potential is corrected for aberration and delocalisation and also not affected by possible transfer gaps since several images with different defocus are processed. CIP on the other side considers only one image and applies corrections on the averaged image amplitudes and phases. The result of the latter is a pseudo-potential map of one projected unit cell. The result can be further improved by crystal tilt compensation and search for the most likely projected symmetry. In conclusion one can say that the exit-wave function reconstruction method has most advantages for determining the (aperiodic) atomic structure of defects and small clusters and CIP is the method of choice if the periodic structure is in focus of the investigation or when defocus series of HREM images cannot be obtained, e.g. due to beam damage of the sample. However, a recent study on the catalyst related material Cs0.5[Nb2.5W2.5O14] shows the advantages when both methods are linked in one study. Brief history of crystallographic image processing 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 K8−xNb16−xW12+xO80 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 Ti11Se4, 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 HFe2Al9Si4O4 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