Total variation

Definition 1.1. The total variation of a real-valued (or more generally complex-valued) function ${\displaystyle f}$, defined on an interval ${\displaystyle [a,b]\subset \mathbb {R} }$ is the quantity

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

where the supremum runs over the set of all partitions ${\displaystyle {\mathcal {P}}=\left\{P=\{x_{0},\dots ,x_{n_{P}}\}\mid P{\text{ is a partition of }}[a,b]\right\}}$ of the given interval.

Definition 1.2. Let Ω be an open subset of Rⁿ. Given a function f belonging to L¹(Ω), the total variation of f in Ω is defined as

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

where

• ${\displaystyle C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}$ is the set of continuously differentiable vector functions of compact support contained in ${\displaystyle \Omega }$,
• ${\displaystyle \Vert \;\Vert _{L^{\infty }(\Omega )}}$ is the essential supremum norm, and
• ${\displaystyle \operatorname {div} }$ is the divergence operator.

This definition does not require that the domain ${\displaystyle \Omega \subseteq \mathbb {R} ^{n}}$ of the given function be a bounded set.

Following Saks (1937, p. 10), consider a signed measure ${\displaystyle \mu }$ on a measurable space ${\displaystyle (X,\Sigma )}$: then it is possible to define two set functions ${\displaystyle {\overline {\mathrm {W} }}(\mu ,\cdot )}$ and ${\displaystyle {\underline {\mathrm {W} }}(\mu ,\cdot )}$, respectively called upper variation and lower variation, as follows

${\displaystyle {\overline {\mathrm {W} }}(\mu ,E)=\sup \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma }$

${\displaystyle {\underline {\mathrm {W} }}(\mu ,E)=\inf \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma }$

clearly

${\displaystyle {\overline {\mathrm {W} }}(\mu ,E)\geq 0\geq {\underline {\mathrm {W} }}(\mu ,E)\qquad \forall E\in \Sigma }$

Definition 1.3. The variation (also called absolute variation) of the signed measure ${\displaystyle \mu }$ is the set function

${\displaystyle |\mu |(E)={\overline {\mathrm {W} }}(\mu ,E)+\left|{\underline {\mathrm {W} }}(\mu ,E)\right|\qquad \forall E\in \Sigma }$

and its total variation is defined as the value of this measure on the whole space of definition, i.e.

${\displaystyle \|\mu \|=|\mu |(X)}$

Saks (1937, p. 11) uses upper and lower variations to prove the Hahn–Jordan decomposition: according to his version of this theorem, the upper and lower variation are respectively a non-negative and a non-positive measure. Using a more modern notation, define

${\displaystyle \mu ^{+}(\cdot )={\overline {\mathrm {W} }}(\mu ,\cdot )\,,}$

${\displaystyle \mu ^{-}(\cdot )=-{\underline {\mathrm {W} }}(\mu ,\cdot )\,,}$

Then ${\displaystyle \mu ^{+}}$ and ${\displaystyle \mu ^{-}}$ are two non-negative measures such that

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

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

The last measure is sometimes called, by abuse of notation, total variation measure.

If the measure ${\displaystyle \mu }$ is complex-valued i.e. is a complex measure, its upper and lower variation cannot be defined and the Hahn–Jordan decomposition theorem can only be applied to its real and imaginary parts. However, it is possible to follow Rudin (1966, pp. 137–139) and define the total variation of the complex-valued measure ${\displaystyle \mu }$ as follows

Definition 1.4. The variation of the complex-valued measure ${\displaystyle \mu }$ is the set function

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

where the supremum is taken over all partitions ${\displaystyle \pi }$ of a measurable set ${\displaystyle E}$ into a countable number of disjoint measurable subsets.

This definition coincides with the above definition ${\displaystyle |\mu |=\mu ^{+}+\mu ^{-}}$ for the case of real-valued signed measures.

The variation so defined is a positive measure (see Rudin (1966, p. 139)) and coincides with the one defined by 1.3 when ${\displaystyle \mu }$ is a signed measure: its total variation is defined as above. This definition works also if ${\displaystyle \mu }$ is a vector measure: the variation is then defined by the following formula

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

where the supremum is as above. This definition is slightly more general than the one given by Rudin (1966, p. 138) since it requires only to consider finite partitions of the space ${\displaystyle X}$: this implies that it can be used also to define the total variation on finite-additive measures.

The total variation of any probability measure is exactly one, therefore it is not interesting as a means of investigating the properties of such measures. However, when μ and ν are probability measures, the total variation distance of probability measures can be defined as ${\displaystyle \|\mu -\nu \|}$ where the norm is the total variation norm of signed measures. Using the property that ${\displaystyle (\mu -\nu )(X)=0}$, we eventually arrive at the equivalent definition

${\displaystyle \|\mu -\nu \|=|\mu -\nu |(X)=2\sup \left\{\,\left|\mu (A)-\nu (A)\right|:A\in \Sigma \,\right\}}$

and its values are non-trivial. The factor ${\displaystyle 2}$ above is usually dropped (as is the convention in the article total variation distance of probability measures). Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event. For a categorical distribution it is possible to write the total variation distance as follows

${\displaystyle \delta (\mu ,\nu )=\sum _{x}\left|\mu (x)-\nu (x)\right|\;.}$

It may also be normalized to values in ${\displaystyle [0,1]}$ by halving the previous definition as follows

${\displaystyle \delta (\mu ,\nu )={\frac {1}{2}}\sum _{x}\left|\mu (x)-\nu (x)\right|}$[2]

The total variation of a ${\displaystyle C^{1}({\overline {\Omega }})}$ function ${\displaystyle f}$ can be expressed as an integral involving the given function instead of as the supremum of the functionals of definitions 1.1 and 1.2.

Theorem 1. The total variation of a differentiable function ${\displaystyle f}$, defined on an interval ${\displaystyle [a,b]\subset \mathbb {R} }$, has the following expression if ${\displaystyle f'}$ is Riemann integrable

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

If ${\displaystyle f}$ is differentiable and monotonic, then the above simplifies to

${\displaystyle V_{a}^{b}(f)=|f(a)-f(b)|}$

For any differentiable function ${\displaystyle f}$, we can decompose the domain interval ${\displaystyle [a,b]}$, into subintervals ${\displaystyle [a,a_{1}],[a_{1},a_{2}],\dots ,[a_{N},b]}$ (with ${\displaystyle a<a_{1}<a_{2}<\cdots <a_{N}<b}$) in which ${\displaystyle f}$ is locally monotonic, then the total variation of ${\displaystyle f}$ over ${\displaystyle [a,b]}$ can be written as the sum of local variations on those subintervals:

${\displaystyle {\begin{aligned}V_{a}^{b}(f)&=V_{a}^{a_{1}}(f)+V_{a_{1}}^{a_{2}}(f)+\,\cdots \,+V_{a_{N}}^{b}(f)\\[0.3em]&=|f(a)-f(a_{1})|+|f(a_{1})-f(a_{2})|+\,\cdots \,+|f(a_{N})-f(b)|\end{aligned}}}$

Theorem 2. Given a ${\displaystyle C^{1}({\overline {\Omega }})}$ function ${\displaystyle f}$ defined on a bounded open set ${\displaystyle \Omega \subseteq \mathbb {R} ^{n}}$, with ${\displaystyle \partial \Omega }$ of class ${\displaystyle C^{1}}$, the total variation of ${\displaystyle f}$ has the following expression

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

The first step in the proof is to first prove an equality which follows from the Gauss–Ostrogradsky theorem.

Under the conditions of the theorem, the following equality holds:

${\displaystyle \int _{\Omega }f\operatorname {div} \varphi =-\int _{\Omega }\nabla f\cdot \varphi }$

From the Gauss–Ostrogradsky theorem:

${\displaystyle \int _{\Omega }\operatorname {div} \mathbf {R} =\int _{\partial \Omega }\mathbf {R} \cdot \mathbf {n} }$

by substituting ${\displaystyle \mathbf {R}$ :=f\mathbf {\varphi } } , we have:

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

where ${\displaystyle \mathbf {\varphi } }$ is zero on the border of ${\displaystyle \Omega }$ by definition:

${\displaystyle \int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=0}$

${\displaystyle \int _{\Omega }\partial _{x_{i}}\left(f\mathbf {\varphi } _{i}\right)=0}$

${\displaystyle \int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f+f\partial _{x_{i}}\mathbf {\varphi } _{i}=0}$

${\displaystyle \int _{\Omega }f\partial _{x_{i}}\mathbf {\varphi } _{i}=-\int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f}$

${\displaystyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f}$

Under the conditions of the theorem, from the lemma we have:

${\displaystyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\leq \left|\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\right|\leq \int _{\Omega }\left|\mathbf {\varphi } \right|\cdot \left|\nabla f\right|\leq \int _{\Omega }\left|\nabla f\right|}$

in the last part ${\displaystyle \mathbf {\varphi } }$ could be omitted, because by definition its essential supremum is at most one.

On the other hand, we consider ${\displaystyle \theta _{N}:=-\mathbb {I} _{\left[-N,N\right]}\mathbb {I} _{\{\nabla f\neq 0\}}{\frac {\nabla f}{\left|\nabla f\right|}}}$ and ${\displaystyle \theta _{N}^{*}}$ which is the up to ${\displaystyle \varepsilon }$ approximation of ${\displaystyle \theta _{N}}$ in ${\displaystyle C_{c}^{1}}$ with the same integral. We can do this since ${\displaystyle C_{c}^{1}}$ is dense in ${\displaystyle L^{1}}$. Now again substituting into the lemma:

${\displaystyle {\begin{aligned}&\lim _{N\to \infty }\int _{\Omega }f\operatorname {div} \theta _{N}^{*}\\[4pt]&=\lim _{N\to \infty }\int _{\{\nabla f\neq 0\}}\mathbb {I} _{\left[-N,N\right]}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\[4pt]&=\lim _{N\to \infty }\int _{\left[-N,N\right]\cap {\{\nabla f\neq 0\}}}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\[4pt]&=\int _{\Omega }\left|\nabla f\right|\end{aligned}}}$

This means we have a convergent sequence of ${\textstyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } }$ that tends to ${\textstyle \int _{\Omega }\left|\nabla f\right|}$ as well as we know that ${\textstyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } \leq \int _{\Omega }\left|\nabla f\right|}$. Q.E.D.

It can be seen from the proof that the supremum is attained when

${\displaystyle \varphi \to {\frac {-\nabla f}{\left|\nabla f\right|}}.}$

The function ${\displaystyle f}$ is said to be of bounded variation precisely if its total variation is finite.

The total variation is a norm defined on the space of measures of bounded variation. The space of measures on a σ-algebra of sets is a Banach space, called the ca space, relative to this norm. It is contained in the larger Banach space, called the ba space, consisting of finitely additive (as opposed to countably additive) measures, also with the same norm. The distance function associated to the norm gives rise to the total variation distance between two measures μ and ν.

For finite measures on R, the link between the total variation of a measure μ and the total variation of a function, as described above, goes as follows. Given μ, define a function ${\displaystyle \varphi \colon \mathbb {R} \to \mathbb {R} }$ by

${\displaystyle \varphi (t)=\mu ((-\infty ,t])~.}$

Then, the total variation of the signed measure μ is equal to the total variation, in the above sense, of the function ${\displaystyle \varphi }$. In general, the total variation of a signed measure can be defined using Jordan's decomposition theorem by

${\displaystyle \|\mu \|_{TV}=\mu _{+}(X)+\mu _{-}(X)~,}$

for any signed measure μ on a measurable space ${\displaystyle (X,\Sigma )}$.

• Caselles, Vicent; Chambolle, Antonin; Novaga, Matteo (2007), The discontinuity set of solutions of the TV denoising problem and some extensions, SIAM, Multiscale Modeling and Simulation, vol. 6 n. 3, archived from the original on 2011-09-27 (a work dealing with total variation application in denoising problems for image processing).

• Rudin, Leonid I.; Osher, Stanley; Fatemi, Emad (1992), "Nonlinear total variation based noise removal algorithms", Physica D: Nonlinear Phenomena, Physica D: Nonlinear Phenomena 60.1: 259-268, 60 (1–4): 259–268, Bibcode:1992PhyD...60..259R, doi:10.1016/0167-2789(92)90242-F.

• Blomgren, Peter; Chan, Tony F. (1998), "Color TV: total variation methods for restoration of vector-valued images", IEEE Transactions on Image Processing, Image Processing, IEEE Transactions on, vol. 7, no. 3: 304-309, 7 (3): 304, Bibcode:1998ITIP....7..304B, doi:10.1109/83.661180, PMID 18276250.

• Tony F. Chan and Jackie (Jianhong) Shen (2005), Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods, SIAM, ISBN 0-89871-589-X (with in-depth coverage and extensive applications of Total Variations in modern image processing, as started by Rudin, Osher, and Fatemi).