Th`ese Marouan BOUALI - Sites personnels de TELECOM ParisTech

86 4. A Variational approach for the destriping issue
Using the same notations for C 1 , C 2 , C 3 , C 4 as in section 4.2, estimates for u, g 1 and g 2
are obtained via the fixed point iterative scheme :
(
) (
u n+1
1
i,j
=
1+ 1
f i,j − gn 1,i+1,j − gn 1,i−1,j
(C
2λh 2 1 + C 2 + C 3 + C 4 )
2h
− gn 2,i,j+1 − gn 2,i,j−1
+ 1
)
2h 2λh 2 (C 1u n i+1,j + C 2 u n i−1,j + C 3 u n i,j+1 + C 4 u n i,j−1)
⎛
g1,i,j n+1 = ⎝
4λ
h 2 µ
⎞
(
2λ
u
n
√
⎠ i+1,j − u n i−1,j
g1,i,j n 2 + g2,i,j n 2 2h
− f i+1,j − f i−1,j
2h
+ gn 1,i+1,j − gn 1,i−1,j
h 2 + 1
4h 2 (g 2,i+1,j+1 + g 2,i−1,j−1 − g 2,i+1,j−1 − g 2,i−1,j+1 )
⎞
(
⎝
2λ
u
n
√
⎠ i,j+1 − u n i,j−1
− f i,j+1 − f i,j−1
g1,i,j n 2 + g2,i,j n 2 2h
2h
⎛
g2,i,j n+1 =
4λ
h 2 µ
+ gn 2,i,j+1 − gn 2,i,j−1
h 2 + 1
4h 2 (g 1,i+1,j+1 + g 1,i−1,j−1 − g 1,i+1,j−1 − g 1,i−1,j+1 )
4.3.3 Osher-Solé-Vese’s Model
)
)
(4.56)
The model proposed by Osher, Solé and Vese in [Osher et al., 2002], provides another
practical approximation of (4.45). If the texture component can be written as v = f − u =
div(⃗g) with ⃗g ∈ L ∞ (Ω) 2 then the Hodge decomposition of ⃗g leads to :
⃗g = ∇P + ⃗ Q (4.57)
where ⃗ Q is a divergence free vector. Considering the divergence of the previous equation,
the texture component f − u can be written as :
f − u = div(⃗g) =div(∇P + ⃗ Q)=div(∇P )=△P (4.58)
From the previous equation, P can be expressed as P = △ −1 (f − u). Osher et al. then
suggest to replace the L ∞ -norm by the L 2 -norm of |⃗g|. Neglecting Q ⃗ in the Hodge decomposition
of ⃗g, Osher et al. consider the simple minimization problem :
∫ ∫
inf
u∈BV (Ω)
Ω
|∇u| + λ
inf
u∈BV (Ω)
Ω
Ω
|∇(△ −1 )(f − u)| 2 dx dy (4.59)
Using the norm in the space H −1 defined by ‖v‖ H −1 = ∫ Ω |∇(△−1 )(v)| 2 dx dy, the previous
minimization problem can be simplified to :
∫
|∇u| + λ‖f − u‖ 2 H−1 (4.60)