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.^[1][2][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 ${\displaystyle B}$ and ${\displaystyle C}$ remain, through which the Mitchell–Netravali filters can be configured. The filters are defined as follows:

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

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 ${\displaystyle P_0}$, ${\displaystyle P_1}$, ${\displaystyle P_2}$, ${\displaystyle P_3}$ the color value is then calculated ${\displaystyle P(d)}$ as follows:

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

${\displaystyle P}$ lies between ${\displaystyle P_1}$ and ${\displaystyle P_2}$; ${\displaystyle d}$ is the distance between ${\displaystyle P_1}$ and ${\displaystyle P}$.

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 ${\displaystyle B+2C=1}$ (dashed line) and especially ${\displaystyle B=C=\frac{1}{3}}$ as a satisfactory compromise.^[1][4]

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

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

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