Towards Generalized Noise-Level Dependent Crystallographic Symmetry Classifications of More or Less Periodic Crystal Patterns

Abstract

Geometric Akaike Information Criteria (G-AICs) for generalized noise-level dependent crystallographic symmetry classifications of two-dimensional (2D) images that are more or less periodic in either two or one dimensions as well as Akaike weights for multi-model inferences and predictions are reviewed. Such novel classifications do not refer to a single crystallographic symmetry class exclusively in a qualitative and definitive way. Instead, they are quantitative, spread over a range of crystallographic symmetry classes, and provide opportunities for inferences from all classes (within the range) simultaneously. The novel classifications are based on information theory and depend only on information that has been extracted from the images themselves by means of maximal likelihood approaches so that these classifications are objective. This is in stark contrast to the common practice whereby arbitrarily set thresholds are employed to force crystallographic symmetry classifications into apparently definitive/exclusive states, while the geometric feature extraction results on which they depend are never definitive in the presence of generalized noise, i.e. in all real world applications. Thus, there is unnecessary subjectivity in the currently practiced ways of making crystallographic symmetry classifications, which can be overcome by the approach outlined in this review.

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

“…In addition, fully objective, i.e. completely researcher independent approaches the crystallographic symmetry classifications become available when geometric Akaike Information Criteria are utilized (as in the author's more recent work on this subject [16][17][18][19][20][21][22][23]).…”

“…Better "symmetrizations" of noisy 2D periodic images have always been the desired outcome of the standard crystallographic image processing method that is used by the 2D crystallography, transmission electron microscopy, and scanning probe microscopy communities [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and works in Fourier space. These symmetrizations combine the noiseremoval feature of traditional Fourier filtering with the crystallographic averaging over all asymmetric units in the whole image, have been utilized for over 50 years, and contributed to the Nobel Prize in Chemistry to Sir Aaron Klug 8 in the year 1982.…”

“…The author's information theory based approach to plane symmetry group classifications [17,21,22] enables advancements on the state of the art in the crystallographic image processing field because it allows for the identification of the most meaningful separation of structure and generalized noise at the level of the asymmetric units. This kind of identification is, in turn, needed for the most meaningful symmetrization of a noisy 2D periodic image.…”

“…The derived mathematical framework is nowadays utilized for the classification of (i) highly magnified but necessarily noisy experimental images from crystals that are sufficiently thin (so that quasi-kinematic imaging conditions prevail) and crystal surfaces as recorded with modern transmission electron and scanning probe microscopes [7][8][9][10][11][12][13][14][15][16][17][18], of (ii) both perfectly 2D periodic and noisy synthetic images [18][19][20][21][22][23][24][25][26], as well as of (iii) more or less 2D periodic images from rugs, windows, a honeycomb, both a tiled floor and wall, metal gates, a metallic anti-skid surface [26], industrially manufactured wallpapers [27][28][29], textiles, and decorative ceramic tiles [29][30][31][32][33], medieval Islamic building ornaments [34], and images of (iv) many other objects and scenes [35].…”

“…Similarly obvious is the fact that members of the communities that traditionally perform direct space classifications may benefit from becoming informed about the opportunities that the Fourier/reciprocal space offers with respect to crystallographic symmetry classifications of more or less 2D periodic images [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Members of the 2D crystallography, transmission electron microscopy, and scanning probe microscopy communities are conversely bound to benefit from an analysis of what has been achieved so far in direct space classification of the crystallographic symmetries of more or less 2D periodic images [26][27][28][29][30][31][32][33][34][35].…”