Multidimensional sampling: Difference between revisions

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==Reconstruction==

[[Image:Unaliased_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:

{{NumBlk|:|<math>\hat f_s(\xi)\ \stackrel{\mathrm{def}}{=} \sum_{y \in \Gamma} \hat f\left(\xi - y\right) = \sum_{x \in \Lambda} |\Lambda|f(x) \ e^{-i 2\pi \langle x, \xi \rangle},</math>|{{EquationRef|Eq.1}}}}

where <math>|\Lambda| </math> represents the volume of the [[parallelepiped]] formed by the vectors {''v''₁, ..., ''v''_''n''}. 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 wavenumber-limited spectrum <math>\hat f(.\cdot)</math> is supported on the set <math>\Omega</math> then the function <math>\hat f_s(.\cdot)</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 the reconstructed field matches the original field and can be expressed in terms of the samples as

{{NumBlk|:|<math>f(x) = \sum_{y \in \Lambda} |\Lambda| f(y) \check \chi_\Omega(y - x)</math>,|{{EquationRef|Eq.2}}}}

where <math>\check \chi_\Omega(.\cdot)</math> is the inverse Fourier transform of the [[Indicator function|characteristic function]] of the set <math>\Omega</math>. This interpolation formula is the higher-dimensional equivalent of the [[Whittaker–Shannon interpolation formula]].

As an example suppose that <math>\Omega</math> is a circular disc. Figure 3 illustrates the support of <math>\hat f_s(.\cdot)</math> when the conditions of the Petersen-Middleton theorem are met. We see that the spectral repetitions do not overlap and hence the original spectrum can be exactly recovered.

==Implications==