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80 4. A Variational approach for the destriping issue
To simplifie notations, we introduce :
1
C 1 = √
(D +x u n i,j )2 +(D 0y u n i,j )2 + ɛ 2
1
C 2 = √
(D −x u n i,j )2 +(D 0y u n i−1,j )2 + ɛ 2
(4.30)
1
C 3 = √
(D 0x u n i,j )2 +(D +y u n i,j )2 + ɛ 2
1
C 4 = √
(D 0x u n i,j−1 )2 +(D −y u n i,j )2 + ɛ 2
The solution of (4.24) is obtained with the following iterative scheme :
u n+1
i,j
= 2λh2 f i,j + C 1 u n i+1,j + C 2u i−1,j + C 3 u n i,j + C 4u n i,j−1
2λh 2 + C 1 + C 2 + C 3 + C 4
(4.31)
To satisfy Neumann boundary condition ∂u
∂n = 0, u i,j is extended by reflection outside the
domain Ω.
Since it’s introduction in 1992, total variation regularization has grown very popular in the
field of image processing. Nevertheless, for denoising purposes, its application is limited
to the removal of isotropic noises such as gaussian or speckle noise [Sheng et al., 2005].
Given the unidirectionality of stripe noise, tackling the striping issue with TV regularization
might not be appropriate. The wavelet analysis conducted in chapter 2 underscored
another geometrical feature of striping ; on most of MODIS emisssive bands, the amplityde
of stripe noise is of the same order as the edges of the image. Discontinuities due to
striping are perceived as image sharp structures and are therefore preserved by the TV
model. Distinction between stripe noise and image edges can not be achieved directly with
the TV model and regardless the choice of the lagrange multiplier λ, reduction of striping
effect is inevitably followed by a loss of contrast (figure 4.2).
Nonetheless, an alternative use of ROF model is conceivable. Recently, a variational approach
was proposed in the context of image destriping and inpainting. It is based on a
maximum a posteriori (MAP) algorithm applied to a modified image formation model
where the noisy observation f is given by :
f = Au + B + n (4.32)
where Au is a point to point multiplication. The degradation process is assumed to be
linear and includes the parameters A and B associated with gain and offset values for
every pixel. A and B are matrices of the same size as the image. n is assumed to be zero
mean gaussian noise. The true image u is deduced from a MAP estimate :
û = argmax
u
p(u|f) (4.33)