Content deleted Content added
Shantham11 (talk | contribs) ←Created page with 'The Nyquist–Shannon sampling theorem for sampling one-dimensional bandlimited functions can be generalized to a theorem on sampling wavenumber-limited ...' |
Shantham11 (talk | contribs) →Aliasing: Adding text on the illustration. |
||
Line 28:
===Aliasing===
{{main|Aliasing}}
The theorem gives conditions on sampling lattices for perfect reconstruction of the sampled. If the lattices are not fine enough to satisfy the Petersen Middleton condition, then the field cannot be reconstructed exactly from the samples in general. In this case we say that the samples may be [[Aliasing|aliased]].
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 on the right. The top 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 lower image 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 hence does not lead to aliasing.
[[File:Moire pattern of bricks.jpg|thumb|205px|Properly sampled image of brick wall.]]
[[File:Moire pattern of bricks small.jpg|thumb|205px|Spatial aliasing in the form of a [[Moiré pattern]].]]
===Optimal sampling lattices===
One of the objects of interest in designing a sampling scheme for bandlimited 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>.
| |||