(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).