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

# Functional:

We want to solve the problem:

 (2.9)

where

 (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

 (2.11)

The solution of (2.11) is given by:

 (2.12)

where is the orthogonal projection on .

being fixed, one solves

 (2.13)

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 .

 (2.19)

where

 (2.20)

We want to find:

 (2.21)

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:

 (2.22)

Our limit problem is:

 (2.23)

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:

 (2.24)

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).

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