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Crystallographic symmetry classifications from real-world images withperiodicities in two dimensions (2D) are of interest to crystallographers andpractitioners of computer vision studies alike. Currently, theseclassifications are typically made by both communities in a subjective mannerthat relies on arbitrary thresholds for judgments, and are reported under thepretense of being definitive, which is impossible. Moreover, the computervision community tends to use direct space methods to make such classificationsinstead of more powerful and computationally efficient Fourier space methods.This is because the proper functioning of those methods requires more periodicrepeats of a unit cell motif than are commonly present in images analyzed bythe computer vision community. We demonstrate a novel approach to planesymmetry group classifications that is enabled by Kenichi Kanatani's GeometricAkaike Information Criterion and associated Geometric Akaike weights. Ourapproach leverages the advantages of working in Fourier space, is well suitedfor handling the hierarchic nature of crystallographic symmetries, and yieldsprobabilistic results that are generalized noise level dependent. The latterfeature means crystallographic symmetry classifications can be updated whenless noisy image data and more accurate processing algorithms become available.We demonstrate the ability of our approach to objectively estimate the planesymmetry and pseudosymmetries of sets of synthetic 2D-periodic images withvarying amounts of red-green-blue and spread noise. Additionally, we suggest asimple solution to the problem of too few periodic repeats in an input imagefor practical application of Fourier space methods. In doing so, we effectivelysolve the decades-old and heretofore intractable problem from computer visionof symmetry detection and classification from images in the presence of noise.

Objective, Probabilistic, and Generalized Noise Level Dependent  Classifications of sets of more or less 2D Periodic Images into Plane  Symmetry Groups