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Concept# Total variation

Summary

In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an 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 functions of bounded variation.
The concept of total variation for functions of one real variable was first introduced by Camille Jordan in the paper . 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.
The total variation of a real-valued (or more generally complex-valued) function , defined on an interval is the quantity
where the supremum runs over the set of all partitions of the given interval.
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
where
is the set of continuously differentiable vector functions of compact support contained in ,
is the essential supremum norm, and
is the divergence operator.
This definition does not require that the domain of the given function be a bounded set.
Following , consider a signed measure on a measurable space : then it is possible to define two set functions and , respectively called upper variation and lower variation, as follows
clearly
The variation (also called absolute variation) of the signed measure is the set function
and its total variation is defined as the value of this measure on the whole space of definition, i.e.
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.

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Signed measure

In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values. There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. Signed measures are usually only allowed to take finite real values, while some textbooks allow them to take infinite values. To avoid confusion, this article will call these two cases "finite signed measures" and "extended signed measures".

Bounded variation

In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value.

Total variation denoising

In signal processing, particularly , total variation denoising, also known as total variation regularization or total variation filtering, is a noise removal process (filter). It is based on the principle that signals with excessive and possibly spurious detail have high total variation, that is, the integral of the absolute is high. According to this principle, reducing the total variation of the signal—subject to it being a close match to the original signal—removes unwanted detail whilst preserving important details such as .

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