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. Crystallographic image processing of high-resolution TEM images The principal steps for solving a structure of an inorganic crystal from HREM images by CIP are as follows (for a detailed discussion see ). Selecting the area of interest and calculation of the Fourier transform (= power spectrum consisting of a 2D periodic array of complex numbers) Determining the defocus value and compensating for the contrast changes imposed by the objective lens (done in Fourier space) Indexing and refining the lattice (done in Fourier space) Extracting amplitudes and phase values at the refined lattice positions (done in Fourier space) Determining the origin of the projected unit cell and determining the projected (plane group) symmetry Imposing constrains of the most likely plane group symmetry on the amplitudes an phases. At this step the image phases are converted into the phases of the structure factors. Calculating the pseudo-potential map by Fourier synthesis with corrected (structure factor) amplitudes and phases (done in real space) Determining 2D (projected) atomic co-ordinates (done in real space) A few computer programs are available which assist to perform the necessary steps of processing. The most popular programs used by materials scientists (electron crystallographers) are CRISP, VEC, and the EDM package. There is also the recently developed crystallographic image processing program EMIA, but so far there do not seem to be reports by users of this program. Structural biologists achieve resolutions of a few ångströms (up from a to few nanometers in the past when samples used to be negatively stained) for membrane forming proteins in regular two-dimensional arrays, but prefer the usage of the programs 2dx, EMAN2, and IPLT. These programs are based on the Medical Research Council (MRC) image processing programs and possess additional functionality such as the "unbending" of the image. As the name suggests, unbending of the image is conceptually equivalent to "flattening out and relaxing to equilibrium positions" one building block thick samples so that all 2D periodic motifs are as