查看“︁Mitchell–Netravali滤波器”︁的源代码

'''Mitchell–Netravali滤波器'''或'''BC–样条'''是一组主要用于[[计算机图形学]]的{{link-en|重建滤波器|Reconstruction filter}}，例如，可被用于[[抗锯齿]]或[[图像缩放|缩放]][[位图|光栅图形]]。它们在[[圖像軟件|图像编辑程序]]中也被称为[[双三次插值|双三次]][[滤波器]]，因为它们是[[二维空间|二维]][[样条插值|三次样条]]<ref name="mitchell-netravali">{{cite conference |last1=Mitchell |first1=Don |last2=Netravali |first2=Arun |author-link2=Arun Netravali |title=Reconstruction Filters in Computer-Graphics |url=https://www.cs.utexas.edu/~fussell/courses/cs384g-fall2013/lectures/mitchell/Mitchell.pdf |conference=[[ACM SIGGRAPH]] |location=[[Atlanta]] |book-title=Proceedings of the 15th annual conference on computer graphics and interactive techniques (SIGGRAPH '88) |volume=22 |issue=4 |pages=221–228 |publisher=[[Association for Computing Machinery]] |publication-place=[[New York City]] |doi=10.1145/378456.378514 |isbn=0897912756 |issn=0097-8930 |date=June 1998 |access-date=25 October 2020 |citeseerx=10.1.1.582.7394 |archive-date=2023-02-07 |archive-url=https://web.archive.org/web/20230207222335/https://www.cs.utexas.edu/~fussell/courses/cs384g-fall2013/lectures/mitchell/Mitchell.pdf |dead-url=no }}</ref><ref name="pbrt">{{cite book |last1=Pharr |first1=Matt |last2=Jakob |first2=Wenzel |last3=Humphreys |first3=Greg |title=Physically Based Rendering: From Theory to Implementation |url=http://www.pbr-book.org/ |chapter=Sampling and Reconstruction |chapter-url=http://www.pbr-book.org/3ed-2018/Sampling_and_Reconstruction.html |edition=3rd |pages=279–367 |isbn=978-0-12-800645-0 |publisher=[[Morgan Kaufmann Publishers]] |location=[[San Francisco]] |date=November 2016 |access-date=25 October 2020 |archive-date=2023-06-02 |archive-url=https://web.archive.org/web/20230602140119/https://www.pbr-book.org/ |dead-url=no }}</ref><ref name="theussl-thesis">{{cite thesis |type=Diploma thesis |last=Theußl |first=Thomas |title=Sampling and Reconstruction in Volume Visualization |url=http://www.cg.tuwien.ac.at/~theussl/DA/node11.html |chapter=The eighties: an image processing view |chapter-url=http://www.cg.tuwien.ac.at/~theussl/DA/node11.html |publisher=[[TU Wien]] |date=29 December 1999 |url-status=yes |archive-url=https://web.archive.org/web/20140824074425/http://www.cg.tuwien.ac.at/~theussl/DA/node11.html |archive-date=2014-08-24 |access-date=2023-04-08 }}</ref>。 

== 定义 ==
[[Image:Mitchell Filter.svg|thumb|Mitchell–Netravali滤波器的图，带有参数B = C = 1/3 。 ]]

Mitchell–Netravali滤波器是作为对{{link-en|重建滤波器|Reconstruction filter}}失真（artifacts）调查的一部分而被设计的。滤波器是具有四像素宽[[支撑集|支撑]]的分段[[三次函數|三次滤波器]]。从这个系列中排除不合适的滤波器后，例如[[连续函数|不连续]]的曲线，仍然存在两个参数B和C ，通过它们可以配置Mitchell–Netravali滤波器。滤波器定义如下：

:<math>
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{其他情况}
\end{cases}
</math>

通过{{link-en|分离滤波器|Separable filter|分离}}构造 Mitchell–Netravali滤波器的二维版本有可能的。在这种情况下，滤波器可以被一维滤波器的一系列插值代替。从四个相邻像素的颜色值<math>P_0</math>, <math>P_1</math>, <math>P_2</math>, <math>P_3</math>然后计算颜色值<math>P(d)</math>如下：

:<math>\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}</math>

<math>P</math>介于<math>P_1</math>和<math>P_2</math> ; <math>d</math>是<math>P_1</math>和<math>P</math>之间的距离。

== 主观效果 ==
如下图所示，参数B和C的某些选择可能会导致各种失真。研究人员推荐参数数值来自<math>B+2C=1</math> （虚线）家族和尤其是<math>\textstyle B=C=\frac{1}{3}</math>作为一个令人满意的妥协<ref name="mitchell-netravali"/><ref name="imagemagick6-resampling-filters"/>。

[[Image:Mitchell-Netravali artifacts.svg|thumb|none|300px|使用各种 Mitchell-Netravali 滤波器重建图像的主观外观。]]

== 实现 ==

以下参数导致常见[[圖像軟件|图像编辑程序]]中使用的众所周知的三次样条

{| class="wikitable"
!''B''
!''C''
!三次样条
!常见的实现
|-
|style="text-align:right;"|0
|style="text-align:center;"|''Any''
|[[Cubic_Hermite_spline#Cardinal_spline|Cardinal splines]]
|
|-
|style="text-align:right;"|0
|style="text-align:right;"|0.5
|[[Cubic_Hermite_spline#Catmull–Rom_spline|Catmull-Rom spline]] 
|[[GIMP]]中的双三次滤波器
|-
|style="text-align:right;"|0
|style="text-align:right;"|0.75
|''Unnamed''
|[[Adobe Photoshop]]中的双三次滤镜<ref name="entropymine-bicubic">{{cite web |type=Project |last=Summers |first=Jason |title=What is bicubic resampling? |url=https://entropymine.com/imageworsener/bicubic/ |date=September 2011 |website=Entropymine |access-date=25 October 2020 |archive-date=2023-04-08 |archive-url=https://web.archive.org/web/20230408213911/https://entropymine.com/imageworsener/bicubic/ |dead-url=no }}</ref>
|-
|style="text-align:right;"|1/3
|style="text-align:right;"|1/3
|Mitchell–Netravali
|[[ImageMagick]]中的 Mitchell滤波器<ref name="imagemagick6-resampling-filters">{{cite techreport |last=Thyssen |first=Anthony |type=Manual |title=Examples of ImageMagick Usage |url=https://legacy.imagemagick.org/Usage/ |chapter=Resampling Filters |chapter-url=https://legacy.imagemagick.org/Usage/filter/ |publisher=[[ImageMagick]] |access-date=25 October 2020}}</ref>
|-
|style="text-align:right;"|1
|style="text-align:right;"|0
|[[B样条]]
|[[Paint.NET]]中的双三次滤波器
|}

== 例子 ==

{{Gallery
|width=512
|height=256
|lines=4
|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|在[[GIMP]](Catmull-Rom) 中使用双三次滤波器进行放大
|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.|在[[Paint.NET]]（B样条）中使用双三次滤波器进行放大
}}
{{Clear}}

== 参阅 ==
* [[振鈴效應]]
* [[各向异性过滤]]

== 参考资料 ==
{{reflist}}

[[Category:数字信号处理]]