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[[Image:Total variation.gif|right|frame|As the green ball travels on the graph of the given function, the length of the path travelled by that ball's projection on the ''y''-axis, shown as a red ball, is the total variation of the function.]]

In [[mathematics]], the '''total variation''' identifies several slightly different concepts, related to the ([[local property|local]] or global) structure of the [[codomain]] of a [[Function (mathematics)|function]] or a [[measure (mathematics)|measure]]. For a [[real number|real-valued]] [[continuous function]] ''f'', defined on an [[interval (mathematics)|interval]] [''a'', ''b''] ⊂ ℝ, its total variation on the interval of definition is a measure of the one-dimensional [[arclength]] of the curve with parametric equation ''x'' ↦ ''f''(''x''), for ''x'' ∈ [''a'', ''b''].

==Historical note==
The concept of total variation for functions of one real variable was first introduced by [[Camille Jordan]] in the paper {{Harv|Jordan|1881}}.<ref>According to {{Harvtxt|Golubov|Vitushkin|2001}}.</ref> He used the new concept in order to prove a convergence theorem for [[Fourier series]] of [[discontinuous function|discontinuous]] [[periodic function]]s whose variation is [[Bounded variation|bounded]]. The extension of the concept to functions of more than one variable however is not simple for various reasons.

==Definitions==

===Total variation for functions of one real variable===
{{EquationRef|1|Definition 1.1.}} The '''total variation''' of a [[real number|real]]-valued (or more generally [[complex number|complex]]-valued) [[function (mathematics)|function]] ''<math>f</math>'', defined on an [[interval (mathematics)|interval]] <math> [a , b] \subset \mathbb{R}</math> is the quantity

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

where the [[supremum]] runs over the [[Set (mathematics)|set]] of all [[partition of an interval|partitions]] <math> \scriptstyle \mathcal{P} =\left\{P=\{ x_0, \dots , x_{n_P}\}|P\text{ is a partition of } [a,b] \right\} </math> of the given [[interval (mathematics)|interval]].

===Total variation for functions of ''n'' > 1 real variables===
{{EquationRef|2|Definition 1.2.}} Let '''Ω''' be an [[open subset]] of ℝ<sup>n</sup>. Given a function ''f'' belonging to L<sup>1</sup>('''Ω'''), the '''total variation''' of ''f'' in '''Ω''' is defined as

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

where  <math> \scriptstyle C_c^1(\Omega,\mathbb{R}^n)</math> is the [[Set (mathematics)|set]] of [[Smooth function|continuously differentiable]] [[Vector valued function|vector functions]] of [[support (mathematics)#Compact support|compact support]] contained  in <math>\Omega</math>, and <math> \scriptstyle \Vert\;\Vert_{L^\infty(\Omega)}</math> is the [[essential supremum]] [[Norm (mathematics)|norm]]. Note that this definition ''does not require'' that the [[Domain of a function|domain]] <math>\Omega \subseteq \mathbb{R}^n</math> of the given function is a [[bounded set]].

===Total variation in measure theory===

====Classical total variation definition====
Following {{Harvtxt|Saks|1937|p=10}}, consider a [[signed measure]] ''<math>\mu</math>'' on a [[sigma-algebra|measurable space]] <math>(X,\Sigma)</math>: then it is possible to define two [[set function]]s <math>\scriptstyle\overline{\mathrm{W}}(\mu,\cdot)</math> and <math>\scriptstyle\underline{\mathrm{W}}(\mu,\cdot)</math>, respectively called '''upper variation''' and '''lower variation''', as follows

:<math>\overline{\mathrm{W}}(\mu,E)=\sup\left\{\mu(A)|A\in\Sigma\text{ and }A\subset E \right\}\qquad\forall E\in\Sigma</math>
:<math>\underline{\mathrm{W}}(\mu,E)=\inf\left\{\mu(A)|A\in\Sigma\text{ and }A\subset E \right\}\qquad\forall E\in\Sigma</math>

clearly

:<math>\overline{\mathrm{W}}(\mu,E)\geq 0\geq \underline{\mathrm{W}}(\mu,E)\qquad\forall E\in\Sigma</math>

{{EquationRef|3|Definition 1.3.}} The '''variation''' (also called '''absolute variation''') of the signed measure ''<math>\mu</math>'' is the set function

:<math>|\mu|(E)=\overline{\mathrm{W}}(\mu,E)+\left|\underline{\mathrm{W}}(\mu,E)\right|\qquad\forall E\in\Sigma</math>

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

:<math>\|\mu\|=|\mu|(X)</math>

====Modern definition of total variation norm====
{{Harvtxt|Saks|1937|p=11}} uses upper and lower variations to prove the [[Hahn decomposition theorem|Hahn&ndash;Jordan decomposition]]: according to his version of this theorem, the upper and lower variation are respectively a [[non-negative]] and a [[non-positive]] [[Measure (mathematics)|measure]]. Using a more modern notation, define

:<math>\mu^+(\cdot)=\overline{\mathrm{W}}(\mu,\cdot)\,,</math>
:<math>\mu^-(\cdot)=-\underline{\mathrm{W}}(\mu,\cdot)\,,</math>

Then ''<math>\mu^+</math>'' and ''<math>\mu^-</math>'' are two non-negative [[measure (mathematics)|measure]]s such that

:<math>\mu=\mu^+-\mu^-\,</math>
:<math>|\mu|=\mu^++\mu^-\,</math>

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

====Total variation norm of complex measures====
If the measure ''<math>\mu</math>'' is [[Complex number|complex-valued]] i.e. is a [[complex measure]], its upper and lower variation cannot be defined and the Hahn&ndash;Jordan decomposition theorem can only be applied to its real and imaginary parts. However, it is possible to follow {{Harvtxt|Rudin|1966|pp=137&ndash;139}} and define the total variation of the complex-valued measure ''<math>\mu</math>'' as follows

{{EquationRef|4|Definition 1.4.}} The '''variation''' of the complex-valued measure ''<math>\mu</math>'' is the [[set function]]

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

where the [[supremum]] is taken over all partitions ''<math>\pi</math>'' of a [[measurable set]] ''<math>E</math>'' into a finite number of disjoint measurable subsets.

This definition coincides with the above definition ''<math>|\mu|=\mu^++\mu^-\,</math>'' for the case of real-valued signed measures.

====Total variation norm of vector-valued measures====

The variation so defined is a [[positive measure]] (see {{Harvtxt|Rudin|1966|p=139}}) and coincides with the one defined by {{EquationNote|3|1.3}} when ''<math>\mu</math>'' is a [[signed measure]]: its total variation is defined as above. This definition works also if ''<math>\mu</math>'' is a [[vector measure]]: the variation is then defined by the following formula

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

where the supremum is as above. Note also that this definition is slightly more general than the one given by {{Harvtxt|Rudin|1966|p=138}} since it requires only to consider ''finite partitions'' of the space ''<math>X</math>'': this implies that it can be used also to define the total variation on [[Sigma additivity|finite-additive measures]].

====Total variation of probability measures====
{{unreferenced section|date=May 2012}}
{{main|Total variation distance of probability 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 measure]]s, the '''[[total variation distance of probability measures]]''' can be defined as ''<math>\| \mu - \nu \|</math>'' where the norm is the total variation norm of signed measures.  Using the property that ''<math>(\mu-\nu)(X)=0</math>'', we eventually arrive at the equivalent definition

:<math>\|\mu-\nu\| = |\mu-\nu|(X)=\sup\left\{\,\left|\mu(A)-\nu(A)\right| : A\in \Sigma\,\right\}</math>

and its values are non-trivial. Informally, this is the largest possible difference between the probabilities that the two [[probability distribution]]s can assign to the same event. For a [[categorical distribution]] it is possible to write the total variation distance as follows

:<math>\delta(\mu,\nu) = \frac 1 2 \sum_x \left| \mu(x) - \nu(x) \right|\;.</math>

The total variational distance for categorical probability distributions is called [[statistical distance]]: sometimes, in the definition of this distance, the factor <math>\scriptstyle\frac 1 2</math> is omitted.

==Basic properties==

===Total variation of differentiable functions===
The total variation of a [[differentiable function]] <math>f</math> can be expressed as an [[integral]] involving the given function instead of as the [[supremum]] of the [[functional (mathematics)|functional]]s of definitions {{EquationNote|1|1.1}} and {{EquationNote|2|1.2}}.

====The form of the total variation of a differentiable function of one variable====
{{EquationRef|5|Theorem 1.}} The '''total variation''' of a [[differentiable function]] ''<math>f</math>'', defined on an [[interval (mathematics)|interval]] <math> [a , b] \subset \mathbb{R}</math>, has the following expression if f' is Riemann integrable

:<math> V^a_b(f) = \int _a^b |f'(x)|\mathrm{d}x</math>

====The form of the total variation of a differentiable function of several variables====
{{EquationRef|6|Theorem 2.}} Given a [[differentiable function]] '''<math>f</math>''' defined on a [[bounded set|bounded]] [[open set]] <math>\Omega \subseteq \mathbb{R}^n</math>, the '''total variation of <math>f</math>''' has the following expression

:<math>V(f,\Omega) = \int\limits_\Omega\left|\nabla f(x)\right|\mathrm{d}x</math>

Here <math>|.|</math> denotes the <math>l_2</math>-norm.

=====Proof=====
The first step in the proof is to first prove an equality which follows from the [[Gauss-Ostrogradsky theorem]].

=====Lemma=====
Under the conditions of the theorem, the following equality holds:
: <math> \int\limits_\Omega f\,\mathrm{div}\varphi = -\int_\Omega\nabla f\cdot\varphi </math>

======Proof of the lemma======
From the [[Gauss-Ostrogradsky theorem]]:
: <math> \int\limits_\Omega \text{div}\mathbf R = \int\limits_{\partial\Omega}\mathbf R\cdot \mathbf n </math>
by substituting <math>\mathbf R:= f\mathbf\varphi</math>, we have:

:<math> \int\limits_\Omega\text{div}\left(f\mathbf\varphi\right) =
\int\limits_{\partial\Omega}\left(f\mathbf\varphi\right)\cdot\mathbf n </math>
where <math>\mathbf\varphi </math> is zero on the border of <math>\Omega</math> by definition:
:<math> \int\limits_\Omega\text{div}\left(f\mathbf\varphi\right)=0</math>
:<math> \int\limits_\Omega \partial_{x_i} \left(f\mathbf\varphi_i\right)=0</math>
:<math> \int\limits_\Omega \mathbf\varphi_i\partial_{x_i} f + f\partial_{x_i}\mathbf\varphi_i=0</math>
:<math> \int\limits_\Omega f\partial_{x_i}\mathbf\varphi_i = - \int\limits_\Omega \mathbf\varphi_i\partial_{x_i} f </math>
:<math> \int\limits_\Omega f\text{div} \mathbf\varphi = - \int\limits_\Omega \mathbf\varphi\cdot\nabla f </math>

=====Proof of the equality=====
Under the conditions of the theorem, from the lemma we have:
:<math> \int\limits_\Omega f\text{div} \mathbf\varphi = - \int\limits_\Omega \mathbf\varphi\cdot\nabla f \leq \left| \int\limits_\Omega \mathbf\varphi\cdot\nabla f \right|\leq \int\limits_\Omega \left|\mathbf\varphi\right|\cdot\left|\nabla f\right|\leq \int\limits_\Omega \left|\nabla f\right| </math>
in the last part <math>\mathbf\varphi</math> could be omitted, because by definition its essential supremum is at most one.

On the other hand we consider <math>\theta_n:=\mathbb I_{\left[-N,N\right]}\frac{\nabla f}{\left|\nabla f\right|}</math> and <math>\theta^*_n</math> which is the up to <math>\varepsilon</math> approximation of <math>\theta</math> in <math> C^1_c</math> with the same integral. We can do this since <math> C^1_c</math> is dense in <math> L^1 </math>. Now again substituting into the lemma:

:<math>\lim\limits_{N\rightarrow\infty}\int\limits_\Omega f\text{div}\theta^*_n =
\lim\limits_{N\rightarrow\infty}\int\limits_\Omega\mathbb I_{\left[-N,N\right]}\nabla f\cdot\frac{\nabla f}{\left|\nabla f\right|}=
\lim\limits_{N\rightarrow\infty}\int\limits_{\mathbb I_{\left[-N,N\right]}} \nabla f\cdot\frac{\nabla f}{\left|\nabla f\right|} = \int\limits_\Omega\left|\nabla f\right|
</math>
This means we have a convergent sequence of <math>\int\limits_\Omega f\text{div}\mathbf\varphi</math> that tends to <math>\int\limits_\Omega\left|\nabla f\right|</math> as well as we know that <math>\int\limits_\Omega f\text{div}\mathbf\varphi \leq \int\limits_\Omega\left|\nabla f\right| </math>. q.e.d.

It can be seen from the proof that the supremum is attained when
: <math>\varphi\to \frac{-\nabla f}{\left|\nabla f\right|}.</math>

The [[Function (mathematics)|function]] ''<math>f</math>'' is said to be of [[bounded variation]] precisely if its total variation is finite.

===Total variation of a measure===
The total variation is a [[norm (mathematics)|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 measure|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 <math>\scriptstyle\varphi\colon \mathbb{R}\to \mathbb{R}</math> by
:<math>\varphi(t) = \mu((-\infty,t])~.</math>
Then, the total variation of the signed measure ''μ'' is equal to the total variation, in the above sense, of the function ''φ''. In general, the total variation of a signed measure can be defined using [[Hahn decomposition theorem|Jordan's decomposition theorem]] by
:<math>\|\mu\|_{TV} = \mu_+(X) + \mu_-(X)~,</math>
for any signed measure ''μ'' on a measurable space <math>(X,\Sigma)</math>.

== Applications ==
Total variation can be seen as a [[non-negative]] [[real number|real]]-valued [[functional (mathematics)|functional]] defined on the space of [[real number|real-valued]] [[function (mathematics)|function]]s (for the case of functions of one variable) or on the space of [[integrable function]]s (for the case of functions of several variables). As a functional, total variation finds applications in several branches of mathematics and engineering, like [[optimal control]], [[numerical analysis]], and [[calculus of variations]], where the solution to a certain problem has to [[Maxima and minima|minimize]] 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 to [[differential equation]]s. Applications of  total variation to this problems are detailed in the article "''[[total variation diminishing]]''"

* '''Image denoising''': in [[image processing]], denoising is a collection of methods used to reduce the [[Electronic noise|noise]] in an [[image]] reconstructed from data obtained by electronic means, for example [[data transmission]] or [[Sensor|sensing]]. [[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 {{Harv|Rudin|Osher|Fatemi|1992}} and {{Harv|Caselles|Chambolle|Novaga|2007}}. A sensible extension of this model to colour images, called Colour TV, can be found in {{Harv|Blomgren|Chan|1998}}.

== See also ==
* [[Bounded variation]]
* [[Total variation diminishing]]
* [[Total variation denoising]]
* [[Quadratic variation]]
* [[Total variation distance of probability measures]]
* [[Kolmogorov–Smirnov test]]

==Notes==
{{more footnotes|date=February 2012}}
{{Reflist|2}}

==Historical references==
*{{Citation
| last = Arzelà
| first = Cesare
| author-link = Cesare Arzelà
| title = Sulle funzioni di due variabili a variazione limitata (On functions of two variables of bounded variation)
| journal = Rendiconto delle sessioni della [[Reale Accademia delle scienze dell'Istituto di Bologna]]
| series = Nuova serie
| volume = IX
| issue = 4
| pages = 100–107
| language = Italian
| date = 7 May 1905
| url = http://www.archive.org/details/rendicontodelle04bologoog
| archiveurl = http://www.archive.org/stream/rendicontodelle04bologoog#page/n121/mode/2up
| archivedate = 2007-08-07
| doi =
| jfm = 36.0491.02
}}.
*{{springer
| title= Arzelà variation
| id= a/a013470
| last= Golubov
| first= Boris I.
}}.
*{{springer
| title= Fréchet variation
| id= f/f041400
| last= Golubov
| first= Boris I.
}}.
*{{springer
| title= Hardy variation
| id= h/h046400
| last= Golubov
| first= Boris I.
}}.
*{{springer
| title= Pierpont variation
| id= p/p072720
| last= Golubov
| first= Boris I.
}}.
*{{springer
| title= Vitali variation
| id= h/h046400
| last= Golubov
| first= Boris I.
}}.
*{{springer
| title= Tonelli plane variation
| id= t/t092990
| last= Golubov
| first= Boris I.
}}.
*{{springer
| title= Variation of a function
| id= V/v096110
| last= Golubov
| first= Boris I.
| last2= Vitushkin
| first2= Anatolii G.
| author2-link= Anatolii Georgievich Vitushkin
}}
*{{Citation
| last = Jordan
| first = Camille
| author-link = Camille Jordan
| title = Sur la série de Fourier
| journal = Comptes rendus hebdomadaires des séances de l'Académie des sciences
| language = French
| volume = 92
| pages = 228–230
| year = 1881
| url = http://gallica.bnf.fr/ark:/12148/bpt6k7351t/f227
| jfm = 13.0184.01
}} (available at [[Gallica]]). This is, according to Boris Golubov, the first paper on functions of bounded variation.
*{{Citation
| last = Hahn
| first = Hans
| author-link = Hans Hahn (mathematician)
| title = Theorie der reellen Funktionen
| place = Berlin
| publisher = [[Springer Verlag]]
| year = 1921
| language = German
| pages = VII+600
| url = http://www.archive.org/details/theoriederreelle01hahnuoft
| archiveurl = http://www.archive.org/details/theoriederreelle01hahnuoft
| archivedate = 2008-12-31
| doi =
| jfm = 48.0261.09
| isbn = 
}}.
* {{Citation
| last = Vitali
| first = Giuseppe
| author-link= Giuseppe Vitali
| title = Sui gruppi di punti e sulle funzioni di variabili reali (On groups of points and functions of real variables)
| journal = [[Atti dell'Accademia delle Scienze di Torino]]
| origyear = 17 dicembre 1907
| year = 1908
| volume = 43
| language = Italian
| pages = 75–92
| url = http://www.archive.org/details/attidellarealeac43real
| archiveurl = http://www.archive.org/stream/attidellarealeac43real#page/228/mode/2up
| archivedate = 2009-03-31
| jfm= 39.0101.05
}}. The paper containing the first proof of [[Vitali covering theorem]].

==References==
*{{Citation
| last = Adams
| first = C. Raymond
| last2 = Clarkson
| first2 = James A.
| title = On definitions of bounded variation for functions of two variables
| journal = [[Transactions of the American Mathematical Society]]
| volume = 35
| pages = 824–854
| year = 1933
| url = http://www.ams.org/journals/tran/1933-035-04/S0002-9947-1933-1501718-2/home.html
| doi = 10.1090/S0002-9947-1933-1501718-2
| jfm = 59.0285.01
| mr = 1501718 
| zbl = 0008.00602
}}.
*{{Citation
| last = Cesari
| first = Lamberto
| author-link = Lamberto Cesari
| title = Sulle funzioni a variazione limitata (On the functions of bounded variation)
| journal = [[Annali della Scuola Normale Superiore]]
| series = II
| volume = 5
| issue = 3-4
| language = Italian
| pages = 299&ndash;313
| date =
| year = 1936
| url = http://www.numdam.org/item?id=ASNSP_1936_2_5_3-4_299_0
| doi =
| jfm = 62.0247.03 
| mr = 1556778
| zbl = 0014.29605
}}. Available at [http://www.numdam.org Numdam].
*{{Citation
| last = Saks
| first = Stanisław
| author-link = Stanislaw Saks
| title = Theory of the Integral
| place = [[Warszawa]]-[[Lwów]]
| publisher = G.E. Stechert & Co.
| year = 1937
| series = [http://matwbn.icm.edu.pl/ksspis.php?wyd=10&jez=pl Monografie Matematyczne]
| volume = 7
| edition = 2nd
| pages = VI+347
| url = http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez=pl
| jfm = 63.0183.05 
| mr = 1556778
| zbl = 0017.30004
}}. (available at the [http://matwbn.icm.edu.pl/ Polish Virtual Library of Science]). English translation from the original French by [[Laurence Chisholm Young]], with two additional notes by [[Stefan Banach]].
*{{Citation
| last = Rudin
| first = Walter
| author-link = Walter Rudin
| title = Real and Complex Analysis
| place = New York
| publisher = [[McGraw-Hill]]
| series = McGraw-Hill Series in Higher Mathematics
| year = 1966
| edition = 1st
| pages = xi+412
| mr = 210528
| zbl = 0142.01701
| isbn = 
}}.

== External links ==
<!-- Most of these are not External links in the sense of [[WP:EL]] -->

'''One variable'''
* "[http://planetmath.org/encyclopedia/TotalVariation.html Total variation]" on [[Planetmath]].

'''One and more variables'''
*[http://www.encyclopediaofmath.org/index.php/Function_of_bounded_variation Function of bounded variation] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

'''Measure theory'''
*{{MathWorld
|author=Rowland, Todd
|title=Total Variation
|urlname=TotalVariation
}}.
*{{PlanetMath|urlname=JordanDecomposition|title=Jordan decomposition}}.
*[http://www.encyclopediaofmath.org/index.php/Jordan_decomposition_%28of_a_signed_measure%29 Jordan decomposition] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

=== Applications ===
*{{Citation
 | last = Caselles | first = Vicent | last2 = Chambolle | first2 = Antonin
 | last3 = Novaga | first3 = Matteo
 | title = The discontinuity set of solutions of the TV denoising problem and some extensions
 | url = http://cvgmt.sns.it/papers/caschanov07/
 | publisher = [[Society for Industrial and Applied Mathematics|SIAM]], Multiscale Modeling and Simulation, vol. 6 n. 3,
 | year = 2007}} (a work dealing with total variation application in denoising problems for [[image processing]]).

*{{Citation
 | last = Rudin | first = Leonid I. | last2 = Osher | first2 = Stanley
 | last3 = Fatemi | first3 = Emad 
 | title = Nonlinear total variation based noise removal algorithms
 | url = http://www.sciencedirect.com/science/article/pii/016727899290242F
 | publisher = Physica D: Nonlinear Phenomena 60.1: 259-268
 | year = 1992}}.

*{{Citation
 | last = Blomgren | first = Peter | last2 = Chan | first2 = Tony F. 
 | title = Color TV: total variation methods for restoration of vector-valued images
 | url = http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=661180&tag=1
 | publisher = Image Processing, IEEE Transactions on, vol. 7, no. 3: 304-309
 | year = 1998}}.

*[[Tony F. Chan]] and Jackie (Jianhong) Shen (2005), [http://jackieneoshen.googlepages.com/ImagingNewEra.html ''Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods''], [[Society for Industrial and Applied Mathematics|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).

{{DEFAULTSORT:Total Variation}}
[[Category:Mathematical analysis]]
[[Category:Probability theory]]