Objective crystallographic symmetry classifications of a noisy crystal pattern with strong Fedorov-type pseudosymmetries and its optimal image-quality enhancement

Abstract

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 Computer Program for Objective Point Symmetry Classifications of Electron Diffraction Spot Patterns with Apparent Hexagonal or Rectangular-Centered Lattice Metric

“…For the purpose of the classification/quantification, digital input data consist in [1][2][3][4]6,7] of the pixelwise sums of approximately Gaussian distributed noise and a strictly 2D periodic unknown underlying signal. Structural defects within the crystal or on the crystal surfaces, instrumental image recording noise, slight deviations from zero-crystal-tilt conditions in transmission electron microscopy, and small inaccuracies in the algorithmic processing of the digital data all contribute to a single generalized noise term.…”

“…Underestimation is, therefore, less harmful than overestimation as far as space group symmetry classifications are concerned. The geometric form of information theory [1][2][3][4] that was adapted here for the purpose of point symmetry classifications/quantifications of electron diffraction spot pattern is known to "err on the side of caution" in this respect. Over time and with better experimental data, the entries in the major crystallographic databases will become more accurate when objective crystallographic symmetry classifications [1][2][3][4] have been generalized to three dimensions and become mainstream [9].…”

“…Lukas von Koch 1*,2 and Peter Moeck 2* 1 Westside Christian High School, Portland, Oregon, USA, *lukasvonkoch@gmail.com 2 Department of Physics, Portland State University, Portland, Oregon, USA, *pmoeck@pdx.edu A Python program has been written to perform complementary calculations to the recently developed information-theoretic approach to crystallographic symmetry classifications/ quantifications [1][2][3][4] in two dimensions (2D) from digital atomic or molecular resolution transmission electron and scanning probe microscope images. This program is briefly described below.…”

“…Structural defects within the crystal or on the crystal surfaces, instrumental image recording noise, slight deviations from zero-crystal-tilt conditions in transmission electron microscopy, 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 [1,3] (or 2D Bravais lattice type [4] and 2D point symmetry group [6,7]) that are "anchored" to the least broken symmetries [1] are identified as genuine in the presence of generalized noise. More severely broken symmetries that are not anchored in this sense are identified as pseudo-symmetries.…”

“…(A table with the confidence levels [1,2] of higher symmetric symmetry classifications is the second output). The data columns of the first result table are as follows: sums of normalized squared residuals, geometric Akaike Information Criteria (G-AIC) values, likelihoods to be the Kullback-Leibler [8] (K-L) best geometric model of the experimental data, geometric Akaike weights [1][2][3], and traditional R sym values. The third, fourth, and fifth columns of these tables are of particular importance to our study (and our scientific progress with respect to reliance on R sym values for classifications of electron diffraction spot patterns into point symmetry groups).…”