![]() |
(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).