08 June 2026

Bounded Variation: The Mathematical Foundation of TV Regularization

A collection of useful preliminary definitions etc.

mathimaginginverse-problems

Bounded Variation (BV): Definitions and Key Facts

Definition of Bounded Variation

Let be open and let .

The total variation of in is defined by

We say that has bounded variation if

The space of functions of bounded variation is

Distributional Characterization

A function belongs to if and only if its distributional derivative can be represented by a finite vector-valued Radon measure.

More precisely,

where

The distributional derivative is defined by the integration-by-parts identity

Writing

each component is a signed Radon measure.

Total Variation Measure

The total variation measure associated with is denoted by .

For every Borel set ,

The measure is positive and records the magnitude of the distributional derivative.

Fundamental Identity

A key result is

Indeed,

so the definition of total variation coincides exactly with the definition of the total variation measure evaluated on .

Relation to Sobolev Spaces

Every Sobolev function belongs to :

If , then the distributional derivative is absolutely continuous with respect to Lebesgue measure and

where denotes -dimensional Lebesgue measure.

The inclusion is strict:

A classical example of a function in but not in is the Heaviside function.

Useful Facts

  • if and only if its distributional derivative is a finite -valued Radon measure.

  • The derivative of a function need not be a function; it may contain singular parts concentrated on lower-dimensional sets.

  • Functions in are allowed to have jump discontinuities.

  • The total variation of a function is exactly the mass of the measure :

  • Every Sobolev function belongs to :

  • The inclusion is strict:

  • The Heaviside function belongs to but not to .

Geometric Intuition

Bounded variation measures the total amount of oscillation of a function.

Unlike Sobolev spaces, permits jump discontinuities while still retaining a meaningful notion of derivative through Radon measures. This makes the natural setting for:

  • functions with jumps,
  • image processing and total variation regularization,
  • sets of finite perimeter,
  • geometric measure theory,
  • conservation laws and weak solutions of PDEs.