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

Peter Moeck|arXiv (Cornell University)|Apr 27, 2022

Advanced Electron Microscopy Techniques and ApplicationsBiochemistry, Genetics and Molecular Biology48 references11 citations

TL;DR

This paper presents an information-theoretic method for objectively classifying crystallographic symmetries in noisy 2D crystal patterns, distinguishing genuine symmetries from Fedorov-type pseudosymmetries without human supervision. By analyzing Fourier coefficients and applying statistical model selection, the method enables optimal image enhancement through symmetry-based averaging, significantly improving signal-to-noise ratio and resolution beyond visual or conventional software-based classification.

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), a.k.a. 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 in 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.

Motivation & Objective

• To develop an objective, automated method for classifying crystallographic symmetries in noisy 2D periodic patterns.
• To distinguish genuine symmetries from Fedorov-type pseudosymmetries without relying on subjective human interpretation.
• To eliminate the need for manually set thresholds in geometric model selection during symmetry classification.
• To enable optimal crystallographic image processing through symmetry-based averaging that enhances signal-to-noise ratio and resolution.
• To provide a statistically sound alternative to visual classification and existing crystallographic image processing software like CRISP.

Proposed method

• The method uses information theory to evaluate competing symmetry models based on the residual sum of squared errors in the Fourier domain.
• It applies a hierarchical model selection process over plane symmetry groups and projected Laue classes using the Akaike Information Criterion (AIC) or similar.
• The approach operates exclusively in Fourier space, filtering out non-structure-bearing coefficients to isolate periodic, symmetry-related features.
• It computes the likelihood of each symmetry model by comparing the residual error of the structure-bearing Fourier coefficients to the noise floor.
• The method selects the symmetry group that minimizes information loss, ensuring statistical robustness under approximately Gaussian noise.
• The final image enhancement is achieved by averaging over all asymmetric units in the selected symmetry group, leveraging the periodicity of the structure.

Experimental results

Research questions

• RQ1 Can information-theoretic methods reliably classify crystallographic symmetries in noisy 2D patterns where human experts and software like CRISP fail?
• RQ2 How does the proposed method outperform visual classification and CRISP in distinguishing genuine symmetries from Fedorov-type pseudosymmetries?
• RQ3 To what extent can symmetry-based averaging enhance signal-to-noise ratio and structural resolution in experimental images?
• RQ4 Is it possible to automate crystallographic symmetry classification without human supervision or subjective thresholding?
• RQ5 How does the performance of the method degrade under increasing noise levels compared to direct-space approaches?

Key findings

• The information-theoretic method correctly classified the true plane symmetry group (p4) for synthetic patterns with strong pseudosymmetries, while CRISP incorrectly assigned p4gm due to pseudosymmetry confusion.
• The method achieved significant signal-to-noise ratio enhancement through symmetry-based averaging, even in patterns with high noise levels.
• For the noise-free and high-noise patterns, the method's classification accuracy surpassed both human experts and CRISP, especially in resolving pseudosymmetry artifacts.
• The approach eliminated the need for visual inspection or manual thresholding in model selection, enabling fully automated processing.
• The method demonstrated superior statistical robustness compared to direct-space approaches, which become increasingly prone to misclassification as noise increases.
• The results suggest that information-theoretic symmetry classification should be extended to 3D crystallography to reduce subjectivity in X-ray and neutron diffraction analysis.