查看“︁总变差”︁的源代码

{{NoteTA |G1=Math}}
[[Image:Total variation.gif|right|frame|当绿点遍历整个函数时，绿点在''y''-轴上的投影红点走过的路程就是该函数的总变分.]]

在[[数学]]中，'''总变差'''（{{lang-en|Total variation}}）就是一[[函数]]其数值变化的差的总和。

==定义==
===矢量空间===
[[实数|实值]]函数<math>f</math>定义在[[区间]]<math>[a,b]\subset \mathbb R</math>的总变差是一维参数曲线<math>x\mapsto f(x),x\in[a,b]</math>的[[弧长]]。
[[连续]][[可微]][[函数]]的总变差，可由如下的[[积分]]给出

:<math> V^a_b(f) = \int _a^b |f'(x)|\, \mathrm dx.</math>

任意实值或[[虚数|虚值]]函数<math>f</math>定义在区间<math>[a,b]</math>上的总变差，由

:<math> V^a_b(f)=\sup_P \sum_{i=0}^{n_P-1} | f(x_{i+1})-f(x_i) |, \,</math>

定义。其中<math>P</math>为区间<math>[a,b]</math>中的所有分划.

定义在有界[[区域]]<math>\Omega \subset \mathbb{R}^n</math>上的[[实数|实值]][[可积函数]]<math>f</math>的'''总变差''',定义为

:<math> V(f,\Omega):=\sup\left\{\int_\Omega f\mathrm{div}\varphi\colon \varphi\in  C_c^1(\Omega,\mathbb{R}^n),\ \Vert \varphi\Vert_{L^\infty(\Omega)}\le 1\right\}, </math>

其中  <math> C_c^1(\Omega,\mathbb{R}^n)</math>是Ω中的[[紧支集]]上全体[[连续可微]][[向量函数]]构成的[[集合 (数学)|集合]],  <math> \Vert\;\Vert_{L^\infty(\Omega)}</math>是[[本质上确界]][[范数]]。

若<math>f</math>可微，上式可简化为

:<math>V(f,\Omega) = \int\limits_\Omega\left|\nabla f\right| .</math>

===度量空间===
在一个[[度量空间]]<math>(\Omega,\Sigma)</math>上，集函数<math>\mu : \Sigma \rightarrow \R</math>，其总变差为：

:<math>|\mu|(E)=\sup_\pi \sum_{A\isin\pi} |\mu(A)|\qquad\forall E\in\Sigma</math>

其中''<math>\pi</math>''为''<math>E</math>''的划分。
如果<math>\mu</math>是[[符号测度]]，通过[[汉分解定理]]可知：
:<math>|\mu|=\mu^++\mu^-\,</math>

==可微定义的证明==
首先需要利用[[高斯散度定理]]证明一个等式.

===引理===
在假设条件下，下面的等式成立：
: <math> \int\limits_\Omega f\,\mathrm{div}\varphi = -\int_\Omega\nabla f\cdot\varphi </math>
==== 引理证明 ====

由[[高斯散度定理]]<math> \int\limits_\Omega \text{div}\mathbf R = \int\limits_{\partial\Omega}\mathbf R\cdot \mathbf n </math>.
将<math>\mathbf R:= f\mathbf\varphi</math>代入，可得

: <math> \int\limits_\Omega\text{div}\left(f\mathbf\varphi\right) = \int\limits_{\partial\Omega}\left(f\mathbf\varphi\right)\cdot\mathbf n </math>

由于在<math>\Omega</math>的边界上<math>\mathbf\varphi = 0</math>,从而

: <math> \int\limits_\Omega\text{div}\left(f\mathbf\varphi\right) =  \int\limits_{\partial\Omega}\left(f\mathbf\varphi\right)\cdot\mathbf n = 0</math>

注意到<math> \text{div}\left(f\mathbf\varphi\right) = f\text{div}\mathbf\varphi + \nabla f\cdot \varphi</math>代入上式，移项即得

: <math> \int\limits_\Omega f\,\mathrm{div}\varphi = -\int_\Omega\nabla f\cdot\varphi</math>.

如果[[函数]]''<math>f</math>''的总变差有限，则称函数''<math>f</math>''为[[有界变差]]函数.

== 参阅 ==

* [[有界变差]]
* {{le|总变差递减|Total variation diminishing}}
* [[总变差规则化]]
* [[二次变差]]

== 外部链接 ==
=== 理论 ===

'''单变量'''
* Boris I. Golubov (and comments of [[Anatolii Georgievich Vitushkin]]) "''[http://eom.springer.de/V/v096110.htm Variation of a function] {{Wayback|url=http://eom.springer.de/V/v096110.htm |date=20111015012310 }}''", [[Springer-Verlag]] Online Encyclopaedia of Mathematics.
* "[https://web.archive.org/web/20070930231726/http://planetmath.org/encyclopedia/TotalVariation.html Total variation]" on [[Planetmath]].

'''多变量'''
* Comments of [[Anatolii Georgievich Vitushkin]] on the preceding article of Boris I. Golubov "''[http://eom.springer.de/V/v096110.htm Variation of a function] {{Wayback|url=http://eom.springer.de/V/v096110.htm |date=20111015012310 }}''", [[Springer-Verlag]] Online Encyclopaedia of Mathematics.
* Boris I. Golubov "[http://eom.springer.de/a/a013470.htm Arzelà variation] {{Wayback|url=http://eom.springer.de/a/a013470.htm |date=20111015034819 }}", "[http://eom.springer.de/f/f041400.htm Fréchet variation] {{Wayback|url=http://eom.springer.de/f/f041400.htm |date=20111015034829 }}", "[http://eom.springer.de/h/h046400.htm Hardy variation] {{Wayback|url=http://eom.springer.de/h/h046400.htm |date=20111015034850 }}", "[http://eom.springer.de/p/p072720.htm Pierpont variation] {{Wayback|url=http://eom.springer.de/p/p072720.htm |date=20111015034924 }}", "[http://eom.springer.de/t/t092990.htm Tonelli plane variation] {{Wayback|url=http://eom.springer.de/t/t092990.htm |date=20111015034941 }}", "[http://eom.springer.de/v/v096790.htm Vitali variation] {{Wayback|url=http://eom.springer.de/v/v096790.htm |date=20111015034948 }}", voices from the [[Springer-Verlag]] Online Encyclopaedia of Mathematics.

'''测度论'''
* Rowland, Todd. "[http://mathworld.wolfram.com/TotalVariation.html Total Variation] {{Wayback|url=http://mathworld.wolfram.com/TotalVariation.html |date=20200319043456 }}". From [[MathWorld]]--A [[Wolfram Research|Wolfram]] Web Resource, created by [[Eric W. Weisstein]].
* "[http://planetmath.org/encyclopedia/JordanDecomposition.html Jordan decomposition] {{Wayback|url=http://planetmath.org/encyclopedia/JordanDecomposition.html |date=20160307083143 }}" on [[Planetmath]].

'''概率论'''
* M. Denuit and S. Van Bellegem "[https://web.archive.org/web/20110706132304/http://www.stat.ucl.ac.be/ISpub/dp/2000/dp0034.ps On the stop-loss and total variation distances between random sums]", [https://web.archive.org/web/20091214175351/http://www.stat.ucl.ac.be/ISpub/ discussion paper] 0034 of the [https://web.archive.org/web/20100402092508/http://www.stat.ucl.ac.be/ Statistic Institute] of the "[[Université Catholique de Louvain]]".

=== 应用 ===
*{{Harvard reference
 | Surname1 = Caselles
 | Given1 = Vicent
 | Surname2 = Chambolle
 | Given2 = Antonin
 | Surname3 = Novaga
 | Given3 = Matteo
 | Title = [http://cvgmt.sns.it/papers/caschanov07/ The discontinuity set of solutions of the TV denoising problem and some extensions]
 | Publisher = [[SIAM]], Multiscale Modeling and Simulation, vol. 6 n. 3,
 | Year = 2007}} (a work dealing with total variation application in denoising problems for [[image processing]]).

*[[Tony F. Chan]] and Jackie (Jianhong) Shen (2005), [https://web.archive.org/web/20080117220948/http://jackieneoshen.googlepages.com/ImagingNewEra.html ''Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods''], [[SIAM]], ISBN 089871589X (with in-depth coverage and extensive applications of Total Variations in modern image processing, as started by Rudin, Osher, and Fatemi).

[[Category:数学分析]]
[[Category:概率论]]