(2.10) |

*The parameter controls the norm of the residual .
The parameter controls the norm of . The larger is, the more contains information.
*

**Remark:**

*We have
.
*

**Principal:**
We solve the two following problems:

* being fixed, one solves
*

*The solution of (2.11) is given by:
*

(2.12) |

* being fixed, one solves
*

*The solution to(2.13) is given by:
*

(2.14) |

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

*1) Initialization:
*

(2.15) |

*2) Iterations:
*

(2.16) |

(2.17) |

*3) Stopping test:
we stop when
*

(2.18) |

**Discretization:**

*Our image is a two dimensional vector of size
. We denote by the Euclidean space
.
*

*We want to find:
*

* Lemme
2.2. There exists a unique couple
minimizing on
.*

**Convergence of the algorithm:**

* Proposition
2.3. The sequence
converges to the minimum of on
.*

**Recall of Meyer's problem:**
We thus consider:

**Link with Meyer's problem:**

*Our limit problem is:
*

*
Let us set
in Meyer's problem (2.22). Then we can choose
so that Meyer's problem (2.22) and our limit problem (2.23) have the same solutions.
*

**Role of :**

*We recall that our problem is:
*

*
Let us denote by
the solution of our problem (2.24).
Then
converges to
(when goes to 0) solution of our limit problem
(2.23).
*