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]

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

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

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:

${\displaystyle {\begin{aligned}P(d)&\textstyle =\left((-{\frac {1}{6}}B-C)P_{0}+(-{\frac {3}{2}}B-C+2)P_{1}+({\frac {3}{2}}B+C-2)P_{2}+({\frac {1}{6}}B+C)P_{3}\right)d^{3}\\&\textstyle +\left(({\frac {1}{2}}B+2C)P_{0}+(2B+C-3)P_{1}+(-{\frac {5}{2}}B-2C+3)P_{2}-CP_{3}\right)d^{2}\\&\textstyle +\left((-{\frac {1}{2}}B-C)P_{0}+({\frac {1}{2}}B+C)P_{2}\right)d\\&\textstyle +{\frac {1}{6}}BP_{0}+(-{\frac {1}{3}}B+1)P_{1}+{\frac {1}{6}}BP_{2}\\\end{aligned}}}$

${\displaystyle P}$ lies between ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$; ${\displaystyle d}$ is the distance between ${\displaystyle P_{1}}$ and ${\displaystyle P}$.

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

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[5]
1/3	1/3	Mitchell–Netravali	Mitchell filter in ImageMagick[4]
1	0	B-spline	Bicubic filter in Paint.net

Examples

• 
  Magnification with a bicubic filter in GIMP (Catmull-Rom)
• 
  Magnification with a bicubic filter in Paint.NET (B-spline)

See also

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