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Crystallographic symmetry classifications of noisy 2D periodic images in the presence of pseudo-symmetries of the Fedorov type

Crystallographic symmetry classifications (CSCs) are a crucial part of the crystallographic processing of noisy 2D periodic images from crystals and crystal surfaces. Based on these classifications, the images can subsequently be corrected for having been recorded under suboptimal imaging conditions. For images that have been recorded within the weak-phase-object approximation of transmission electron microscopy, these corrections allow for the extraction of good estimates of the phase angles of the projected structure factors [1]. Good estimates of these phase angles are crucial for the solving of crystal structures by means of electron crystallography [1]. In case of scanning probe microscopy, crystallographic image processing allows for the correction of the effects of a blunt scanning probe tip [2,3]. Severe problems in the CSC of more or less 2D periodic images may arise from the co-existence of a large amount of imaging noise and pseudosymmetries of the Fedorov type [4]. Pseudo-symmetries of that particular type are compatible with a Bravais lattice that does not necessarily need to be the prevailing Bravais lattice of the hypothetical noise-free version of the image. In noisy 2D periodic images, these kinds of pseudo-symmetries are, therefore, difficult to distinguish from the genuine symmetries of that hypothetical image so that mis-classifications may result.