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Recent advances in scanning tunneling and transmission electron microscopies(STM and STEM) have allowed routine generation of large volumes of imaging datacontaining information on the structure and functionality of materials. Theexperimental data sets contain signatures of long-range phenomena such asphysical order parameter fields, polarization and strain gradients in STEM, orstanding electronic waves and carrier-mediated exchange interactions in STM,all superimposed onto scanning system distortions and gradual changes ofcontrast due to drift and/or mis-tilt effects. Correspondingly, while the humaneye can readily identify certain patterns in the images such as latticeperiodicities, repeating structural elements, or microstructures, theirautomatic extraction and classification are highly non-trivial and universalpathways to accomplish such analyses are absent. We pose that the mostdistinctive elements of the patterns observed in STM and (S)TEM images aresimilarity and (almost-) periodicity, behaviors stemming directly from theparsimony of elementary atomic structures, superimposed on the gradual changesreflective of order parameter distributions. However, the discovery of theseelements via global Fourier methods is non-trivial due to variability and lackof ideal discrete translation symmetry. To address this problem, we developshift-invariant variational autoencoders (shift-VAE) that allow disentanglingcharacteristic repeating features in the images, their variations, and shiftsinevitable for random sampling of image space. Shift-VAEs balance theuncertainty in the position of the object of interest with the uncertainty inshape reconstruction. This approach is illustrated for model 1D data, andfurther extended to synthetic and experimental STM and STEM 2D data.

Finding simplicity: unsupervised discovery of features, patterns, and order parameters via shift-invariant variational autoencoders