Mitchell–Netravali filters

The Mitchell–Netravali filters or BC-splines are a group of reconstruction filters used primarily in computer graphics, which can be used, for example, for anti-aliasing or for scaling raster graphics. They are also known as bicubic filters in image editing programs because they are bi-dimensional cubic splines.

Definition

Graph of the Mitchell–Netravali filter with parameters B = C = 1/3

The Mitchell–Netravali filters were designed as part of an investigation into artifacts from reconstruction filters. The filters are piece-wise cubic filters with four-pixel wide supports. After excluding unsuitable filters from this family, such as discontinuous curves, two parameters $B$ and $C$ remain, through which the Mitchell–Netravali filters can be configured. The filters are defined as follows:

$k(x)=\frac{1}{6}\{\begin{array}{cc}\begin{array}{c}(12-9B-6C)|x{|}^3+(-18+12B+6C)|x{|}^2\\ +(6-2B)\end{array}& \text{, if }|x|<1\\ \begin{array}{c}(-B-6C)|x{|}^3+(6B+30C)|x{|}^2\\ +(-12B-48C)|x|+(8B+24C)\end{array}& \text{, if }1\le |x|<2\\ 0& \text{otherwise}\end{array}$

It is possible to construct two-dimensional versions of the Mitchell–Netravali filters by separation. In this case the filters can be replaced by a series of interpolations with the one-dimensional filter. From the color values of the four neighboring pixels $P_0$, $P_1$, $P_2$, $P_3$ the color value is then calculated $P(d)$ as follows:

$\begin{array}{cc}P(d)& =((-\frac{1}{6}B-C)P_0+(-\frac{3}{2}B-C+2)P_1+(\frac{3}{2}B+C-2)P_2+(\frac{1}{6}B+C)P_3)d^3\\ & +((\frac{1}{2}B+2C)P_0+(2B+C-3)P_1+(-\frac{5}{2}B-2C+3)P_2-C{P}_3)d^2\\ & +((-\frac{1}{2}B-C)P_0+(\frac{1}{2}B+C)P_2)d\\ & +\frac{1}{6}B{P}_0+(-\frac{1}{3}B+1)P_1+\frac{1}{6}B{P}_2\\ \end{array}$

$P$ lies between $P_1$ and $P_2$; $d$ is the distance between $P_1$ and $P$.

Subjective effects

Various artifacts may result from certain choices of parameters B and C, as shown in the following illustration. The researchers recommended values from the family $B+2C=1$ (dashed line) and especially $B=C=\frac{1}{3}$ as a satisfactory compromise.

Subjective appearance of images reconstructed with various Mitchell–Netravali filters.

Implementations

The following parameters result in well-known cubic splines used in common image editing programs:

B	C	Cubic spline	Common implementations
0	Any	Cardinal splines
0	0.5	Catmull-Rom spline	Bicubic filter in GIMP
0	0.75	Unnamed	Bicubic filter in Adobe Photoshop
1/3	1/3	Mitchell–Netravali	Mitchell filter in ImageMagick
1	0	B-spline	Bicubic filter in Paint.net

See also

• Ringing artifacts
• Anisotropic filtering
• Kernel (image processing)