# Mitchell–Netravali filters

(Redirected from 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

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]

## 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

## References

1. ^ a b Mitchell, Don; Netravali, Arun (June 1998). "Reconstruction Filters in Computer-Graphics" (PDF). Written at Atlanta. Proceedings of the 15th annual conference on computer graphics and interactive techniques (SIGGRAPH '88). ACM SIGGRAPH. Vol. 22. New York City: Association for Computing Machinery. pp. 221–228. CiteSeerX 10.1.1.582.7394. doi:10.1145/378456.378514. ISBN 0897912756. ISSN 0097-8930. Retrieved 25 October 2020.
2. ^ Pharr, Matt; Jakob, Wenzel; Humphreys, Greg (November 2016). "Sampling and Reconstruction". Physically Based Rendering: From Theory to Implementation (3rd ed.). San Francisco: Morgan Kaufmann Publishers. pp. 279–367. ISBN 978-0-12-800645-0. Retrieved 25 October 2020.
3. ^ Theußl, Thomas (29 December 1999). "The eighties: an image processing view". Sampling and Reconstruction in Volume Visualization (Diploma thesis). TU Wien. Archived from the original on 24 August 2014.
4. ^ a b Thyssen, Anthony. "Resampling Filters". Examples of ImageMagick Usage (Manual). ImageMagick. Retrieved 25 October 2020.
5. ^ Summers, Jason (September 2011). "What is bicubic resampling?". Entropymine (Project). Retrieved 25 October 2020.