Mitchell–Netravali filters

Image:Mitchell Filter.svg

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) = \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\le |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 $P\_0$, $P\_1$, $P\_2$, $P\_3$ the color value is then calculated $P(d)$ as follows:

:$\backslash begin\{align\}$

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{align}

$P$ lies between $P\_1$ and $P\_2$; $d$ is the distance between $P\_1$ and $P$.

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 $B+2C=1$ (dashed line) and especially $\backslash textstyle\; B=C=\backslash frac\{1\}\{3\}$ as a satisfactory compromise.

Image:Mitchell-Netravali artifacts.svg

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

{{Gallery

|width=512

|height=256

|align=left

|File:Resize Catmull-Rom.png|alt1=Back of statue facing a city building whose facade is Greek columns covered by a huge U.S. flag|Magnification with a bicubic filter in GIMP (Catmull-Rom)

|File:Resize Cubic B-Spline.png|alt2=Profile of stone face jutting out from a mountainside. Three workers clamber over it, each about the height of the face's upper lip.|Magnification with a bicubic filter in Paint.NET (B-spline)

}}

{{Clear}}

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

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Category:Digital signal processing