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[[Image:Aliased_sampled_spectrum_in_2D.png|thumb|Fig. 4: Support of the sampled spectrum <math>\hat f_s(.)</math> obtained by hexagonal sampling of a two-dimensional function bandlimited to a circular disc. In this example, the sampling lattice is not fine enough and hence the discs overlap in the sampled spectrum leading to aliasing.|right|300px]]
[[File:Moire pattern of bricks small.jpg|thumb|205px|Fig. 5: Spatial aliasing in the form of a [[Moiré pattern]].]]
[[File:Moire pattern of bricks.jpg|thumb|205px|Fig. 6: Properly sampled image of brick wall.]]
The theorem gives conditions on sampling lattices for perfect reconstruction of the sampled. If the lattices are not fine enough to satisfy the Petersen-Middleton condition, then the field cannot be reconstructed exactly from the samples in general. In this case we say that the samples may be [[Aliasing|aliased]].
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where <math>|\Lambda| </math> represents the volume of the [[parallelepiped]] formed by the vectors {''v''<sub>1</sub>, ..., ''v''<sub>''n''</sub>}. This periodic function is often referred to as the sampled spectrum and can be interpreted as the analogue of the [[discrete-time Fourier transform]] (DTFT) in higher dimensions. If the original bandlimited spectrum <math>\hat f(.)</math> is supported on the set <math>\Omega</math> then the function <math>\hat f_s(.)</math> is supported on periodic repetitions of <math>\Omega</math> shifted by points on the reciprocal lattice <math>\Gamma</math>. If the conditions of the Petersen-Middleton theorem are met, then the function <math>\hat f_s(\xi)</math> is equal to <math>\hat f(\xi)</math> for all <math>\xi \in \Omega</math>, and hence the original field can be exactly reconstructed from the samples. In this case there is no aliasing in the reconstruction. As an example suppose that <math>\Omega</math> is a circular disc. Figure 3 illustrates the support of <math>\hat f_s(.)</math> when the conditions of the Petersen-Middleton theorem are met. We see that the spectral repetitions do not overlap. Figure 4 shows the scenario where the conditions are not met. In this case the spectral repetitions overlap leading to aliasing in the reconstruction.
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
===Optimal sampling lattices===
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