Accurate lattice parameters from 2D-periodic images for subsequent Bravais lattice type assignments

Abstract

Three different algorithms, as implemented in three different computer programs, were put to the task of extracting direct space lattice parameters from four sets of synthetic images that were per design more or less periodic in two dimensions (2D). One of the test images in each set was per design free of noise and, therefore, genuinely 2D periodic so that it adhered perfectly to the constraints of a Bravais lattice type, Laue class, and plane symmetry group. Gaussian noise with a mean of zero and standard deviations of 10 and 50% of the maximal pixel intensity was added to the individual pixels of the noise-free images individually to create two more images and thereby complete the sets. The added noise broke the strict translation and site/point symmetries of the noise-free images of the four test sets so that all symmetries that existed per design turned into pseudo-symmetries of the second kind. Moreover, motif and translation-based pseudo-symmetries of the first kind, a.k.a. genuine pseudo-symmetries, and a metric specialization were present per design in the majority of the noise-free test images already. With the extraction of the lattice parameters from the images of the synthetic test sets, we assessed the robustness of the algorithms’ performances in the presence of both Gaussian noise and pre-designed pseudo-symmetries. By applying three different computer programs to the same image sets, we also tested the reliability of the programs with respect to subsequent geometric inferences such as Bravais lattice type assignments. Partly due to per design existing pseudo-symmetries of the first kind, the lattice parameters that the utilized computer programs extracted in their default settings disagreed for some of the test images even in the absence of noise, i.e., in the absence of pseudo-symmetries of the second kind, for any reasonable error estimates. For the noisy images, the disagreement of the lattice parameter extraction results from the algorithms was typically more pronounced. Non-default settings and re-interpretations/re-calculations on the basis of program outputs allowed for a reduction (but not a complete elimination) of the differences in the geometric feature extraction results of the three tested algorithms. Our lattice parameter extraction results are, thus, an illustration of Kenichi Kanatani’s dictum that no extraction algorithm for geometric features from images leads to definitive results because they are all aiming at an intrinsically impossible task in all real-world applications (Kanatani in Syst Comput Jpn 35:1–9, 2004). Since 2D-Bravais lattice type assignments are the natural end result of lattice parameter extractions from more or less 2D-periodic images, there is also a section in this paper that describes the intertwined metric relations/holohedral plane and point group symmetry hierarchy of the five translation symmetry types of the Euclidean plane. Because there is no definitive lattice parameter extraction algorithm, the outputs of computer programs that implemented such al...

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

“…A pseudosymmetric unit cell that is in direct space three times smaller than the real unit cell of the images in Fig. 1 was the final result of this oversight [44]. Figure 2 reveals the prevalent genuine symmetries and pseudo-symmetries visually at the single unit cell level in direct space.…”

“…Note that it is for analytical approaches (other than the author's information theory based approach) typically more difficult to classify images such as the one on the right hand side of Fig. 3 with respect to its Bravais lattice type correctly due to the added noise (as demonstrated for other sets of synthetic images in [44]). The presence of noise does, on the other hand, not present challenges to the author's information theory based approach to crystallographic translation symmetry type classifications [18].…”

“…3 were extracted with the electron crystallography program CRISP [13,14] and are given in Table II. Within typical error bars of the extraction of these lattice parameters [44], the corresponding Bravais lattice could be of the oblique, rectangular primitive, rectangular centered (when the lattice parameters refer to one lattice point), or square type. In crystallographic standard terms [1], an ambiguity such as this is said to be due to a metric specialization [44,71].…”

“…There were incorrect usages of crystallographic standard terms for the 2D Bravais lattice types [1,3,44] as well as misleading statements about the nature of some of their samples [50], the existence of clearly discernable Moiré fringes in the amplitude map of the discrete Fourier transform of one of their images [16], and a figure which implied that classification probabilities by their DCNN could be negative or in excess of 100 % in the original version of the paper by Vasudevan and co-workers [46]. The crystallographic misnomers have been corrected to a large extent in the current (December 18, 2018) on-line version of their paper, i.e.…”

“…Deep convolutional neural networks were recently also applied to synthetic "images" that represent crystallographic symmetry information in 3D [47,48]. Just as in the case of the neural network for 2D Bravais lattice type classifications [46] (as discussed in the fifth section of this paper), pseudosymmetries present challenges to these networks in conjunction with generalized crystal structure data recording and processing noise [44], but were neglected in both of these studies. Because noisy images with G-AIC based labels were not part of the training sets of these neural networks that worked with crystallographic 3D data [47,48] (as judged from the available information in these papers), the classification performance of these networks must also be limited.…”