Mitchell-Netravali filter

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 antialiasing or for scaling raster graphics . They are also known as bicubic filters in image editing programs because they belong to the cubic splines .

definition

Graph of the Mitchell-Netravali filter with parameter selection (⅓, ⅓)

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 carriers . After excluding unsuitable filters from this family, such as discontinuous curves, there are two parameters and that can be used to configure the Mitchell-Netravali filters. The filters are defined as follows: ${\displaystyle B}$ ${\displaystyle C}$

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

It is possible to construct two-dimensional versions of the Mitchell-Netravali filters by separation (see: Reconstruction filters: Construction of two-dimensional filters ). 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 adjacent pixels , , , then calculates the color value as follows: ${\displaystyle P_{0}}$ ${\displaystyle P_{1}}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{3}}$ ${\displaystyle P(d)}$

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

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

Special cases

Subjective appearance of images reconstructed with Mitchell-Netravali filters. The dashed line marks a family of parameters recommended by the developers as a good compromise.

Depending on the choice of parameters B and C , various artifacts can occur (see picture on the right). The developers suggested the parameters from the family and especially as a good compromise. ${\displaystyle B+2C=1}$ ${\displaystyle \textstyle B=C={\frac {1}{3}}}$

The choice of certain parameters leads to well-known cubic splines:

• B = 1, C = 0 is the cubic B-spline (used as a bicubic filter e.g. in Paint.NET );
• B = 0 is the family of cardinal splines ;
• B = 0, C = 0.5 is the Catmull-Rom spline (used as a bicubic filter e.g. in GIMP ).

Magnification with bicubic filter in GIMP ( B = 0, C = 0.5)

Magnification with bicubic filter in Paint.NET ( B = 1, C = 0)

literature

• Don Mitchell, Arun Netravali : Reconstruction Filters in Computer Graphics. ACM SIGGRAPH Computer Graphics 22, 4 (Aug. 1988): 221-228, ISSN  0097-8930
• Matt Pharr, Greg Humphreys: Physically Based Rendering. From Theory to Implementation, pp. 279-367. Morgan Kaufmann, London 2004, ISBN 0-12-553180-X ( PDF, 7 MB )

Web links

• The eighties: an image processing view. ( Memento from August 24, 2014 in the Internet Archive ) - Thomas Teußl: Sampling and Reconstruction in Volume Visualization (diploma thesis).