Implementation considerations

   
Choice of low-pass filters

In the previous sections we have ignored the problem of implementing a filter with an infinite length. In fact, in most cases it is not possible to implement an infinite filter unless it has a very special form. In particular, it is not possible to implement the ideal low-pass filter. All we can do is to make approximations to it, choosing finite filters with a similar frequency response. Fortunately, our eye is not that sensitive to small amounts of aliasing and we can get away with rather simple approximations for images. However, we should be careful when we choose a filter: some of them produce better results than others. For example, it turns out that the obvious choice-truncated sinc filter-generates excessive ringing. A number of filters produce adequate results. A classical example is the Lanczos filter

$$h(t) = \left\{ \begin{array}{ll} sinc(t) sinc(t/3)&\mbox{ if... ...t t\vert < 3$}\\ 0& \mbox{otherwise}\\ \end{array} \right.$$

Another example is a cubic filter proposed by D. Mitchell:

$$h(t) = \left\{ \begin{array}{ll} 7 \vert t\vert^3 - 12\vert ... ... t\vert < 2$} \\ 0 &\mbox{otherwise}\\ \end{array} \right.$$

The frequency response of the first filter is somewhat better, but it is more expensive to compute than the second one.

Note that when you use a filter, it is desirable to scale it in such a way that when you apply it to a constant, you get the same constant. This is achieved by a scaling factor equal to the reciprocal of the integral of the filter. In the case of Mitchell's filter the scaling factor should be 1/6.