总变差

在数学中，总变差（）就是一函数其数值变化的差的总和。

定义
矢量空间
实值函数f定义在区间[a,b]\subset \mathbb R的总变差是一维参数曲线x\mapsto f(x),x\in[a,b]的弧长。
连续可微函数的总变差，可由如下的积分给出

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

任意实值或虚值函数f定义在区间[a,b]上的总变差，由

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

定义。其中P为区间[a,b]中的所有分划.

定义在有界区域\Omega \subset \mathbb{R}^n上的实值可积函数f的总变差,定义为

: 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\},

其中 C_c^1(\Omega,\mathbb{R}^n)是Ω中的紧支集上全体连续可微向量函数构成的集合, \Vert\;\Vert_{L^\infty(\Omega)}是本质上确界范数。

若f可微，上式可简化为

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

度量空间
在一个度量空间(\Omega,\Sigma)上，集函数\mu : \Sigma \rightarrow \R，其总变差为：

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

其中\pi为E的划分。
如果\mu是符号测度，通过汉分解定理可知：
:|\mu|=\mu^++\mu^-\,

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：
: \int\limits_\Omega f\,\mathrm{div}\varphi = -\int_\Omega\nabla f\cdot\varphi
引理证明
由高斯散度定理 \int\limits_\Omega \text{div}\mathbf R = \int\limits_{\partial\Omega}\mathbf R\cdot \mathbf n .
将\mathbf R:= f\mathbf\varphi代入，可得

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

由于在\Omega的边界上\mathbf\varphi = 0,从而

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

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

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

如果函数f的总变差有限，则称函数f为有界变差函数.

参阅

• 有界变差

*

• 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] ", 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] ", Springer-Verlag Online Encyclopaedia of Mathematics.
• Boris I. Golubov "[http://eom.springer.de/a/a013470.htm Arzelà variation] ", "[http://eom.springer.de/f/f041400.htm Fréchet variation] ", "[http://eom.springer.de/h/h046400.htm Hardy variation] ", "[http://eom.springer.de/p/p072720.htm Pierpont variation] ", "[http://eom.springer.de/t/t092990.htm Tonelli plane variation] ", "[http://eom.springer.de/v/v096790.htm Vitali variation] ", voices from the Springer-Verlag Online Encyclopaedia of Mathematics.

测度论

• Rowland, Todd. "[http://mathworld.wolfram.com/TotalVariation.html Total Variation] ". From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein.
• "[http://planetmath.org/encyclopedia/JordanDecomposition.html Jordan decomposition] " 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

• (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).