Position of the problem

Rudin-Osher-Fatemi's model:

(Physica D. 1992)

Definition 1.1.  $BV(\Omega)$ is the subspace of functions $u$ in $L^{1}(\Omega)$ such that the following quantity is finite:

$$J(u) = \sup \left\{ \int_{\Omega} u(x) \textup{div\,}\xi (x) dx \... ... C_{c}^{1}(\Omega; {\mathbb{R}}^{2}), \Vert\xi \Vert _{\infty} \leq 1 \right \}$$ (1.1)

$BV(\Omega)$ embedded with the norm: $\Vert u\Vert _{BV}=\Vert u\Vert _{1}+J(u)$ is a Banach space.

Remark: if $u \in W^{1,1}(\Omega)$, then

$$J(u)=\int_{\Omega}\vert\nabla u\vert$$

In the ROF model, one seeks to minimize:

$$\inf_{(u,v) \in BV \times L^2 / f=u+v} \left( J(u) + \frac{1}{2 \lambda} \Vert v\Vert _{2}^{2} \right)$$ (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:

$$u=f- P_{\lambda K} (f)$$ (1.3)

where $P$ is the orthogonal projection on $\lambda K$, and where $K$ is the closure in $L^2$ of the set:

$$\{\textup{div\,}(g)/ g \in C_{c}^{1}(\Omega,{\mathbb{R}}^{2}), \Vert g\Vert _{\infty}\leq 1 \}$$ (1.4)

Meyer's model :

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

$$\inf_{(u,v) \in BV \times G / f=u+v} \left( J(u) + \lambda \Vert v\Vert _{G} \right)$$ (1.5)

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

Definition 1.2. $G$ is the Banach space composed of the distributions $v$ which can be written

$$v=\partial_{1}g_{1}+\partial_{2}g_{2}=\textup{div\,}(g)$$ (1.6)

with $g_{1}$ and $g_{2}$ in $L^{\infty}$.

$$\Vert v\Vert _{G} = \inf \left \{ \Vert g \Vert _{\infty} / v=\te... ..., \vert g(x)\vert=\sqrt{\vert g_{1}\vert^{2}+\vert g_{2}\vert^{2}}(x) \right \}$$ (1.7)

Exemple:

Images	$TV$	$L^2$	$G$
textured image	1 000 000	9 500	360
geometric image	64 600	9 500	2000

Figure 1: Images

Remarks:

Lemma 1.1.  $u \mapsto \frac{J(u)^2}{2}$ and $v \mapsto \frac{\Vert v\Vert _{G}^2}{2}$ are dual (in the sens of the Legendre-Fenchel duality).

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

$$X_{0} = \{ v \in X \ / \ \sum_{i,j} v_{i,j} = 0 \}$$ (1.8)