  During the last decade, automatic data analysis methods concerning different
aspects of crystal analysis have been developed, e.g., unsupervised primitive
unit cell extraction and automated crystal distortion and defects detection.
However, an automatic, unsupervised motif extraction method is still not widely
available yet. Here, we propose and demonstrate a novel method for the
automatic motif extraction in real space from crystalline images based on a
variational approach involving the unit cell projection operator. Due to the
non-convex nature of the resulting minimization problem, a multi-stage
algorithm is used. First, we determine the primitive unit cell in form of two
lattice vectors. Second, a motif image is estimated using the unit cell
information. Finally, the motif is determined in terms of atom positions inside
the unit cell. The method was tested on various synthetic and experimental
HAADF STEM images. The results are a representation of the motif in form of an
image, atomic positions, primitive unit cell vectors, and a denoised and a
modeled reconstruction of the input image. The method was applied to extract
the primitive cells of complex $\mu$-phase structures
Nb$_\text{6.4}$Co$_\text{6.6}$ and Nb$_\text{7}$Co$_\text{6}$, where subtle
differences between their interplanar spacings were determined.
