynikov, P., Electron Crystallography: Electron Microscopy and Electron Diffraction, IUCr Texts on Crystallography 16, Oxford University Press, 2011.

[2] Moeck, P. (2018) Symmetry, (special issue on Mathematical Crystallography, http://www.mdpi.com/journal/symmetry/special_issues/Mathematical_Crystallography), open access, submitted, earlier version at https://arxiv.org/abs/1801.01202, see also book chapter in: Méndez-Vilas, A. (ed.), Microscopy Book Series No. 7, 503–514 (2017), http://www.microscopy7.org/book/503-514.pdf.

[3] Hamilton, W.C. (1965) Acta Cryst. 18, 502–510.

Keywords: Geometric Akaike Information criteria, crystallographic symmetry classifications, genuine pseudo-symmetry