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

Crystallographic symmetry classifications from real-world images with periodicities in two dimensions (2D) are of interest to crystallographers and practitioners of computer vision studies alike. Currently, these classifications are typically made by both communities in a subjective manner that relies on arbitrary thresholds for judgments, and are reported under the pretense of being definitive, which is impossible. Moreover, the computer vision community tends to use direct space methods to make such classifications instead of more powerful and computationally efficient Fourier space methods. This is because the proper functioning of those methods requires more periodic repeats of a unit cell motif than are commonly present in images analyzed by the computer vision community. We demonstrate a novel approach to plane symmetry group classifications that is enabled by Kenichi Kanatani's Geometric Akaike Information Criterion and associated Geometric Akaike weights. Our approach leverages the advantages of working in Fourier space, is well suited for handling the hierarchic nature of crystallographic symmetries, and yields probabilistic results that are generalized noise level dependent. The latter feature means crystallographic symmetry classifications can be updated when less noisy image data and more accurate processing algorithms become available. We demonstrate the ability of our approach to objectively estimate the plane symmetry and pseudosymmetries of sets of synthetic 2D-periodic images with varying amounts of red-green-blue and spread noise. Additionally, we suggest a simple solution to the problem of too few periodic repeats in an input image for practical application of Fourier space methods. In doing so, we effectively solve the decades-old and heretofore intractable problem from computer vision of symmetry detection and classification from images in the presence of noise.