Total variation

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

{{EquationRef|1|Definition 1.1.}} The total variation of a real-valued (or more generally complex-valued) function $f$, defined on an interval $[a\; ,\; b]\; \backslash subset\; \backslash mathbb\{R\}$ is the quantity

:$V\_a^b(f)=\backslash sup\_\{\backslash mathcal\{P\}\}\; \backslash sum\_\{i=0\}^\{n\_P-1\}\; |\; f(x\_\{i+1\})-f(x\_i)\; |,$

where the supremum runs over the set of all partitions $\backslash mathcal\{P\}\; =\; \backslash left\backslash \{P=\backslash \{\; x\_0,\; \backslash dots\; ,\; x\_\{n\_P\}\backslash \}\; \backslash mid\; P\backslash text\{\; is\; a\; partition\; of\; \}\; [a,b]\; \backslash right\backslash \}$ of the given interval. Which means that .

{{citation needed section|date=September 2022}}

{{EquationRef|2|Definition 1.2.}}{{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 }} 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

:$V(f,\backslash Omega):=\backslash sup\backslash left\backslash \{\backslash int\_\backslash Omega\; f(x)\; \backslash operatorname\{div\}\; \backslash phi(x)\; \backslash ,\; \backslash mathrm\{d\}x\; \backslash colon\; \backslash phi\backslash in\; C\_c^1(\backslash Omega,\backslash mathbb\{R\}^n),\backslash \; \backslash Vert\; \backslash phi\backslash Vert\_\{L^\backslash infty(\backslash Omega)\}\backslash le\; 1\backslash right\backslash \},$

where

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

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

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

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

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

clearly

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

{{EquationRef|3|Definition 1.3.}} The variation (also called absolute variation) of the signed measure $\backslash mu$ is the set function

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

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

:$\backslash |\backslash mu\backslash |=|\backslash mu|(X)$

{{Harvtxt|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

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

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

Then $\backslash mu^+$ and $\backslash mu^-$ are two non-negative measures such that

:$\backslash mu=\backslash mu^+-\backslash mu^-$

:$|\backslash mu|=\backslash mu^++\backslash mu^-$

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

If the measure $\backslash mu$ 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 {{Harvtxt|Rudin|1966|pp=137–139}} and define the total variation of the complex-valued measure $\backslash mu$ as follows

{{EquationRef|4|Definition 1.4.}} The variation of the complex-valued measure $\backslash mu$ is the set function

:$|\backslash mu|(E)=\backslash sup\_\backslash pi\; \backslash sum\_\{A\backslash isin\backslash pi\}\; |\backslash mu(A)|\backslash qquad\backslash forall\; E\backslash in\backslash Sigma$

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

This definition coincides with the above definition $|\backslash mu|=\backslash mu^++\backslash mu^-$ for the case of real-valued signed 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 $\backslash mu$ is a signed measure: its total variation is defined as above. This definition works also if $\backslash mu$ is a vector measure: the variation is then defined by the following formula

:$|\backslash mu|(E)\; =\; \backslash sup\_\backslash pi\; \backslash sum\_\{A\backslash isin\backslash pi\}\; \backslash |\backslash mu(A)\backslash |\backslash qquad\backslash forall\; E\backslash in\backslash Sigma$

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 $X$: this implies that it can be used also to define the total variation on finite-additive 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 measures, the total variation distance of probability measures can be defined as $\backslash |\; \backslash mu\; -\; \backslash nu\; \backslash |$ where the norm is the total variation norm of signed measures. Using the property that $(\backslash mu-\backslash nu)(X)=0$, we eventually arrive at the equivalent definition

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

and its values are non-trivial. The factor $2$ 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

:$\backslash delta(\backslash mu,\backslash nu)\; =\; \backslash sum\_x\; \backslash left|\; \backslash mu(x)\; -\; \backslash nu(x)\; \backslash right|\backslash ;.$

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

:$\backslash delta(\backslash mu,\backslash nu)\; =\; \backslash frac\{1\}\{2\}\backslash sum\_x\; \backslash left|\; \backslash mu(x)\; -\; \backslash nu(x)\; \backslash right|${{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}}

The total variation of a $C^1(\backslash overline\{\backslash Omega\})$ function $f$ can be expressed as an integral involving the given function instead of as the supremum of the functionals of definitions {{EquationNote|1|1.1}} and {{EquationNote|2|1.2}}.

{{EquationRef|5|Theorem 1.}} The total variation of a differentiable function $f$, defined on an interval $[a\; ,\; b]\; \backslash subset\; \backslash mathbb\{R\}$, has the following expression if $f\text{'}$ is Riemann integrable

:$V\_a^b(f)\; =\; \backslash int\; \_a^b\; |f\text{'}(x)|\backslash mathrm\{d\}x$

If $f$ is differentiable and monotonic, then the above simplifies to

:$V\_a^b(f)\; =\; |f(a)\; -\; f(b)|$

For any differentiable function $f$, we can decompose the domain interval $[a,b]$, into subintervals $[a,a\_1],\; [a\_1,a\_2],\; \backslash dots,\; [a\_N,b]$ (with

:$$

\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}

{{EquationRef|6|Theorem 2.}} Given a $C^1(\backslash overline\{\backslash Omega\})$ function $f$ defined on a bounded open set $\backslash Omega\; \backslash subseteq\; \backslash mathbb\{R\}^n$, with $\backslash partial\; \backslash Omega$ of class $C^1$, the total variation of $f$ has the following expression

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

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:

: $\backslash int\_\backslash Omega\; f\backslash operatorname\{div\}\backslash varphi\; =\; -\backslash int\_\backslash Omega\backslash nabla\; f\backslash cdot\backslash varphi$

From the Gauss–Ostrogradsky theorem:

: $\backslash int\_\backslash Omega\; \backslash operatorname\{div\}\backslash mathbf\; R\; =\; \backslash int\_\{\backslash partial\backslash Omega\}\backslash mathbf\; R\backslash cdot\; \backslash mathbf\; n$

by substituting $\backslash mathbf\; R:=\; f\backslash mathbf\backslash varphi$, we have:

:$\backslash int\_\backslash Omega\backslash operatorname\{div\}\backslash left(f\backslash mathbf\backslash varphi\backslash right)\; =$

\int_{\partial\Omega}\left(f\mathbf\varphi\right)\cdot\mathbf n

where $\backslash mathbf\backslash varphi$ is zero on the border of $\backslash Omega$ by definition:

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

:$\backslash int\_\backslash Omega\; \backslash partial\_\{x\_i\}\; \backslash left(f\backslash mathbf\backslash varphi\_i\backslash right)=0$

:$\backslash int\_\backslash Omega\; \backslash mathbf\backslash varphi\_i\backslash partial\_\{x\_i\}\; f\; +\; f\backslash partial\_\{x\_i\}\backslash mathbf\backslash varphi\_i=0$

:$\backslash int\_\backslash Omega\; f\backslash partial\_\{x\_i\}\backslash mathbf\backslash varphi\_i\; =\; -\; \backslash int\_\backslash Omega\; \backslash mathbf\backslash varphi\_i\backslash partial\_\{x\_i\}\; f$

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

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

:$\backslash int\_\backslash Omega\; f\backslash operatorname\{div\}\; \backslash mathbf\backslash varphi$

= - \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 $\backslash mathbf\backslash varphi$ could be omitted, because by definition its essential supremum is at most one.

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

:$\backslash begin\{align\}$

&\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}

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

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

: $\backslash varphi\backslash to\; \backslash frac\{-\backslash nabla\; f\}\{\backslash left|\backslash nabla\; f\backslash right|\}.$

The function $f$ 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 $\backslash varphi\backslash colon\; \backslash mathbb\{R\}\backslash to\; \backslash mathbb\{R\}$ by

:$\backslash varphi(t)\; =\; \backslash mu((-\backslash infty,t])~.$

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

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

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

Total variation can be seen as a non-negative real-valued functional defined on the space of real-valued functions (for the case of functions of one variable) or on the space of integrable functions (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 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 equations. Applications of total variation to these problems are detailed in the article "total variation diminishing"
• Image denoising:https://arxiv.org/pdf/1603.09599 Retrieved 12/15/2024 in image processing, denoising is a collection of methods used to reduce the noise in an image reconstructed from data obtained by electronic means, for example data transmission or 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}}.

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

{{more footnotes|date=February 2012}}

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{{DEFAULTSORT:Total Variation}}

Category:Mathematical analysis