Editing Total variation

{{Short description|Measure of local oscillation behavior}}
{{distinguish|Total variation distance of probability measures}}
{{primary sources|date=February 2012}}

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'']. Functions whose total variation is finite are called ''[[Bounded variation|functions of bounded variation]]''.

==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_{\mathcal{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> \mathcal{P} = \left\{P=\{ x_0, \dots , x_{n_P}\} \mid P\text{ is a partition of } [a,b] \right\} </math> of the given [[interval (mathematics)|interval]]. Which means that <math>a = x_{0} < x_{1} <  ...  < x_{n_{P}} = b</math>.

===Total variation for functions of ''n'' > 1 real variables ===
{{citation needed section|date=September 2022}}
{{EquationRef|2|Definition 1.2.}}<ref name="10.1093/oso/9780198502456.001.0001">{{cite book |last1=Ambrosio |first1=Luigi |last2=Fusco |first2=Nicola |last3=Pallara |first3=Diego |title=Functions of Bounded Variation and Free Discontinuity Problems |date=2000 |publisher=Oxford University Press |isbn=9780198502456 |url=https://doi.org/10.1093/oso/9780198502456.001.0001}|pages=119|doi=10.1093/oso/9780198502456.001.0001 }}</ref> 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) \operatorname{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> 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>, 
* <math> \Vert\;\Vert_{L^\infty(\Omega)}</math> is the [[essential supremum]] [[Norm (mathematics)|norm]], and 
* <math>\operatorname{div}</math> is the [[divergence]] operator. 
This definition ''does not require'' that the [[Domain of a function|domain]] <math>\Omega \subseteq \mathbb{R}^n</math> of the given function be 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>\overline{\mathrm{W}}(\mu,\cdot)</math> and <math>\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)\mid 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)\mid 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 countable 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. 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)=2 \sup\left\{\,\left|\mu(A)-\nu(A)\right| : A\in \Sigma\,\right\}</math>

and its values are non-trivial. The factor <math>2</math> 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 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) = \sum_x \left| \mu(x) - \nu(x) \right|\;.</math>

It may also be normalized to values in  <math>[0, 1]</math> by halving the previous definition as follows

:<math>\delta(\mu,\nu) = \frac{1}{2}\sum_x \left| \mu(x) - \nu(x) \right|</math><ref>{{cite web|last1=Gibbs|first1=Alison|author2=Francis Edward Su|title=On Choosing and Bounding Probability Metrics|url=https://www.math.hmc.edu/~su/papers.dir/metrics.pdf|access-date=8 April 2017|pages=7|date=2002}}</ref>

==Basic properties==

===Total variation of differentiable functions===
The total variation of a <math>C^1(\overline{\Omega})</math>  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 <math>f'</math> is Riemann integrable

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

If <math> f</math> is differentiable and [[Monotonic function|monotonic]], then the above simplifies to
:<math> V_a^b(f) = |f(a) - f(b)|</math>

For any differentiable function <math>f</math>, we can decompose the domain interval <math>[a,b]</math>, into subintervals <math>[a,a_1], [a_1,a_2], \dots, [a_N,b]</math> (with <math>a<a_1<a_2<\cdots<a_N<b </math>) in which <math>f</math> is locally monotonic, then the total variation of <math> f</math> over <math>[a,b]</math> can be written as the sum of local variations on those subintervals:
:<math>
\begin{align}
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{align}</math>

====The form of the total variation of a differentiable function of several variables====
{{EquationRef|6|Theorem 2.}} Given a <math>C^1(\overline{\Omega})</math> function <math>f</math> defined on a [[bounded set|bounded]] [[open set]] <math>\Omega \subseteq \mathbb{R}^n</math>, with <math>\partial \Omega </math> of class <math>C^1</math>,  the '''total variation of <math>f</math>''' has the following expression

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

=====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_\Omega f\operatorname{div}\varphi = -\int_\Omega\nabla f\cdot\varphi </math>

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

:<math> \int_\Omega\operatorname{div}\left(f\mathbf\varphi\right) =
\int_{\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_\Omega\operatorname{div}\left(f\mathbf\varphi\right)=0</math>
:<math> \int_\Omega \partial_{x_i} \left(f\mathbf\varphi_i\right)=0</math>
:<math> \int_\Omega \mathbf\varphi_i\partial_{x_i} f + f\partial_{x_i}\mathbf\varphi_i=0</math>
:<math> \int_\Omega f\partial_{x_i}\mathbf\varphi_i = - \int_\Omega \mathbf\varphi_i\partial_{x_i} f </math>
:<math> \int_\Omega f\operatorname{div} \mathbf\varphi = - \int_\Omega \mathbf\varphi\cdot\nabla f </math>

=====Proof of the equality=====
Under the conditions of the theorem, from the lemma we have:
:<math> \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| </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]}\mathbb I_{\{\nabla f\ne 0\}}\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_N</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>\begin{align}
&\lim_{N\to\infty}\int_\Omega f\operatorname{div}\theta^*_N \\[4pt]
&= \lim_{N\to\infty}\int_{\{\nabla f\ne 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\ne 0\}}} \nabla f\cdot\frac{\nabla f}{\left|\nabla f\right|} \\[4pt]
&= \int_\Omega\left|\nabla f\right|
\end{align}</math>
This means we have a convergent sequence of <math display="inline">\int_\Omega f \operatorname{div} \mathbf\varphi</math> that tends to <math display="inline">\int_\Omega\left|\nabla f\right|</math> as well as we know that <math display="inline">\int_\Omega f\operatorname{div}\mathbf\varphi \leq \int_\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>\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 <math>\varphi</math>. 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 these problems are detailed in the article "''[[total variation diminishing]]''"
* '''Image denoising''':<ref>https://arxiv.org/pdf/1603.09599 Retrieved 12/15/2024</ref> 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]]
* [[p-variation]]
* [[Total variation diminishing]]
* [[Total variation denoising]]
* [[Quadratic variation]]
* [[Total variation distance of probability measures]]
* [[Kolmogorov–Smirnov test]]
* [[Anisotropic diffusion]]

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

==Historical references==
{{sfn whitelist|CITEREFGolubovVitushkin2001}}
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}}.
*{{springer
| title= Arzelà variation
| id= a/a013470
| last= Golubov
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}}.
*{{springer
| title= Fréchet variation
| id= f/f041400
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}}.
*{{springer
| title= Hardy variation
| id= h/h046400
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}}.
*{{springer
| title= Pierpont variation
| id= p/p072720
| last= Golubov
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}}.
*{{springer
| title= Vitali variation
| id= h/h046400
| last= Golubov
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}}.
*{{springer
| title= Tonelli plane variation
| id= t/t092990
| last= Golubov
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}}.
*{{springer
| title= Variation of a function
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}}
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| 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
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* {{Citation
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| volume = 43
| language = it
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| archive-url = https://archive.org/stream/attidellarealeac43real#page/228/mode/2up
| archive-date = 2009-03-31
| jfm= 39.0101.05
}}. The paper containing the first proof of [[Vitali covering theorem]].

==References==
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*{{Citation
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*{{Citation
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*{{Citation
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== 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
 |last1        = Caselles
 |first1       = 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
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20110927172158/http://cvgmt.sns.it/papers/caschanov07/
|archive-date = 2011-09-27
 }} (a work dealing with total variation application in denoising problems for [[image processing]]).

*{{Citation
 | last1 = Rudin | first1 = Leonid I. | last2 = Osher | first2 = Stanley
 | last3 = Fatemi | first3 = Emad 
 | title = Nonlinear total variation based noise removal algorithms
 | journal = Physica D: Nonlinear Phenomena | volume = 60 | issue = 1–4 | pages = 259–268 | publisher = Physica D: Nonlinear Phenomena 60.1: 259-268
 | year = 1992| doi = 10.1016/0167-2789(92)90242-F | bibcode = 1992PhyD...60..259R }}.

*{{Citation
 | last1 = Blomgren | first1 = Peter | last2 = Chan | first2 = Tony F. 
 | title = Color TV: total variation methods for restoration of vector-valued images
 | journal = IEEE Transactions on Image Processing | volume = 7 | issue = 3 | pages = 304–309 | publisher = Image Processing, IEEE Transactions on, vol. 7, no. 3: 304-309
 | year = 1998| bibcode = 1998ITIP....7..304B | doi = 10.1109/83.661180 | pmid = 18276250 }}.

*[[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''], [[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]]