Digital darkfield tableaus

Like wavelet decompositions, power spectra "tiled with direct space maps" are versions of a lattice image intermediate between direct and reciprocal space [1]. They generally take about the same time to calculate as a power spectrum, and contain useful information of all sorts about "where periodicities lie". Questions of this sort will be of increasing importance e.g. in subångstrom resolution pictures. The information is useful in spite of the fact that members of the mathematical harmonic analysis community have long moved past the sharp edged Fourier apertures that electron optical darkfield requires [2]. By illustrating useful applications for sharp-edged tilings, we might thus inspire the math community to look at the much wider range of pattern recognition problems that today's high resolution electron microscopists face.