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suivant: Numerical results: monter: demosar précédent: Position of the problem


We want to solve the problem:

$\displaystyle \inf_{(u,v) \in BV \times G_{\mu}} \left( J(u) + \frac{1}{2 \lambda} \Vert f-u-v\Vert _{2}^{2} \right)$ (2.9)


$\displaystyle G_{\mu}=\left\{ v \in G / \Vert v\Vert _{G}\leq \mu \right\}$ (2.10)

The parameter $ \lambda$ controls the $ L^2$ norm of the residual $ f-u-v$. The parameter $ \mu$ controls the $ \vert\vert.\vert\vert _{G}$ norm of $ v$. The larger $ \mu$ is, the more $ v$ contains information.


We have $ G_{\mu}= \mu K$.

Principal: We solve the two following problems:

$ \bullet$ $ v$ being fixed, one solves

$\displaystyle \inf_{u \in BV } \left( J(u) + \frac{1}{2 \lambda} \Vert f-u-v\Vert _{2}^{2} \right)$ (2.11)

The solution of (2.11) is given by:

$\displaystyle \hat{u}=f-v-P_{G_{\lambda}}(f-v)$ (2.12)

where $ P_{G_{\lambda}}$ is the orthogonal projection on $ G_{\lambda}$.

$ \bullet$ $ u$ being fixed, one solves

$\displaystyle \inf_{v \in G_{\mu}} \Vert f-u-v\Vert _{2}^{2}$ (2.13)

The solution to(2.13) is given by:

$\displaystyle \hat{v}=P_{G_{\mu}}(f-u)$ (2.14)

Algorithm: To solve problem (2.9), we iteratively solve problems (2.11) and (2.13).

1) Initialization:

$\displaystyle u_{0}=v_{0}=0$ (2.15)

2) Iterations:

$\displaystyle v_{n+1}=P_{G_{\mu}}(f-u_{n})$ (2.16)

$\displaystyle u_{n+1}=f-v_{n+1}-P_{G_{\lambda}}(f-v_{n+1})$ (2.17)

3) Stopping test: we stop when

$\displaystyle \max(\vert u_{n+1}-u_{n}\vert,\vert v_{n+1}-v_{n}\vert)\leq \epsilon$ (2.18)


Our image is a two dimensional vector of size $ N \times N$. We denote by $ X$ the Euclidean space $ {\mathbb{R}}^{N \times N}$.

$\displaystyle F(u,v)= \frac{1}{2 \lambda} \Vert f-u-v\Vert _{2}^{2}+J(u)+J^{*}\left(\frac{v}{\mu}\right)$ (2.19)


$\displaystyle J^{*}\left(\frac{v}{\mu}\right)=\left\{ \begin{array}{ll} 0 & \mb...
...$\Vert v\Vert _{G}\leq \mu$} \\ + \infty & \mbox{otherwise} \end{array} \right.$ (2.20)

We want to find:

$\displaystyle \inf_{(u,v) \in X \times X} F(u,v)$ (2.21)

Lemme 2.2. There exists a unique couple $ (\hat{u},\hat{v}) \in X \times G_{\mu}^{d}$ minimizing $ F$ on $ X \times X$.

Convergence of the algorithm:

Proposition 2.3. The sequence $ F(u_{n}, v_{n})$ converges to the minimum of $ F$ on $ X \times X$.

Recall of Meyer's problem: We thus consider:

$\displaystyle \inf_{(u,v) \in X \times G^{d} / f=u+v} J(u)+ \alpha \Vert v\Vert _{G^{d}}$ (2.22)

Link with Meyer's problem:

Our limit problem is:

$\displaystyle \inf_{(u,v) \in X \times X / f=u+v} J(u)+J^{*}\left(\frac{v}{\mu}\right)$ (2.23)

$ \Longrightarrow$ Let us set $ \alpha > 0$ in Meyer's problem (2.22). Then we can choose $ \mu \geq 0$ so that Meyer's problem (2.22) and our limit problem (2.23) have the same solutions.

Role of $ \lambda$:

We recall that our problem is:

$\displaystyle \inf_{(u,v) \in X \times X} \frac{1}{2 \lambda} \Vert f-u-v\Vert _{2}^{2}+J(u)+J^{*}\left(\frac{v}{\mu}\right)$ (2.24)

$ \Longrightarrow$ Let us denote by $ (u_{\lambda}, v_{\lambda})$ the solution of our problem (2.24). Then $ (u_{\lambda}, v_{\lambda})$ converges to $ (u_{0}, v_{0}) \in X \times X$ (when $ \lambda$ goes to 0) solution of our limit problem (2.23).

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suivant: Numerical results: monter: demosar précédent: Position of the problem
Jean-Francois.Aujol 2003-06-30