Multidimensional sampling: Difference between revisions

Revision as of 12:14, 8 May 2012

Revision as of 12:16, 8 May 2012

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===Aliasing===

{{main|Aliasing}}

[[Image:Unaliased_sampled_spectrum_in_2D.png|thumb|Fig. 3: Support of the sampled spectrum <math>\hat f_s(.)</math> obtained by hexagonal sampling of a two-dimensional function bandlimitedwavenumber-limited to a circular disc. In this example the discs are disjoint and hence there is no aliasing.|right|300px]]

[[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 bandlimitedwavenumber-limited 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]].]]

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{{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 bandlimitedwavenumber-limited 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 in Figure 5. The image shows the effects of aliasing when the sampling theorem's condition is not satisfied. If the lattice of pixels is not fine enough for the scene, aliasing occurs as evidenced by the appearance of the [[Moiré pattern]] in the image obtained. The image in Figure 6 is obtained when a smoothened version of the scene is sampled with the same lattice. In this case the conditions of the theorem are satisfied and no aliasing occurs.

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===Optimal sampling lattices===

One of the objects of interest in designing a sampling scheme for bandlimitedwavenumber-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>. 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]]. As a consequence, hexagonal lattices are preferred for sampling [[Isotropy|isotropic fields]] in <math>\Re^2</math>.

==Applications==