Symmetric cubic filters

Symmetric cubic filters are piecewise cubic polynomials defined on the intervals [-2;-1], [-1;0], [0;1], and [1;2].Outside the interval [-2;2] these filters are defined to be zero. This definition implies that the number of involved sample points increases to four instead of two (linear interpolation). Mitchell and Netravali [39] derived a family of such cubic filters dependent on two variables (called BC-cubic splines) in the following way.

A symmetric cubic filter is in its general form given by

$$\mathrm{h}(x)=\left \{ \begin{array}{ll} P\vert x\vert^3+Q\v... ...eq \vert x\vert < 2$}\\ 0 & \textrm{else} \end{array} \right.$$ (4.7)

The eight parameters ($P \ldots W$) can be reduced to two by requiring the filter to be smooth, i.e., value and first derivative are continuous. Further, if all the samples are a constant value the reconstruction should be a flat signal and if the filter is applied to a sample point it should return this value, i.e., the filter should be one at zero and zero at all other integer positions. This results in the following family of cubic filters

Figure 4.5: Examples of BC-cubic splines on top and their frequency responses below.

Figure 4.6: Examples of some cardinal splines on top and their frequency responses below.

Definition 4 (BC-cubic splines)  

$$\mathrm{h_{BC}}(x)=\frac{1}{6}\left \{ \begin{array}{ll} (12... ...\vert x\vert+(8B+24C)\\ 0 & \textrm{else} \end{array} \right.$$ (4.8)

Some of the values for B and C correspond to well-known filters, e.g., B=1 and C=0 corresponds to the cubic B-spline, and C=0 results in the family of Duff's tensioned B-splines [12]. Some examples of BC-cubic splines can be seen in Fig. 4.5 and their corresponding frequency responses below.

Setting B=0 and C=-a results in the family of the cardinal splines (with $C=\frac{1}{2}$ or $a=-\frac{1}{2})$ being the Catmull-Rom spline) which were derived by Keys [25] in 1981. They are quite popular (probably because they depend on only one variable), therefore we define them separately.

Definition 5 (cardinal cubic splines)  

$$\mathrm{h_a}(x)=\left \{ \begin{array}{ll} (a+2)\vert x\vert... ...eq \vert x\vert < 2$}\\ 0 & \textrm{else} \end{array} \right.$$ (4.9)

As already noted, only for $a=-\frac{1}{2}$ the Taylor series expansion of the interpolating function agrees with the first three terms of the original function. A fourth-order convergence could be achieved by increasing the filter length to the interval [-3,3]. Some examples of cardinal splines can be seen in Fig. 4.6 on top and the corresponding frequency responses below.

The frequency response of the BC-cubic filter (and therefore of the cardinal cubic splines too) is analytically given by [39]

$$\begin{eqnarray*} \mathcal{F}(\mathrm{h_{BC}})(x) &=& \frac{3-3B}{(\pi \omega)^2... ...(2\omega) + 2\mathrm{sinc}(4\omega)] + B\mathrm{sinc^4}(\omega) \end{eqnarray*}$$

Compared to the frequency responses of nearest neighbor or linear interpolation the frequency responses of BC-cubic and cardinal splines approximate a box considerably closer but the filters are on the other hand computationally more expensive to evaluate.