Multidimensional sampling: Difference between revisions

Revision as of 04:14, 10 January 2013

Revision as of 04:16, 10 January 2013

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[[Image:Hexagonal_sampling_lattice.png|thumb|Fig. 1: A hexagonal sampling lattice <math>\Lambda</math> and its basis vectors ''v''₁ and ''v''₂|right|200px]]

[[Image:Reciprocal_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''₁ and ''u''₂ (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>ƒ(.\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>

where ''x'' and ''ξ'' are ''n''-dimensional [[vector (mathematics)|vectors]], and <math>\langle x,\xi \rangle</math> is the [[inner product]] of the vectors. The function ''ƒ<math>f(.\cdot)''</math> is said to be wavenumber-limited to a set <math>\Omega</math> if the Fourier transform satisfies <math>\hat{f}(\xi) = 0</math> for <math>\xi \notin \Omega</math>.

Similarly, the configuration of uniformly spaced sampling points in one-dimension can be generalized to a [[Lattice (group)|lattice]] in higher dimensions. A lattice is a collection of points <math>\Lambda \subset \Re^n</math> of the form

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==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==

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===Aliasing===

{{main|Aliasing}}

[[Image:Aliased_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]].]]