Total variation
In
Historical note
The concept of total variation for functions of one real variable was first introduced by
Definitions
Total variation for functions of one real variable
Definition 1.1. The total variation of a real-valued (or more generally complex-valued) function , defined on an interval is the quantity
where the
Total variation for functions of n > 1 real variables
Definition 1.2.
where
- is the continuously differentiable vector functions of compact supportcontained 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.
Total variation in measure theory
Classical total variation definition
Following Saks (1937, p. 10), consider a signed measure on a
clearly
Definition 1.3. 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.
Modern definition of total variation norm
Then and are two non-negative measures such that
The last measure is sometimes called, by abuse of notation, total variation measure.
Total variation norm of complex measures
If the measure 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 Rudin (1966, pp. 137–139) and define the total variation of the complex-valued measure as follows
Definition 1.4. The variation of the complex-valued measure is the set function
where the
This definition coincides with the above definition for the case of real-valued signed measures.
Total variation norm of vector-valued measures
The variation so defined is a
where the supremum is as above. This definition is slightly more general than the one given by Rudin (1966, p. 138) since it requires only to consider finite partitions of the space : this implies that it can be used also to define the total variation on
Total variation 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 where the norm is the total variation norm of signed measures. Using the property that , we eventually arrive at the equivalent definition
and its values are non-trivial. The factor 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
It may also be normalized to values in by halving the previous definition as follows
Basic properties
Total variation of differentiable functions
The total variation of a function can be expressed as an
.The form of the total variation of a differentiable function of one variable
Theorem 1. The total variation of a differentiable function , defined on an interval , has the following expression if is Riemann integrable
If is differentiable and monotonic, then the above simplifies to
For any differentiable function , we can decompose the domain interval , into subintervals (with ) in which is locally monotonic, then the total variation of over can be written as the sum of local variations on those subintervals:
The form of the total variation of a differentiable function of several variables
Theorem 2. Given a function defined on a bounded open set , with of class , the total variation of has the following expression
- .
Proof
The first step in the proof is to first prove an equality which follows from the
Lemma
Under the conditions of the theorem, the following equality holds:
Proof of the lemma
From the
by substituting , we have:
where is zero on the border of by definition:
Proof of the equality
Under the conditions of the theorem, from the lemma we have:
in the last part could be omitted, because by definition its essential supremum is at most one.
On the other hand, we consider and which is the up to approximation of in with the same integral. We can do this since is dense in . Now again substituting into the lemma:
This means we have a convergent sequence of that tends to as well as we know that . Q.E.D.
It can be seen from the proof that the supremum is attained when
The function is said to be of bounded variation precisely if its total variation is finite.
Total variation of a measure
The total variation is a
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 by
Then, the total variation of the signed measure μ is equal to the total variation, in the above sense, of the function . In general, the total variation of a signed measure can be defined using Jordan's decomposition theorem by
for any signed measure μ on a measurable space .
Applications
Total variation can be seen as a
- 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: in 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 (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).
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
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (February 2012) |
- ^ According to Golubov & Vitushkin (2001).
- ISBN 9780198502456.
- ^ Gibbs, Alison; Francis Edward Su (2002). "On Choosing and Bounding Probability Metrics" (PDF). p. 7. Retrieved 8 April 2017.
Historical references
- JFM 36.0491.02, archived from the originalon 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
- Gallica). This is, according to Boris Golubov, the first paper on functions of bounded variation.
- JFM 48.0261.09.
- Vitali covering theorem.
References
- Adams, C. Raymond; Clarkson, James A. (1933), "On definitions of bounded variation for functions of two variables", Zbl 0008.00602.
- Zbl 0014.29605. Available at Numdam.
- 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.
- Zbl 0017.30004.. (available at the Polish Virtual Library of Science). English translation from the original French by Laurence Chisholm Young, with two additional notes by Stefan Banach.
- Zbl 0142.01701.
External links
One variable
- "Total variation" on PlanetMath.
One and more variables
Measure theory
- Rowland, Todd. "Total Variation". MathWorld..
- Jordan decomposition at PlanetMath..
- Jordan decomposition at Encyclopedia of Mathematics
Applications
- Caselles, Vicent; Chambolle, Antonin; Novaga, Matteo (2007), The discontinuity set of solutions of the TV denoising problem and some extensions, image 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, .
- 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, PMID 18276250.
- 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).