Reconstruction filter

In the signal processing, a reconstruction filter is a filter that determines, as in the transformation of a continuous signal into a discrete individual samples are to be interpolated.

At uniformly sampled above the Nyquist frequency of the sinc function, the theoretically ideal reconstruction filter as its Fourier transform is a square-wave function ( ideal low-pass filter ), which is completely isolated in the frequency range of the useful signal. The original signal can thus be fully restored.

In the electronics, a reconstruction filter is applied to the output of a digital to analog converter.

Reconstruction filters in computer graphics

In computer graphics, no distinction is made between reconstruction filters and antialiasing filters. Instead of producing a continuous signal, the color of a pixel is calculated, for example, scaling of images, or in order to apply anti- aliasing determined in the vicinity of the pixel color values. The reconstruction-filter is used is a two-dimensional function, or distribution, which is centered on the pixel to be calculated. Since in computer graphics, the scanned image description contains almost always higher frequencies due to the edges of objects than can be detected by scanning the low-pass filter ( sinc filter) is not ideal.

Construction of two-dimensional filter

When comparing different reconstruction filter, the one-dimensional filter can first be considered. There are two ways in which a two-dimensional can be formed from a one-dimensional reconstruction filter, namely, by means of radial symmetry, and by separation.

In the design by radial symmetry a two-dimensional reconstruction filter from the surface of revolution of a one-dimensional filter is generated. The filter value only depends on the distance from the center of the filter. To apply radially symmetrical reconstruction filter, therefore the Euclidean distance of the sample has to be calculated.

In the design by two-dimensional separation of a reconstruction filter is produced by the one-dimensional filter displaced over the X and Y -axis and the product of two functions thus generated will be formed. The generated separable filters are well suited for grid-like arranged samples. In this case, the calculation of the filter value may be replaced by a series of interpolations with the corresponding one-dimensional reconstruction filter. Here, the value of the one-dimensional filter is first charged to the x- coordinate of the center of the filter in an intermediate step, for each of the overlapping of the filter sampling. The value is then calculated at the filter center of the so- generated vertical points.

Separable filters lead to anisotropic effects: image artifacts caused by separable filters, are not isotropic (all directions uniformly ) distributed, but preferred ( ie horizontally and vertically ) aligned along the filter design Saxony.

Since separable filters, only one sequence must be performed by one-dimensional interpolations and not Euclidean distances are calculated, they are faster to compute than radially symmetric filter.

The Gaussian filter is the only radially symmetrical reconstruction filter which is also separable. For all other filters leads the separable and radially symmetric produce different results.

Known filter

The following table lists the reconstruction filter, which are described, or commonly used in computer graphics.

Uneven sampling

As image descriptions may have an unlimited frequency range in computer graphics, usually a non-uniform sampling is preferred, so that aliasing can be replaced by noise. For non-uniform sampling method are also applied on a signal with unlimited frequency spectrum, there is no ideal reconstruction method. Recent research in the signal processing is believed that in such cases a perfect reconstruction, in practice, not possible. Instead, attempts to minimize the difference between the original signal and the reconstructed signal, and regardless of whether the original signal has an unlimited frequency range or not. These theoretical findings are, however, been little used.

Artifacts in the reconstruction

Reconstruction filter, in addition to post -aliasing lead to a number of other artifacts:

• Sampling frequency ripple arises when the same samples lead to a non-constant reconstructed signal.
• Anisotropic effects result when the reconstruction filter is not radially symmetrical.
• Ringing ( Gibbs phenomenon ) refers to over-or undershoots of hard edges.
• Blur.