Total Variation

Definition

The total variation (TV) is a nonnegative, convex, and lower semicontinuous functional on the space of integrable functions. For a function $u\in{{\mathcal L}^1_{\text{loc}}}(\Omega)$ on a domain $\Omega\subset{\mathbb R}{n}$, it is defined as

$$\begin{array}{rcl} J(u)&:=& \sup\left\{ -\int_\Omega u\cdot{\rm div}({\boldsymbol{\xi}}) {\,{\rm d}x} : {\boldsymbol{\xi}}\in{\mathcal C}_c^\infty(\Omega, {\mathbb R}{n}),\right.\nonumber\\ &&\left.\|{{\boldsymbol{\xi}}}\|_\infty \leq 1 \vphantom{\sup -\int_\Omega}\right\}. \end{array}$$

(1)

For differentiable functions $u\in{\mathcal C}^1(\Omega)$, this definition can be reduced to the familiar expression

$$ J(u) = \int_\Omega |{\nabla u}| {\,{\rm d}x}$$

(2)

with the help of Gauss' integral theorem. A function with J(u) < ∞ is called of bounded variation; the space of all functions on Ω with bounded variation is denoted by ${{\mathcal{BV}}(\Omega)}$.

Background

The total variation is a favorite prior term and regularizer in...