Box restruction filter
Box filter (or rectangular). The classic filter in CG. This filter performs consistently worst of all the filters. The support of this filter is 1.
Bartlett restruction filter
Bartlett filter (or tent/triangle). Convolve two boxes and get triangle (well, pyramid really). Using this filter corresponds to linear interpolation. The support is 2.
Cook restruction filter
Cook filter (truncated Gaussian). Convolve an infinite number of boxes and get the Gaussian. This filter is a Gaussian shifted downwards. The support is 3, as suggested by Cook et al. in their famous paper in stochastic sampling.
Max restruction filter
Max filter. A two-parameter filter (in all the images s=0.4810 t=1.3712). The support is 2t.
Mitchell-Netravali restruction filter
Mitchell-Netravali filter. Another class of two-parameter filters (in all the images b=1/3 c=1/3).
Sinc restruction filter
Sinc or sinus cardinal. The sinc is theoretically the perfect reconstruction filter, since its Fourier transform is a box centered around DC of width 1. It is given by:
It is however IIR, since any signal that has finite extent in the frequency domain must have infinite extent in the spatial domain (and vice versa). The sinc just keeps osciliating with ever decreasing amplitude, so we need to window it. The purpose of windowing is to attenuate the sinc to zero at the extremes of the resulting FIR filter.
There are many choices for windows. These include the rectangular (Fourier) window, the Bartlett window, the Hanning window, the Hamming window, the Blackman window, the Kaiser window, and the Lanzcos window.
Each provide a different trade-off between blurring, ringing and other reconstruction artifacts. The effect of altering the supprt is also shown somewhere on the page.
Fourier (rectangular) window
Using this window is equivalent to not windowing at all. As a result, the frequency response is particularly bad, since using the Fourier window corresponds to multiplying the sinc with a box in the spatial domain (or equivalently to convolving the sinc in the frequency domain (a box) with the box in the frequency domain (which is a sinc). The formula for the Fourier window is simply:
Bartlett window
The Bartlett (or triangular) window is (like like its reconstruction filter counterpart) given by:
This gives the following impulse and frequency response:
After applying the window, the sinc now looks like:
Hanning window
The Hanning window is given by:
This gives the following impulse and frequency response:
After applying the window, the sinc now looks like:
Hamming window
The Hamming window is given by:
This gives the following impulse and frequency response:
After applying the window, the sinc now looks like:
Blackman window
The Blackman window is given by:
This gives the following impulse and frequency response:
After applying the window, the sinc now looks like:
Kaiser window
The Kaiser window is given by:
This window has one parameters. In the following this parameter is 1, 2, 4, 8, and 16. This gives the following impulse and frequency responses:
After applying the window, the sinc now looks like:
