(Physica D. 1992)
Definition
1.1.
is the subspace of functions
in
such that the following quantity is finite:
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(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
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(1.6) |
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(1.7) |
Exemple:
Images | ![]() |
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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:
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(1.8) |