(Physica D. 1992)
Definition 1.1. is the subspace of functions in such that the following quantity is finite:
(1.1) |
Remark: if , then
In the ROF model, one seeks to minimize:
Chambolle's model: A. Chambolle has proposed a projection algorithm to minimize the total variation (MIA 2002).
Proposition 1.1. The solution of (1.2) is given by:
Meyer's model :
Y. Meyer (2001) has proposed the following model:
The Banach space contains signals signals with strong oscillations, and thus in particular textures and noise.
Definition 1.2. is the Banach space composed of the distributions which can be written
(1.6) |
(1.7) |
Exemple:
Images | |||
textured image | 1 000 000 | 9 500 | 360 |
geometric image | 64 600 | 9 500 | 2000 |
Remarks:
Lemma 1.1. and are dual (in the sens of the Legendre-Fenchel duality).
Proposition 1.2. In the discrete case, the space identifies with the following subspace:
(1.8) |