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97
Using the same notations as in sections 4.2, equation (4.90) can be discretized as :
(
) (
)
D +x (u i,j − f i,j )
D +y u i,j
D −x √ + λD −y √ =0
(D+x (u i,j − f i,j )) 2 + ɛ 2 (D+y u i,j ) 2 + ɛ 2
(
)
(u i+1,j − u i,j − f i+1,j + f i,j )
√ − (u i,j − u i−1,j − f i,j + f i−1,j )
√
(D+x (u i,j − f i,j )) 2 + ɛ 2 (D−x (u i,j − f i,j )) 2 + ɛ 2
+λ
(
)
u i,j+1 − u i,j
√
(D+y u i,j ) 2 + ɛ − u i,j − u i,j−1
√ =0
2 (D−y u i,j ) 2 + ɛ 2
(4.91)
We introduce the following linearization :
⎛
⎞
⎝ (un i+1,j − un+1 i,j
− f i+1,j + f i,j )
√
− (un+1 i,j
− u n i−1,j − f i,j + f i−1,j )
√
⎠
(D +x (u n i,j − f i,j)) 2 + ɛ 2 (D −x (u n i,j − f i,j)) 2 + ɛ 2
⎛
⎞
+λ ⎝
un i,j+1 − un+1 i,j
√
− un+1 i,j
− u n i,j−1
√
⎠ =0
(D +y u n i,j )2 + ɛ 2 (D −y u n i,j )2 + ɛ 2
(4.92)
If we denote :
C 1 =
C 2 =
C 3 =
C 4 =
1
√
(D +x (u n i,j − f i,j)) 2 + ɛ 2
1
√
(D −x (u n i,j − f i,j)) 2 + ɛ 2
(4.93)
1
√
(D +y u n i,j )2 + ɛ 2
1
√
(D −y u n i,j )2 + ɛ 2
The destriped image is obtained with a fixed point iterative scheme :
u n+1
i,j
= C 1(u n i+1,j − f i+1,j + f i,j )+C 2 (u n i−1,j + f i,j − f i−1,j )+λC 3 u n i,j+1 + λC 4u n i,j−1
C 1 + C 2 + λC 3 + λC 4
(4.94)
4.6 Optimal regularization
The limitations of standard destriping techniques discussed at the end of chapter 3,
were used as a starting point to establish the requirements of optimal destriping. One of