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Inmathematics, thetotal variation identifies several slightly different concepts, related to the (local or global) structure of thecodomain of afunction or ameasure. For areal-valuedcontinuous functionf, defined on aninterval [a,b] ⊂R, its total variation on the interval of definition is a measure of the one-dimensionalarclength of the curve with parametric equationx ↦f(x), forx ∈ [a,b]. Functions whose total variation is finite are calledfunctions of bounded variation.

The concept of total variation for functions of one real variable was first introduced byCamille Jordan in the paper (Jordan 1881).^[1] He used the new concept in order to prove a convergence theorem forFourier series ofdiscontinuousperiodic functions whose variation isbounded. The extension of the concept to functions of more than one variable however is not simple for various reasons.

Definition 1.1. Thetotal variation of areal-valued (or more generallycomplex-valued)function${\displaystyle f}$, defined on aninterval${\displaystyle [a,b]\subset \mathbb{R}}$ is the quantity

${\displaystyle V_a^b(f)=\underset{\mathcal{P}}{sup}\sum _{i=0}^{n_P-1}|f(x_{i+1})-f(x_i)|,}$

where thesupremum runs over theset of allpartitions${\displaystyle \mathcal{P}=\left\{P=\{x_0,\dots ,x_{n_P}\}\mid P\text{ is a partition of }[a,b]\right\}}$ of the giveninterval. Which means that.

Definition 1.2.^[2] LetΩ be anopen subset ofRⁿ. Given a functionf belonging toL¹(Ω), thetotal variation off inΩ is defined as

${\displaystyle V(f,\mathrm{\Omega }):=sup\left\{{\int }_{\mathrm{\Omega }}f(x)\mathrm{div}\varphi (x)\mathrm{d}x:\varphi \in C_c^1(\mathrm{\Omega },{\mathbb{R}}^n),\Vert \varphi {\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le 1\right\},}$

where

• ${\displaystyle C_c^1(\mathrm{\Omega },{\mathbb{R}}^n)}$ is theset ofcontinuously differentiablevector functions ofcompact support contained in${\displaystyle \mathrm{\Omega }}$,
• ${\displaystyle \Vert {\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}}$ is theessential supremumnorm, and
• ${\displaystyle \mathrm{div}}$ is thedivergence operator.

This definitiondoes not require that thedomain${\displaystyle \mathrm{\Omega }\subseteq {\mathbb{R}}^n}$ of the given function be abounded set.

FollowingSaks (1937, p. 10), consider asigned measure${\displaystyle \mu }$ on ameasurable space${\displaystyle (X,\mathrm{\Sigma })}$: then it is possible to define twoset functions${\displaystyle \overline{\mathrm{W}}(\mu ,\cdot )}$ and${\displaystyle \underset{\_}{\mathrm{W}}(\mu ,\cdot )}$, respectively calledupper variation andlower variation, as follows

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

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

clearly

${\displaystyle \overline{\mathrm{W}}(\mu ,E)\ge 0\ge \underset{\_}{\mathrm{W}}(\mu ,E)\mathrm{\forall }E\in \mathrm{\Sigma }}$

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

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

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

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

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

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

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

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

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

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

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

If the measure${\displaystyle \mu }$ iscomplex-valued i.e. is acomplex 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 followRudin (1966, pp. 137–139) and define the total variation of the complex-valued measure${\displaystyle \mu }$ as follows

Definition 1.4. Thevariation of the complex-valued measure${\displaystyle \mu }$ is theset function

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

where thesupremum is taken over all partitions${\displaystyle \pi }$ of ameasurable 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 apositive measure (seeRudin (1966, p. 139)) and coincides with the one defined by1.3 when${\displaystyle \mu }$ is asigned measure: its total variation is defined as above. This definition works also if${\displaystyle \mu }$ is avector measure: the variation is then defined by the following formula

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

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

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The total variation of anyprobability measure is exactly one, therefore it is not interesting as a means of investigating the properties of such measures. However, when μ and ν areprobability measures, thetotal variation distance of probability measures can be defined as${\displaystyle \Vert \mu -\nu \Vert }$ 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

${\displaystyle \Vert \mu -\nu \Vert =|\mu -\nu |(X)=2sup\left\{\left|\mu (A)-\nu (A)\right|:A\in \mathrm{\Sigma }\right\}}$^[3]

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

${\displaystyle \delta (\mu ,\nu )=\sum _x\left|\mu (x)-\nu (x)\right|.}$^[4]

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

${\displaystyle \delta (\mu ,\nu )=\frac{1}{2}\sum _x\left|\mu (x)-\nu (x)\right|}$^[5]

The total variation of a${\displaystyle C^1(\overline{\mathrm{\Omega }})}$ function${\displaystyle f}$ can be expressed as anintegral involving the given function instead of as thesupremum of thefunctionals of definitions1.1 and1.2.

Theorem 1. Thetotal variation of adifferentiable function${\displaystyle f}$, defined on aninterval${\displaystyle [a,b]\subset \mathbb{R}}$, has the following expression if${\displaystyle f^{\prime }}$ is Riemann integrable

${\displaystyle V_a^b(f)={\int }_a^b|f^{\prime }(x)|\mathrm{d}x}$

If${\displaystyle f}$ is differentiable andmonotonic, 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) 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:

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

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

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

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

${\displaystyle {\int }_{\mathrm{\Omega }}f\mathrm{div}\phi =-{\int }_{\mathrm{\Omega }}\mathrm{\nabla }f\cdot \phi }$

From theGauss–Ostrogradsky theorem:

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

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

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

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

${\displaystyle {\int }_{\mathrm{\Omega }}\mathrm{div}\left(f\phi \right)=0}$

${\displaystyle {\int }_{\mathrm{\Omega }}{\mathrm{\partial }}_{x_i}\left(f{\phi }_i\right)=0}$

${\displaystyle {\int }_{\mathrm{\Omega }}{\phi }_i{\mathrm{\partial }}_{x_i}f+f{\mathrm{\partial }}_{x_i}{\phi }_i=0}$

${\displaystyle {\int }_{\mathrm{\Omega }}f{\mathrm{\partial }}_{x_i}{\phi }_i=-{\int }_{\mathrm{\Omega }}{\phi }_i{\mathrm{\partial }}_{x_i}f}$

${\displaystyle {\int }_{\mathrm{\Omega }}f\mathrm{div}\phi =-{\int }_{\mathrm{\Omega }}\phi \cdot \mathrm{\nabla }f}$

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

${\displaystyle {\int }_{\mathrm{\Omega }}f\mathrm{div}\phi =-{\int }_{\mathrm{\Omega }}\phi \cdot \mathrm{\nabla }f\le \left|{\int }_{\mathrm{\Omega }}\phi \cdot \mathrm{\nabla }f\right|\le {\int }_{\mathrm{\Omega }}\left|\phi \right|\cdot \left|\mathrm{\nabla }f\right|\le {\int }_{\mathrm{\Omega }}\left|\mathrm{\nabla }f\right|}$

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

On the other hand, we consider${\displaystyle {\theta }_N:=-{\mathbb{I}}_{\left[-N,N\right]}{\mathbb{I}}_{\{\mathrm{\nabla }f\ne 0\}}\frac{\mathrm{\nabla }f}{\left|\mathrm{\nabla }f\right|}}$ and${\displaystyle {\theta }_N^{\ast }}$ which is the up to${\displaystyle \epsilon }$ 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:

This means we have a convergent sequence of${\int }_{\mathrm{\Omega }}f\mathrm{div}\phi$ that tends to${\int }_{\mathrm{\Omega }}\left|\mathrm{\nabla }f\right|$ as well as we know that${\int }_{\mathrm{\Omega }}f\mathrm{div}\phi \le {\int }_{\mathrm{\Omega }}\left|\mathrm{\nabla }f\right|$.Q.E.D.

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

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

Thefunction${\displaystyle f}$ is said to be ofbounded variation precisely if its total variation is finite.

The total variation is anorm defined on the space of measures of bounded variation. The space of measures on a σ-algebra of sets is aBanach space, called theca space, relative to this norm. It is contained in the larger Banach space, called theba space, consisting offinitely additive (as opposed to countably additive) measures, also with the same norm. Thedistance function associated to the norm gives rise to the total variation distance between two measuresμ andν.

For finite measures onR, the link between the total variation of a measureμ and the total variation of a function, as described above, goes as follows. Givenμ, define a function${\displaystyle \phi :\mathbb{R}\to \mathbb{R}}$ by

${\displaystyle \phi (t)=\mu ((-\mathrm{\infty },t]).}$

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

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

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

Total variation can be seen as anon-negativereal-valuedfunctional defined on the space ofreal-valuedfunctions (for the case of functions of one variable) or on the space ofintegrable functions (for the case of functions of several variables). As a functional, total variation finds applications in several branches of mathematics and engineering, likeoptimal control,numerical analysis, andcalculus of variations, where the solution to a certain problem has tominimize its value. As an example, use of the total variation functional is common in the following two kind of problems

• Numerical analysis of differential equations: it is the science of finding approximate solutions todifferential equations. Applications of total variation to these problems are detailed in the article "total variation diminishing"
• Image denoising:^[6] inimage processing, denoising is a collection of methods used to reduce thenoise in animage reconstructed from data obtained by electronic means, for exampledata transmission orsensing. "Total variation denoising" is the name for the application of total variation to image noise reduction; further details can be found in the papers of (Rudin, Osher & Fatemi 1992) and (Caselles, Chambolle & Novaga 2007). A sensible extension of this model to colour images, called Colour TV, can be found in (Blomgren & Chan 1998).

• Bounded variation
• p-variation
• Total variation diminishing
• Total variation denoising
• Quadratic variation
• Total variation distance of probability measures
• Kolmogorov–Smirnov test
• Anisotropic diffusion

This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(February 2012) (Learn how and when to remove this message)

• Arzelà, Cesare (7 May 1905),"Sulle funzioni di due variabili a variazione limitata (On functions of two variables of bounded variation)",Rendiconto delle Sessioni della Reale Accademia delle Scienze dell'Istituto di Bologna, Nuova serie (in Italian),IX (4):100–107,JFM 36.0491.02, archived fromthe original on 2007-08-07.
• Golubov, Boris I. (2001) [1994],"Arzelà variation",Encyclopedia of Mathematics,EMS Press.
• Golubov, Boris I. (2001) [1994],"Fréchet variation",Encyclopedia of Mathematics,EMS Press.
• Golubov, Boris I. (2001) [1994],"Hardy variation",Encyclopedia of Mathematics,EMS Press.
• Golubov, Boris I. (2001) [1994],"Pierpont variation",Encyclopedia of Mathematics,EMS Press.
• Golubov, Boris I. (2001) [1994],"Vitali variation",Encyclopedia of Mathematics,EMS Press.
• Golubov, Boris I. (2001) [1994],"Tonelli plane variation",Encyclopedia of Mathematics,EMS Press.
• Golubov, Boris I.;Vitushkin, Anatoli G. (2001) [1994],"Variation of a function",Encyclopedia of Mathematics,EMS Press
• Jordan, Camille (1881),"Sur la série de Fourier",Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French),92:228–230,JFM 13.0184.01 (available atGallica). This is, according to Boris Golubov, the first paper on functions of bounded variation.
• Hahn, Hans (1921),Theorie der reellen Funktionen (in German), Berlin: Springer Verlag, pp. VII+600,JFM 48.0261.09.
• Vitali, Giuseppe (1908) [17 dicembre 1907],"Sui gruppi di punti e sulle funzioni di variabili reali (On groups of points and functions of real variables)",Atti dell'Accademia delle Scienze di Torino (in Italian),43:75–92,JFM 39.0101.05, archived fromthe original on 2009-03-31. The paper containing the first proof ofVitali covering theorem.

• Adams, C. Raymond; Clarkson, James A. (1933), "On definitions of bounded variation for functions of two variables",Transactions of the American Mathematical Society,35 (4):824–854,doi:10.1090/S0002-9947-1933-1501718-2,JFM 59.0285.01,MR 1501718,Zbl 0008.00602.
• Cesari, Lamberto (1936),"Sulle funzioni a variazione limitata (On the functions of bounded variation)",Annali della Scuola Normale Superiore, II (in Italian),5 (3–4):299–313,JFM 62.0247.03,MR 1556778,Zbl 0014.29605. Available atNumdam.

• Leoni, Giovanni (2017),A First Course in Sobolev Spaces: Second Edition, Graduate Studies in Mathematics, American Mathematical Society, pp. xxii+734,ISBN 978-1-4704-2921-8.
• Saks, Stanisław (1937).Theory of the Integral. Monografie Matematyczne. Vol. 7 (2nd ed.). Warszawa–Lwów: G.E. Stechert & Co. pp. VI+347.JFM 63.0183.05.Zbl 0017.30004.. (available at thePolish Virtual Library of Science). English translation from the original French byLaurence Chisholm Young, with two additional notes byStefan Banach.
• Rudin, Walter (1966),Real and Complex Analysis, McGraw-Hill Series in Higher Mathematics (1st ed.), New York: McGraw-Hill, pp. xi+412,MR 0210528,Zbl 0142.01701.

One variable

• "Total variation atPlanetMath.

One and more variables

• Function of bounded variation atEncyclopedia of Mathematics

Measure theory

• Rowland, Todd."Total Variation".MathWorld..
• Jordan decomposition atPlanetMath..
• Jordan decomposition atEncyclopedia of Mathematics

• 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 fromthe original on 2011-09-27 (a work dealing with total variation application in denoising problems forimage processing).

• Rudin, Leonid I.; Osher, Stanley; Fatemi, Emad (1992), "Nonlinear total variation based noise removal algorithms",Physica D: Nonlinear Phenomena,60 (1–4), Physica D: Nonlinear Phenomena 60.1: 259-268:259–268,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,7 (3), Image Processing, IEEE Transactions on, vol. 7, no. 3: 304-309:304–309,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).

• Mathematical analysis

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