总变差

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.

在数学领域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${\displaystyle f}$定义在区间${\displaystyle [a,b]\subset \mathbb {R} }$的总变差是一维参数曲线${\displaystyle x\mapsto f(x),x\in [a,b]}$的弧长。 连续可微函数的总变差，可由如下的积分给出

${\displaystyle V_{b}^{a}(f)=\int _{a}^{b}|f'(x)|\,\mathrm {d} x.}$

任意实值或虚值函数${\displaystyle f}$定义在区间${\displaystyle [a,b]}$上的总变差，由

${\displaystyle V_{b}^{a}(f)=\sup _{P}\sum _{i=0}^{n_{P}-1}|f(x_{i+1})-f(x_{i})|,\,}$

定义。其中${\displaystyle P}$为区间${\displaystyle [a,b]}$中的所有分划.

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

${\displaystyle V(f,\Omega ):=\sup \left\{\int _{\Omega }f\mathrm {div} \varphi \colon \varphi \in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\ \Vert \varphi \Vert _{L^{\infty }(\Omega )}\leq 1\right\},}$

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

若${\displaystyle f}$可微，上式可简化为

${\displaystyle V(f,\Omega )=\int \limits _{\Omega }\left|\nabla f\right|.}$

度量空间

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

${\displaystyle |\mu |(E)=\sup _{\pi }\sum _{A\in \pi }|\mu (A)|\qquad \forall E\in \Sigma }$

其中${\displaystyle \pi }$为${\displaystyle E}$的划分。 如果${\displaystyle \mu }$是符号测度，通过汉分解定理可知：

${\displaystyle |\mu |=\mu ^{+}+\mu ^{-}\,}$

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：

${\displaystyle \int \limits _{\Omega }f\,\mathrm {div} \varphi =-\int _{\Omega }\nabla f\cdot \varphi }$

引理证明

由高斯散度定理${\displaystyle \int \limits _{\Omega }{\text{div}}\mathbf {R} =\int \limits _{\partial \Omega }\mathbf {R} \cdot \mathbf {n} }$. 将${\displaystyle \mathbf {R}$ :=f\mathbf {\varphi } } 代入，可得

${\displaystyle \int \limits _{\Omega }{\text{div}}\left(f\mathbf {\varphi } \right)=\int \limits _{\partial \Omega }\left(f\mathbf {\varphi } \right)\cdot \mathbf {n} }$

由于在${\displaystyle \Omega }$的边界上${\displaystyle \mathbf {\varphi } =0}$,从而

${\displaystyle \int \limits _{\Omega }{\text{div}}\left(f\mathbf {\varphi } \right)=\int \limits _{\partial \Omega }\left(f\mathbf {\varphi } \right)\cdot \mathbf {n} =0}$

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

${\displaystyle \int \limits _{\Omega }f\,\mathrm {div} \varphi =-\int _{\Omega }\nabla f\cdot \varphi }$.

如果函数${\displaystyle f}$的总变差有限，则称函数${\displaystyle f}$为有界变差函数.

参阅

• 有界变差
• 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) "Variation of a function", Springer-Verlag Online Encyclopaedia of Mathematics.
• "Total variation" on Planetmath.

多变量

• Comments of Anatolii Georgievich Vitushkin on the preceding article of Boris I. Golubov "Variation of a function", Springer-Verlag Online Encyclopaedia of Mathematics.
• Boris I. Golubov "Arzelà variation", "Fréchet variation", "Hardy variation", "Pierpont variation", "Tonelli plane variation", "Vitali variation", voices from the Springer-Verlag Online Encyclopaedia of Mathematics.

测度论

• Rowland, Todd. "Total Variation". From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.
• "Jordan decomposition" on Planetmath.

概率论

• M. Denuit and S. Van Bellegem "On the stop-loss and total variation distances between random sums", discussion paper 0034 of the Statistic Institute of the "Université Catholique de Louvain".

应用

• Caselles, Vicent; Chambolle; Novaga, , SIAM, Multiscale Modeling and Simulation, vol. 6 n. 3, 2007 外部链接存在于|title= (帮助) (a work dealing with total variation application in denoising problems for image processing).

• Tony F. Chan and Jackie (Jianhong) Shen (2005), 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).