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The minimization of (4.60) is obtained with the Euler-Lagrange equation :
( ) ∇u
2λ △ −1 (f − u) =div
|∇u|
which takes a pratical form after application of the Laplacian operator :
( ( )) ∇u
2λ(f − u) =△ div
|∇u|
(4.61)
(4.62)
Equation (4.62) can be solved using a fixed step gradient descent. The estimation of u at
iteration n + 1 is given by :
( ( ))) ∇u
u n+1 = u n − ∆ t
(2λ(f − u n n
) −△ div
|∇u n |
(4.63)
u 0 = f
It is interesting to point out that Osher-Solé-Vese model is very similar to the particular
case of Vese-Osher model where p = 2 and λ = ∞. In fact, the space G p (Ω) coincides
with the Sobolev space W −1,p (Ω) which is the dual of W 1,p′
0 (Ω) where p and p ′ satisfie the
relationship 1 p + 1 p
= 1. If p = 2, then the texture component v lies in G ′ 2 (Ω) = W −1,2 (Ω) =
H −1 (Ω). The decomposition model of Vese and Osher can then be written as :
inf |u| BV (Ω) + λ‖f − (u + v)‖ 2 L 2 (Ω) + µ ‖v‖ H −1 (Ω) (4.64)
(u,v)∈BV (Ω)×H −1 (Ω)
4.3.4 Other u + v models
More image decomposition models were introduced following [Vese and Osher, 2002]
and [Osher et al., 2002]. In [Aujol et al., 2005], the following energy is considered :
(
)
TV(u)+λ‖f − (u + v)‖ 2 L 2 (Ω)
(4.65)
inf
(u,v)∈BV (Ω)×G µ(Ω)
where the space G µ is defined as :
G µ = {v ∈ G(Ω)/‖v‖ G ≤ µ)} (4.66)
The minimization of (4.66) is obtained with the projection algorithm of Chambolle.
[Daubechies and Teschke, 2005] introduced an image decomposition model based on a
wavelet framework as :
inf
(u,v)∈B 1 1 (L1 (Ω))×H −1 (Ω)
(
)
2α|u| B 1
1 (L 1 (Ω)) + ‖f − (u + v)‖2 L 2 (Ω) + ‖‖2 H −1 (Ω)
(4.67)
The space BV (Ω) used in most image decomposition models is replaced by a space suited
for wavelet coefficients, namely the Besov space B 1 1 (Ω).