Digital images from crystals, as projected from the third spatial dimension and recorded in atomic resolution with any kind of real-world microscope, feature necessarily broken symmetries of the translation-periodicity-restricted Euclidean plane. The symmetry breakings are due to both the imaging process and the real structure of the imaged crystal, with the former cause typically dominating. A posterior algorithmic reduction of the symmetry breaking in such images constitutes, thus, often a correction for many of the distortions that were introduced by the imaging processes. Numerically quantified restorations of such symmetries can, therefore, be used to demonstrate the efficacy of a newly implemented posterior correction method for atomic-resolution images from hexagonal crystals. Recently developed information theory based methods are here shown to be suitable for this purpose. Thirteen experimental atomic-resolution images from graphite and monolayer molybdenite (MoS2), as respectively obtained by scanning tunneling microscopy, atomic force microscopy in the torsional resonance mode, and aberration-corrected parallel illumination transmission electron microscopy served as test cases in our larger (in its totality so far unpublished) study, from which we quote here. The source code of the software that was used for our distortion corrections and the whole report on that study are freely available on GitHub.

The computer program "Histropy" is an interactive Python program for the quantification of selected features of two-dimensional (2D) images/patterns (in either JPG/JPEG, PNG, GIF, BMP, or baseline TIF/TIFF formats) using calculations based on the pixel intensities in this data, their histograms, and user-selected sections of those histograms. The histograms of these images display pixel-intensity values along the x-axis (of a 2D Cartesian plot), with the frequency of each intensity value within the image represented along the y-axis. The images need to be of 8-bit or 16-bit information depth and can be of arbitrary size. Histropy generates an image's histogram surrounded by a graphical user interface that allows one to select any range of image-pixel intensity levels, i.e. sections along the histograms' x-axis, using either the computer mouse or numerical text entries. The program subsequently calculates the (so-called Monkey Model) Shannon entropy and root-mean-square contrast for the selected section and displays them as part of what we call a "histogram-workspace-plot." To support the visual identification of small peaks in the histograms, the user can switch between a linear and log-base-10 display scale for the y-axis of the histograms. Pixel intensity data from different images can be overlaid onto the same histogram-workspace-plot for visual comparisons. The visual outputs of the program can be saved as histogram-workspace-plots in the PNG format for future usage. The source code of the program and a brief user manual are published in the supporting materials as well as on GitHub. Instead of taking only 2D images as inputs, the program's functionality could be extended by a few lines of code to other potential uses employing data tables with one or two dimensions in the CSV format.

Journal Article Detecting and Correcting Piezoelectric-tube Actuator Drift Induced Distortion in Atomic-Resolution Scanning Tunneling Microscope Images from Crystal Surfaces Get access Tyler Bortel, Tyler Bortel Nano-Crystallography Group, Department of Physics, Portland State University, OR, USA Search for other works by this author on: Oxford Academic Google Scholar Arthur P Baddorf, Arthur P Baddorf Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, TN, USA Search for other works by this author on: Oxford Academic Google Scholar Rama Vasudevan, Rama Vasudevan Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, TN, USA Search for other works by this author on: Oxford Academic Google Scholar Peter Moeck Peter Moeck Nano-Crystallography Group, Department of Physics, Portland State University, OR, USA Search for other works by this author on: Oxford Academic Google Scholar Microscopy and Microanalysis, Volume 29, Issue Supplement_1, 1 August 2023, Pages 148–149, https://doi.org/10.1093/micmic/ozad067.067 Published: 22 July 2023

The reader is informed about a method for the objective identification of the plane symmetry group of a "noisy" crystal pattern. Without giving numerical details, this information theory based method is applied to two beautiful pieces of graphic art. The plane symmetry group identifications distinguish between genuine symmetries and pseudosymmetries as a byproduct. Pieces of graphic/geometric artworks are ideal for the further refinement of the new method because they are macroscopic and their '"noise content" is chiefly due to the handiwork and employed creative procedures of individual artists. As different graphic techniques/procedures were employed in the creation of the classified crystal patterns, one may glean insights on how well a particular technique or procedure supports the realization of a crystallographic symmetry group in a graphic work of art.

Information-theoretic methods for objective classifications of the crystallographic symmetries in digital images and electron density maps that are more or less translation-periodic in two dimensions (2D) have recently been developed [1, 2].Classifications into 2D Bravais lattice types, projected Laue classes, plane symmetry groups, and associated sets of 2D site symmetries can all be made solely on the basis of the structural information in experimental data.The objectivity of these symmetry classifications is ensured by selecting the Kullback-Leibler-best geometric model for the desired aspect of the 2D periodic signal in noisy image/map data by means of geometric Akaike Information Criteria.The data to be classified is considered to consist of the pixel-wise sums of more or less Gaussian distributed noise and an unknown underlying signal that is strictly 2D periodic.Structural defects in a crystal, instrumental image/map recording noise, and small inaccuracies in the algorithmic processing of the image/map data all contribute to a single generalized noise term.The new methods were applied to freely obtainable [3] transmission electron microscope (TEM) images of the cyclic nucleotidemodulated potassium channel (MloK1) from the bacterium Mesorhizobium loti in both the closed and open conformation.Plane symmetry group p2gg and projected Laue class 2mm were obtained as objective crystallographic symmetry classifications of these TEM images for both conformations [1].These classification results assign site symmetry 2 to unit-cell positions (0,0), (½,½), (½,0), and (0,½) in the orthogonal projection of the quaternary membrane-protein complex shape in both conformations.The likelihood that the membrane protein is a dimer of homodimers is, therefore, higher than it being a homotetramer.There were, however, very strong four-fold-rotational and translational pseudo-symmetries in the classified TEM images [1] as also shown in Fig. 1a.That TEM image stems from an earlier study by other authors [4].Other authors [3][4][5] concluded subjectively that the MloK1 membrane protein features point symmetry 4. As the discussion in the caption of Fig. 1 demonstrates, there are visibly only pseudo-mirror lines in Fig. 1a.This fact confirms the results of the information-theoretic analysis [1].Also in support of the classifications in [1] (and above), it was speculated on the basis of the 3D X-ray crystallography structure of this protein that the postulated four-fold point symmetry of MloK1 macromolecules is broken whenever they are embedded in lipid bilayers [6].

Recently developed information-theory-based methods enable objective classifications and quantifications of the crystallographic [1-5] and non-crystallographic [6] symmetries in noisy experimental data that are deemed to be translation periodic or quasiperiodic in two dimensions (2D).These classifications are objective because they are based solely on the experimental data via the fulfillment or violation of numerical inequalities that are based on pair-wise ratios of geometric Akaike Information Criterion (G-AIC) values for non-disjoint geometric models of the data.Because the models are in minimal supergroup to maximal subgroup relationships with each other as far as their symmetries are concerned, an a priori estimate of the noise level in the experimental study is not required.Confidence level can be assigned for the selection of a geometric model that features the symmetry of a minimal supergroup over a non-disjoint model that features a maximal subgroup.G-AIC values are in essence geometric bias corrected sums of squared residuals of the difference between the experimental data and geometric models of that data.The type of microscope or diffraction apparatus that has been used for the recording of the experimental data is immaterial for the application of the new methods.Pseudo-symmetries [7] can reliably be distinguished from genuine symmetries [1], even in the presence of large amounts of generalized noise [2].Generalized noise includes all effects of (unavoidably) imperfect recordings of experimental data, all kinds of rounding effects and numerical approximations by any kinds of data processing algorithms, and all structural defects in crystalline and quasicrystalline real-world material samples.When there are many noise sources and the effects of none of these sources dominate, the resulting generalized noise is approximately Gaussian distributed (by generalizations of the central limit theorem of statistics [6]).Such an approximate distribution is the precondition for the application of G-AIC framework [8].After an information theoretic symmetry classification has been made, one obtains a good a posteriori estimate of the noise level of the experimental study as a byproduct [1].Conditional symmetry model probabilities, i.e. so called geometric Akaike weights, within model sets [2,3] can also be calculated on the basis of the G-AIC values of the individual geometric models of the experimental data.These weights represent the probability that a certain geometric model of the experimental data is the Kullback-Leibler best model in the selected model set.The mathematical feature that probabilities need to be multiplied when one wants to obtain joint probabilities aids the distinction between genuine symmetries and pseudo-symmetries in experimental data [3] on a quantitative basis.The prevailing common practice in materials science is, by stark contrast, to make crystallographic symmetry classifications on the basis of subjective judgments whenever an unknown crystalline or quasicrystalline sample is involved.Those classifications are bound to be misleading or false on occasions, especially when the images or diffraction patterns feature a comparatively large amount of generalized noise, metric specializations [7], and/or pseudo-symmetries.Note that the information-theoretic methods deliver only probabilistic crystallographic or non-crystallographic symmetry classifications as it is fundamentally unsound to assign abstract mathematical concepts such as a single 2D Bravais lattice type, a crystallographic or non-crystallographic projected Laue class, a point symmetry group, or a plane symmetry group with 100 % certainty to a real-world image or diffraction pattern from a crystal or quasicrystal.The new methods quantify deviations from symmetries, which can be interpreted as providing "error-bars" on symmetry measurements.Experimental atomic-resolution transmission electron microscope images (both in parallel illumination and the scanning probe mode) as well as scanning tunneling microscope images serve as examples for the demonstration of the image-based symmetry classification and quantification methods.Experimental selected-area electron diffraction spot patterns and precession electron diffraction patterns serve as examples for the demonstration of the diffraction-pattern based counterparts [4, 5] of these methods.

Journal Article Information-Theory Based Symmetry Classifications of Sets of S/TEM Zone-Axis Images in Support of Nanocrystallography and Discrete Electron Tomography Get access Peter Moeck Peter Moeck Nano-Crystallography Group, Department of Physics, Portland State University, Portland, OR, USA Corresponding author: pmoeck@pdx.edu Search for other works by this author on: Oxford Academic Google Scholar Microscopy and Microanalysis, Volume 29, Issue Supplement_1, 1 August 2023, Pages 598–599, https://doi.org/10.1093/micmic/ozad067.289 Published: 22 July 2023

It is well known that several percent of the crystal structures in all of the major crystallographic databases are either misleading or incorrect due to space group assignments that were not maximally supported by the recorded experimental data itself [1].Walter C. Hamilton's significance tests have enabled semi-objective space group assignments in the presence of symmetry inclusion relationships since the year 1965 [2].The popularity of such tests is a testament to the fact that the same experimental diffraction data support often refinements in different space groups similarly well.When there are pseudosymmetries in the experimental data, these space groups are typically in minimal supergroup and maximal subgroup relationships with each other.With different space group assignments to the raw data go, of course, differences in the subsequent averaging over the presumed symmetries in the experimental data during the refinement of the crystal structure.This leads, in turn, unavoidably to differently refined crystal structures, which cannot all be correct.Pseudosymmetries are not rare in nature [3] and can in noisy experimental data easily be mistaken for genuine symmetries.Whereas underestimation of the crystallographic symmetry in the experimental raw data means that one does not make the most out of the performed experiment, its overestimation leads to a "washing out" of structural information due to its averaging with noise.The underlying fundamental problem is that in the unavoidable presence of experimental noise, it is always the most general structural model, i.e. the one which is least constrained by symmetries, that fits the experimental data best.This implies that there is only translation symmetry in diffraction data as evidenced by discrete Bragg reflections that are laid out on a reciprocal lattice.Most crystal structures do, however, feature more than translation symmetry (as visibly evidenced by their macroscopic morphologies).Null-hypotheses tests, such as the ones used by Hamilton [2], proceed in a way that one may or may not be able to reject the nullhypothesis in favor of the alternative hypothesis with a confidence level that one is free to choose.Whereas the null-hypothesis and the alternatively hypothesis can for logical consistency not simultaneously be true (if the former is rejected), experimental crystallographic data often supports the assignment of a maximal supergroup and one or more of its minimal supergroups.There is then just more or less evidence in support of these individual assignments.Model selection by information theory is much more powerful than null-hypothesis tests because it involves direct quantifications of the evidence in favor of a multitude of individual competing hypotheses/models [4].A procedure for the quantification of crystallographic symmetries in experimental data needs to deal with the well-known crystallographic symmetry hierarchies objectively and implement a solution to the symmetry inclusion problem [1].A solution to the latter problem has been found by Kenichi Kanatani with his reformulation of Hirotugo Akaike's [5] Information Criterion for geometrically constrained information in noisy observational data (for computer vision applications) [6].The "distance" between experimental data that is to be classified with respect to hierarchical geometric constraints and a model for this data is in geometric Akaike Information Criteria (G-AICs) defined as the sum of a squared residual term and a term that is proportional to the square of the noise level in the study as modified by the degrees of freedom in the geometric model (that result from the geometric constraints).The model with the lower G-AIC value is preferred in the information-theoretic sense over the model with the larger G-AIC value for the same experimental data because it has more predictive power and provides a better overall fit when differences in the number of geometric constraints are properly taken into account [6].Analyzing ratios of G-AIC values for non-disjoint geometric models allows for symmetry classifications without an a priori estimate of the noise level in an experimental study [1].A good a posteriori estimate of that noise level can be obtained after the Kullback-Leibler-best geometric model for the desired aspects of the translation periodic signal in the data has been found [7].G-AIC-based methods have recently been applied to the classification of symmetries in noisy data that were translation periodic in two dimensions [7][8][9].These methods should be generalized to translation periodicity in three dimensions so that full objectivity can eventually be brought to the assignment of space groups to experimental data from crystals.Hamilton's venerably significance tests on crystallographic reliability values [2] could then be retired after having served the crystallographic community well for more than half a century.

Journal Article Precession Electron Diffraction for Electron Crystallography Get access Peter Moeck Peter Moeck Department of Physics, Portland State University, Portland, OR, USA Corresponding author: pmoeck@pdx.edu Search for other works by this author on: Oxford Academic Google Scholar Microscopy and Microanalysis, Volume 28, Issue S1, 1 August 2022, Pages 3206–3207, https://doi.org/10.1017/S1431927622011928 Published: 01 August 2022

The recently developed information-theoretic approach to crystallographic symmetry classifications and quantifications in two dimensions (2D) from digital transmission electron and scanning probe microscope images is adapted for the analysis of an experimental selected-area transmission electron diffraction spot pattern. The extracted lattice parameters of this crystal are within experimental error bars consistent with a metric tensor that suggests the presence of hexagonal translation symmetry. The point symmetry of the combined low, medium, and high resolution spots is, however, no higher than 2mm. The likelihood of this electron diffraction pattern belonging to a rectangular-centered crystal rather than a hexagonal crystal is quantified on the basis of its information-theoretic point group symmetry classifications. Presumably due to a slight misorientation away from the exact [001] zone axis combined with the curvature of the Ewald sphere and a real structure that includes intergrowth of quadruple NbO and triple BaNbO3 blocks of varying sizes and orientations, the group of highest resolution spots, i.e. d-spacings between 0.125 to 0.085 nm, feature point symmetry .m. only. The crystallographic Rsym values of traditional classifications into the point groups that are compatible with the experimentally obtained primitive lattice parameters are provided for comparison purposes. As it is common practice in diffraction based crystallography, point symmetry classification and quantification results for the group of highest resolution spots are provided separately from their counterparts for the combined low, medium, and high resolution spots.

Journal Article Distinguishing Between Quaternary Symmetries and Pseudo-symmetries in a Prokaryotic Potassium Channel in Both the Open and Closed Conformation Get access Peter Moeck Peter Moeck Department of Physics, Portland State University, Portland, Oregon, USA Corresponding author: pmoeck@pdx.edu Search for other works by this author on: Oxford Academic Google Scholar Microscopy and Microanalysis, Volume 28, Issue S1, 1 August 2022, Pages 1286–1287, https://doi.org/10.1017/S1431927622005281 Published: 01 August 2022

The recently developed information-theoretic approach to crystallographic symmetry classifications and quantifications in two dimensions (2D) from digital transmission electron and scanning probe microscope images is adapted for the analysis of an experimental electron diffraction spot pattern, for the first time. Digital input data are considered in this approach to consist of the pixel-wise sums of approximately Gaussian distributed noise and an unknown underlying signal that is strictly 2D periodic. Structural defects within the crystals or on the crystal surfaces, instrumental image recording noise, slight deviations from zero-crystal-tilt conditions in transmission electron microscopy, inhomogeneous staining in structural biology studies of intrinsic membrane protein complexes in lipid bilayers, and small inaccuracies in the algorithmic processing of the digital data all contribute to a single generalized noise term. The plane symmetry group and projected Laue class(or 2D Bravais lattice type) that is anchored to the least broken symmetries are identified as genuine in the presence of generalized noise. More severely broken symmetries that are not anchored to the least broken symmetries are identified as pseudo-symmetries. Our point symmetry quantification study of an electron diffraction spot pattern is highly topical because a new contrast mechanism for 4D scanning transmission electron microscopy was recently demonstrated by other authors. The usage of objective symmetry quantifications is bound to become the preeminent condition of the establishment of that contrast mode as an industry-wide standard.

Statistically sound crystallographic symmetry classifications are obtained with information-theory-based methods in the presence of approximately Gaussian distributed noise. A set of three synthetic patterns with strong Fedorov-type pseudosymmetries and varying amounts of noise serve as examples. Contrary to traditional crystallographic symmetry classifications with an image processing program such as CRISP, the classification process does not need to be supervised by a human being and is free of any subjectively set thresholds in the geometric model selection process. This enables crystallographic symmetry classification of digital images that are more or less periodic in two dimensions (2D), also known as crystal patterns, as recorded with sufficient structural resolution from a wide range of crystalline samples with different types of scanning probe and transmission electron microscopes. Correct symmetry classifications enable the optimal crystallographic processing of such images. That processing consists of the averaging over all asymmetric units in all unit cells in the selected image area and significantly enhances both the signal-to-noise ratio and the structural resolution of a microscopic study of a crystal. For sufficiently complex crystal patterns, the information-theoretic symmetry classification methods are more accurate than both visual classifications by human experts and the recommendations of one of the popular crystallographic image processing programs of electron crystallography.

A Python program for calculating the metrics necessary to perform information-theory based symmetry classifications and quantifications of transmission electron diffraction spot patterns is introduced. It is the first of its kind, in that it implements objectivity into crystallographic symmetry classifications and quantifications of approximate zone axis patterns from crystals. The equations by which the program operates as well as the required inputs are given. The results of the program's analysis of an experimental transmission electron diffraction spot pattern from a crystal with a pseudo-hexagonal lattice metric and a rectangular-centered Bravias lattice is used as an example. The program will eventually be appended to allow analysis of the other hierarchical translational pseudo-symmetry and Bravais lattice type combinations. Crystallographic Rsym values of traditional classifications into projected point symmetry groups are provided alongside information-theoretic results of the new program's analysis for comparison purposes.

A geometric form of information theory allows for reasonable, i.e. probabilistic, evidence-ranking based, and generalized noise-level dependent, classifications of the crystallographic and quasicrystallographic symmetries in noisy digital images. Such classifications are based solely on the image pixel intensity values, justifiable assumptions about the aggregate distribution of generalized noise in the images, asymptotic extrapolations to zero-noise images, and rational symmetry model selections with maximized predictive accuracy in the presence of both symmetry-inclusion relations and pseudo-symmetries. Preferring a well developed geometric form of information theory over a theoretically possible geometric-Bayesian approach for these classifications is the only subjective choice made. Using digital data planes and assuming approximately Gaussian distributed generalized noise, reasonable crystallographic and quasicrystallographic symmetry classifications can be made for noisy images from both scanning probe and transmission electron microscopes. A binary type classification of structurally very similar materials into either a quasicrystal or one of its rational/crystalline approximants based on the approximate point symmetries in their noisy digital images is proposed here for the first time.

Methods for objective classifications of more or less 2D periodic patterns into Bravais lattice types, Laue classes, and plane symmetry groups have recently been developed [1,2]. The objectivity of the crystallographic symmetry classifications is ensured by the statistically sound selection of the best geometric model for the 2D periodic signal in the image data on the basis of geometric Akaike Information Criteria [3]. As recently reviewed [4], the new crystallographic image classification methods are the only ones that can be considered to be objective, i.e. researcher and arbitrary thresholds independent. These techniques are analytic in nature rather than based on machine learning. This feature enables them to deal effectively with all types of pseudosymmetries and to obtain geometric Akaike weights, which represent the probabilities of the correctness of particular crystallographic symmetry classifications. (Machine learning systems have so far ignored pseudosymmetries in crystallographic image classification studies.)The digital input images of the more or less 2D periodic patterns are considered to consist of the pixel-wise sums of more or less Gaussian distributed noise and an unknown underlying signal that is strictly 2D periodic. Structural defects in the molecular 2D array, instrumental image recording noise, small systematic errors, and small inaccuracies in the algorithmic processing of the image data all contribute to a single generalized noise term. Because there are many different sources of noise and only small systematic errors that all contribute to the generalized noise term, the central limit theorem justifies the overarching assumption that the generalized noise is approximately Gaussian distributed.Experimental images from transmission electron microscopes that are digital and sufficiently well resolved at the molecular level serve as the input of the crystallographic symmetry classifications in electron-crystallography based structural biology.For good information extraction results and generalized noise suppression, one should record experimental images with a large field of view, containing several hundreds of more or less identical unit cells of the 2D periodic array. The outputs of the techniques are the most probable 2D periodic signal distribution from the underlying molecular array in addition to the most probably plane symmetry group, Laue class, and Bravais lattice type. The generalized noise level is quantified as a byproduct.The new methods are demonstrated on (nominal) zero-tilt transmission electron microscope images from tilt series of 2D crystals of a fragment of beef heart NADH:ubiquinone oxidoreductase and a prokaryotic cyclic nucleotide-modulated potassium channel.