text: Aaron Klug suggested in 1979 that a technique that was originally developed for structure determination of membrane protein structures can also be used for structure determination of inorganic crystals. This idea was picked up by the research group of Sven Hovmöller which proved that the metal framework partial structure of the KNbWO heavy-metal oxide could be determined from single HREM images recorded at Scherzer defocus. (Scherzer defocus ensures within the weak-phase object approximation a maximal contribution  to the image of elastically scattered electrons that were scattered just once while contributions of doubly elastically scattered electrons to the image are optimally suppressed.)
  In later years the methods became more sophisticated so that also non-Scherzer images could be processed. One of the most impressive applications at that time was the determination of the complete structure of the complex compound TiSe, which has been inaccessible by X-ray crystallography. Since CIP on single HREM images works only smoothly for layer-structures with at least one short (3 to 5 Å) crystal axis, the method was extended to work also with data from different crystal orientations (= atomic resolution electron tomography). This approach was used in 1990 to reconstruct the 3D structure of the mineral staurolite HFeAlSiO  and more recently to determine the structures of the huge quasicrystal approximant phase ν-AlCrFe  and the structures of the complex zeolites TNU-9  and IM-5. As mentioned below in the section on crystallographic processing of images that were recorded from 2D periodic arrays with other types of microscopes, the CIP techniques were taken up since 2009 by members of the scanning transmission electron microscopy, scanning probe microscopy and applied crystallography communities.
  Contemporary robotics and computer vision researchers also deal with the topic of "computational symmetry", but have so far failed to utilize the spatial distribution of site symmetries that result from crystallographic  origin conventions. In addition, a well known statistician noted in his comments on "Symmetry as a continuous feature"  that symmetry groups possess inclusion relations (are not disjoint in other words) so that conclusions about which symmetry is most likely present in an image need to be based on "geometric inferences". Such inferences are deeply rooted in information theory, where one is not trying to model empirical data, but extracts and models the information content of the data.
  The key difference between geometric inference and all kinds of traditional statistical inferences is that the former merely states the co-existence of a set of definitive (and exact geometrical) constraints and noise, whereby noise is nothing else but an unknown characteristic of the measurement device and data processing operations. From this follows that "in comparing two" (or more) "geometric models we must take into account the fact that the noise is identical (but unknown) and has the same characteristic for both" (all) "models". Because many of these approaches use linear approximations, the level of random noise needs to be low to moderate, or in other words, the measuring devices must be very well corrected for all kinds of known systematic errors. 
  These kinds of ideas have, however, only been taken up by a tiny minority of researchers within the computational symmetry  and scanning probe microscopy / applied crystallography  communities.
  It is fair to say that the members of computational symmetry community are doing crystallographic image processing under a different name and without utilization of its full mathematical framework (e.g. ignorance to the proper choice of the origin of a unit cell and preference for direct space analyses). Frequently, they are working with artificially created 2D periodic patterns, e.g. wallpapers, textiles, or building decoration in the Moorish/Arabic/Islamic tradition. The goals of these researchers are often related to the identification of point and translation symmetries by computational means and the subsequent classifications of patterns into groups. Since their patterns were artificially created, they do not need to obey all of the restrictions that nature typically imposes on long range periodic ordered arrays of atoms or molecules.
  Computational geometry takes a broader view on this issue and concluded already in 1991 that the problem of testing approximate point symmetries in noisy images is in general NP-hard and later on that it is also NP-complete. For restricted versions of this problem, there exist polynomial time algorithms that solve the corresponding optimization problems for a few point symmetries in 2D.