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==Preliminaries==
[[Image:Hexagonal_sampling_latticeHexagonal sampling lattice.png|thumb|Fig. 1: A hexagonal sampling lattice <math>\Lambda</math> and its basis vectors ''v''<sub>1</sub> and ''v''<sub>2</sub>|right|200px]]
[[Image:Reciprocal_latticeReciprocal lattice.png|thumb|Fig. 2: The reciprocal lattice <math>\Gamma</math> corresponding to the lattice <math>\Lambda</math> of Fig. 1 and its basis vectors ''u''<sub>1</sub> and ''u''<sub>2</sub> (figure not to scale).|right|200px]]
The concept of a [[Bandlimiting|bandlimited]] function in one dimension can be generalized to the notion of a wavenumber-limited function in higher dimensions. Recall that the [[Fourier transform]] of an integrable function <math>f(\cdot)</math> on ''n''-dimensional Euclidean space is defined as:
:<math>\hat{f}(\xi) = \mathcal{F}(f)(\xi) = \int_{\Re^n} f(x) e^{-2\pi i \langle x,\xi \rangle} \, dx</math>
==The theorem==
Let <math>\Lambda</math> denote a lattice in <math>\Re^n</math> and <math>\Gamma</math> the corresponding reciprocal lattice. The theorem of Petersen and Middleton<ref name="petmid62">< /ref> states that a function <math>f(\cdot)</math> that is wavenumber-limited to a set <math>\Omega \subset \Re^n</math> can be exactly reconstructed from its measurements on <math>\Lambda</math> provided that the set <math>\Omega</math> does not overlap with any of its shifted versions <math>\Omega + x </math> where the shift ''x'' is any nonzero element of the reciprocal lattice <math>\Gamma</math>. In other words, <math>f(\cdot)</math> can be exactly reconstructed from its measurements on <math>\Lambda</math> provided that <math>\Omega \cap \{x+y:y\in\Omega\} = \phi </math> for all <math>x \in \Gamma\setminus\{0\}</math>.
==Reconstruction==
[[Image:Unaliased_sampled_spectrum_in_2DUnaliased sampled spectrum in 2D.png|thumb|Fig. 3: Support of the sampled spectrum <math>\hat f_s(\cdot)</math> obtained by hexagonal sampling of a two-dimensional function wavenumber-limited to a circular disc. The blue circle represents the support <math>\Omega</math> of the original wavenumber-limited field, and the green circles represent the repetitions. In this example the spectral repetitions do not overlap and hence there is no aliasing. The original spectrum can be exactly recovered from the sampled spectrum.|right|300px]]
The generalization of the [[Poisson summation formula]] to higher dimensions <ref name="stewei71">E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces", Princeton University Press, Princeton, 1971.</ref> can be used to show that the samples, <math>\{f(x): x \in \Lambda\} </math>, of the function <math>f(\cdot)</math> on the lattice <math>\Lambda</math> are sufficient to create a [[periodic summation]] of the function <math>\hat f(\cdot)</math>. The result is:
===Aliasing===
{{main|Aliasing}}
[[Image:Aliased_sampled_spectrum_in_2DAliased sampled spectrum in 2D.png|thumb|Fig. 4: Support of the sampled spectrum <math>\hat f_s(\cdot)</math> obtained by hexagonal sampling of a two-dimensional function wavenumber-limited to a circular disc. In this example, the sampling lattice is not fine enough and hence the discs overlap in the sampled spectrum. Thus the spectrum within <math>\Omega</math> represented by the blue circle cannot be recovered exactly due to the overlap from the repetitions (shown in green), thus 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]].]]
===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_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>.
==Applications==
==References==
{{Reflist}}
{{DSP}}
[[Category:Digital signal processing]]
[[Category:Theorems in Fourier analysis]]
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