MS29-P01 Towards objective crystallographic symmetry classifications of noisy 2D periodic images Peter Moeck (Nano-Crystallography Group, Department of Physics, Portland State University, Portland, Oregon , United States of America) Andrew Dempsey (Nano-Crystallography Group, Department of Physics, Portland State University, Portland, Oregon , United States of America)email: pmoeck@pdx.eduCrystallographic symmetry classifications of noisy 2D periodic images are currently made on the basis of the three traditional plane symmetry deviation quantifiers of electron crystallography [1]. These quantifiers are, however, “pure distance measures” that are unable to deal with crystallographic supergroup-subgroup relationships and pseudo-symmetries in an objective manner [2]. A consequence of this is that the model with the lowest symmetry, i.e. the one which possesses the highest number of free parameters, fits noisy experimental data best. A version of Hamilton’s well known R-factor ratio test [3] of mainstream 3D crystallography can be applied in principle, but is of limited utility because it is a null hypothesis test.
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 ( 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,, open access, submitted, earlier version at, see also book chapter in: Méndez-Vilas, A. (ed.), Microscopy Book Series No. 7, 503–514 (2017),

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

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