Mitchell–Netravali filters explained

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

B and C remain, through which the Mitchell–Netravali filters can be configured. The filters are defined as follows:k(x) = (1)/(6) \begin{cases} \begin{array}{l} ( 12-9 B -6 C)|x| ^{3} + ( -18+12 B +6 C)|x| ^{2} \\ + ( 6-2 B) \end{array} & ,if |x|< 1 \\ \begin{array}{l} ( - B -6 C)|x| ^{3} + ( 6 B +30 C)|x| ^{2} \\ + ( -12 B -48 C)|x| + ( 8 B +24 C) \end{array} & ,if 1 \le |x|< 2 \\ 0 & 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

P _{0} , P _{1} , P _{2} , P _{3} the color value is then calculated P(d) as follows:\begin{align} P(d) & style = \left(( - (1)/(6) B - C)P _{0} + ( - (3)/(2) B - C +2 )P _{1} + ( (3)/(2) B + C -2 )P _{2} + ( (1)/(6) B + C)P _{3\right)} d ^{3} \\ & style + \left(( (1)/(2) B +2 C)P _{0} + ( 2 B + C -3 )P _{1} + ( - (5)/(2) B -2 C +3 )P _{2} - CP _{3\right)} d ^{2} \\ & style + \left(( - (1)/(2) B - C)P _{0} + ( (1)/(2) B + C)P _{2\right)} d \\ & style + (1)/(6) BP _{0} + ( - (1)/(3) B +1 )P _{1} + (1)/(6) BP _{2} \\ \end{align} 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 +2 C =1 (dashed line) and especially style B = C = (1)/(3) as a satisfactory compromise.^[1]

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		Bicubic filter in GIMP
0	0.75	Unnamed	Bicubic filter in Adobe Photoshop^[4]
1/3	1/3	Mitchell–Netravali	Mitchell filter in ImageMagick^[5]
1	0	B-spline	Bicubic filter in Paint.net

Examples

See also

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

Notes and References

1. Mitchell . Don . Netravali . Arun . Arun Netravali . Reconstruction Filters in Computer-Graphics . . . Proceedings of the 15th annual conference on computer graphics and interactive techniques (SIGGRAPH '88) . 22 . 4 . 221–228 . . . 10.1145/378456.378514 . 0897912756 . 0097-8930 . June 1998 . 25 October 2020. 10.1.1.582.7394 .
2. Book: Pharr . Matt . Jakob . Wenzel . Humphreys . Greg . Physically Based Rendering: From Theory to Implementation . Sampling and Reconstruction . http://www.pbr-book.org/3ed-2018/Sampling_and_Reconstruction.html . 3rd . 279–367 . 978-0-12-800645-0 . . . November 2016 . 25 October 2020.
3. Diploma thesis . Theußl . Thomas . Sampling and Reconstruction in Volume Visualization . The eighties: an image processing view . http://www.cg.tuwien.ac.at/~theussl/DA/node11.html . . 29 December 1999 . dead . https://web.archive.org/web/20140824074425/http://www.cg.tuwien.ac.at/~theussl/DA/node11.html . 24 August 2014.
4. Web site: Project . Summers . Jason . What is bicubic resampling? . September 2011 . Entropymine . 25 October 2020.
5. Thyssen . Anthony . Manual . Examples of ImageMagick Usage . Resampling Filters . https://legacy.imagemagick.org/Usage/filter/ . . 25 October 2020.