Multidimensional sampling: Difference between revisions

Revision as of 14:46, 9 May 2022

Revision as of 17:46, 29 August 2022

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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}}}}