sinc and square sinc

Sampling a continuous signal is not harmless for the signal's spectrum: it multiplies it by a comb function, making it periodic. If Shannon's theorem is not respected, it is even worse since the signal's central spectrum is replicated over itself, creating aliasing artifacts.

If we assume a correctly sampled signal, then the ideal filter in Fourier space is a Rect function, which produces a sinc interpolation kernel in spatial domain.

$$\mbox{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$$

This kernel is far from being ideal in spatial domain, since it has infinite ringing converging towards 0 as 1/x, which is too slow for small kernels such as a $4\times4$ pixel neighbourhood. A square sinc may be a better idea, converging as 1/x², and being always positive. Figure 7 shows both sinc and square sinc interpolation kernels.

  

Figure 7: sinc and square sinc kernels