Total Variation

The total variation of a complex measure is the positive measure which measures the size of without cancellation. In particular, on a measurable set is the supremum of sums of "variations" for any subdivision of . Roughly speaking, a total variation measure is an infinitesimal version of the absolute value.

More precisely,

(1)

where the supremum is taken over all finite partitions of into measurable subsets .

Note that may not be the same as . When already is a positive measure, then . More generally, if is absolutely continuous, that is

(2)

then so is , and the total variation measure can be written as

(3)

The total variation measure can be used to rewrite the original measure, in analogy to the norm of a complex number. The measure has a polar representation

(4)

with .

For a real-valued function on , total variation is defined by

(5)

where the supremum is over all finite partitions of the interval. A function has bounded variation when this quantity is finite. If is continuously differentiable, then

(6)

For a differentiable image on a planar domain , the isotropic total variation is

(7)

Penalizing this quantity favors images that are nearly constant in regions while permitting sharp edges (Rudin et al. 1992). Total variation minimization is therefore used as a regularizer in compressed sensing, including reconstruction of undersampled magnetic resonance images (Lustig et al. 2007).