suivant: Functional: monter: demosar précédent: demosar

# Position of the problem

Rudin-Osher-Fatemi's model:

(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:

 (1.2)

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:

 (1.3)

where is the orthogonal projection on , and where is the closure in of the set:

 (1.4)

Meyer's model :

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

 (1.5)

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)

suivant: Functional: monter: demosar précédent: demosar
Jean-Francois.Aujol 2003-06-30