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A simple illustration of aliasing can be obtained by studying low-resolution images. A gray-scale image can be interpreted as a function in two-dimensional space. An example of aliasing is shown in the images of brick patterns in Figure 5. The image shows the effects of aliasing when the sampling theorem's condition is not satisfied. If the lattice of pixels is not fine enough for the scene, aliasing occurs as evidenced by the appearance of the [[Moiré pattern]] in the image obtained. The image in Figure 6 is obtained when a smoothened version of the scene is sampled with the same lattice. In this case the conditions of the theorem are satisfied and no aliasing occurs.
S. P. Efimov from [[Bauman Moscow State Technical University]] in 1978 y. found an approach to ease the restrictions for spectrum ___domain.<ref>{{cite journal |last1=Efimov |first1=Sergei |title=Reconstruction of a field with finite-spectrum by samples of signals of filters |journal=Problemy Peredaci Informacii |date=1978 |volume=14 |issue=2 |pages=53-60 |url=https://mi.mathnet.ru/eng/ppi/p1534}}</ref> He considered N identical sampling lattices to be shifted arbitrarily to each other. Optimal sampling is valid for spectrum ___domain that shifted versions of is close-packed N times on reciprocal lattice. Therefore, ring can be overlapped by a set of hexagons instead of one. [[JWST]] telescope array consists from 18 hexagons. Sampling on 18 shifted lattices is possible for 2-d Fourier transform of the array signal (i. e. for emitted signal).
===Optimal sampling lattices===
One of the objects of interest in designing a sampling scheme for wavenumber-limited fields is to identify the configuration of points that leads to the minimum sampling density, i.e., the density of sampling points per unit spatial volume in <math>\Re^n</math>. Typically the cost for taking and storing the measurements is proportional to the sampling density employed. Often in practice, the natural approach to sample two-dimensional fields is to sample it at points on a [[Lattice (group)|rectangular lattice]]. However, this is not always the ideal choice in terms of the sampling density. The theorem of Petersen and Middleton can be used to identify the optimal lattice for sampling fields that are wavenumber-limited to a given set <math>\Omega \subset \Re^d</math>. For example, it can be shown that the lattice in <math>\Re^2</math> with minimum spatial density of points that admits perfect reconstructions of fields wavenumber-limited to a circular disc in <math>\Re^2</math> is the hexagonal lattice.<ref name="mer79">D. R. Mersereau, “The processing of hexagonally sampled two-dimensional signals,” Proceedings of the IEEE, vol. 67, no. 6, pp. 930 – 949, June 1979.</ref> As a consequence, hexagonal lattices are preferred for sampling [[Isotropy|isotropic fields]] in <math>\Re^2</math>.
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