Fig. 1

Structural states as scale-invariant features. a Simulated STEM images of bulk SrTiO₃ and SrTiO₃/BaTiO₃ interface with the electron beam propagating along the [100] crystallographic direction. Images are convoluted with a Gaussian function with FWHM of 0.7 Å to account for the finite source size of the electron beam. b Features extracted by the Laplacian of Gaussian detector are shown as an overlay of circles on the images in a. The intensity scale was inverted to improve the visibility. The size of the circle indicates the scale at which the feature was detected. For simplicity in the ensuing analyses, the contrast threshold of the LoG is tuned so that oxygen columns in the right image in b are not detected (see Additional file 1 for all atomic columns). c Close-up of the left image in b indicating both the keypoint, Kp, which describes the atom locally and the descriptor vectors, Ds, which encode the intensity distribution of neighboring columns to provide a nonlocal description of the column. Descriptors for the different atomic columns are shown as 1-dimensional vectors, indicating that columns with the same intensity can have different descriptors due to the different angular configuration of their neighboring atoms. The structural state, in this case an atomic column, is then defined by the pair composed of (Kp, Ds). The implementations of the LoG detector in the Python scikit-image library [41] and SIFT in OpenCV [42] were used throughout