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suivant: Functional: monter: demosar précédent: demosar

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:

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

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

In the ROF model, one seeks to minimize:

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

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

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

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

Meyer's model :

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

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

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

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

$\displaystyle \Vert v\Vert _{G} = \inf \left \{ \Vert g \Vert _{\infty} / v=\te...
..., \vert g(x)\vert=\sqrt{\vert g_{1}\vert^{2}+\vert g_{2}\vert^{2}}(x) \right \}$ (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
\begin{figure}\begin{center}
\begin{tabular}{cc}
\par textured image & geometric...
...igure=geometrique.PS,scale=0.5}\\
\par\end{tabular}\par\end{center}\end{figure}

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:

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


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