On Classification Approaches for Crystallographic Symmetries of Noisy 2D Periodic Patterns

Abstract

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. Pseudosymmetries 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.

Computational scanning tunneling microscope image database

“…They can also be used as an aid to automatic conventional crystallographic image processing of big datasets and to obtain information from noisy images. This work acts as a starting point for identifying the defects in experimental images by providing a collection of ideal STM images for comparison purposes 55 . Ideally one would use an information-theoretic approach, as opposed to deep learning, to enable space group determination with uncertainty quantification, as demonstrated by Moeck, to distinguish the specific subgroups of selected group 56 .…”

Identification of crystal symmetry from noisy diffraction patterns by a shape analysis and deep learning

“…As an important note, our classification task is different from the standard image classification task due to the hierarchical nature of crystal symmetries. Crystallography is inherently hierarchical and continuous for any real material system 44,45 . Under enough noise, for example, a slight tetragonal distortion to a cubic structure will be indistinguishable from a standard cubic structure.…”

Information theory based symmetry classifications of more or less 2D periodic patterns in Islamic building ornaments and Hans Hinterreiter's graphic art

“…He had to concede, however, more than 30 years ago that this "certainly brings a certain subjectivity into the process: we should not shun away from it because such a degree of abstraction from imperfections of certain degrees and kinds underlies the entire practice of natural sciences wherefrom the science of symmetry originated" [2]. Utilizing recently developed information theory-based approaches to crystallographic symmetry classifications in 2D [6,7], one can now replace that subjectivity with the objectivity that comes from calculations that involve all pixel intensity values of digital images of such patterns. These information theory-based approaches to crystallographic symmetry classifications utilize geometric Akaike Information Criteria (G-AICs), which are in essence first-order geometric bias corrected sums of squared residuals between the raw image data and symmetrized versions of this data.…”

“…The identification of the plane symmetry group that can with the highest likelihood, i.e. minimized expected Kullback-Leibler information loss [6,7], be assigned to a noisy digital 2D periodic image of that pattern by an information theory based method enables the most meaningful crystallographic averaging in the spatial frequency domain.…”