(Physica D. 1992)

**Definition
1.1. **
is the subspace of functions in
such that the following quantity is finite:

(1.1) |

embedded with the norm: is a Banach space.

Remark: if , then

In the ROF model, one seeks to minimize:

**Chambolle's model:**
A. Chambolle has proposed a projection algorithm to minimize the total variation (MIA 2002).

**Proposition
1.1. ***The solution of (1.2) is given by:
*

**Meyer's model :**

*Y. Meyer (2001) has proposed the following model:
*

**The Banach space contains signals signals with strong oscillations, and thus in particular textures and noise.**

* Definition
1.2. * is the Banach space composed of the distributions which can be written

(1.6) |

with and in .

(1.7) |

**Exemple:**

Images | |||

textured image | 1 000 000 | 9 500 | 360 |

geometric image | 64 600 | 9 500 | 2000 |

**Remarks:**

**Lemma
1.1. ***
and
are dual (in the sens of the Legendre-Fenchel duality).*

* Proposition
1.2. In the discrete case, the space identifies with the following subspace:
*

(1.8) |