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94 4. A Variational approach for the destriping issue
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Figure 4.7 – Frequency response of the filter ˜H. As the value of λ decreases, only frequencies
close to the vertical axis of the fourier domain are reduced.
where u F , Is
F and H F are the fourier transform of u, I s and H. An estimate of the stripe
free true image u is obtained with a simple inverse fourier transform :
)
u = F −1 (
(ξ
2
x + λξ 2 yH F ).I F s
ξ 2 x + λξ 2 y
(4.83)
Remark : If we ignore the term H ⊗I s in (4.80) and denote ˜H a filter defined in the fourier
domain as :
ξx
˜H 2 =
ξx 2 + λξy
2 (4.84)
destriping via the variational model (4.80) is equivalent to filtering with ˜H. The fourier
response of the filter ˜H is illustrated in figure 4.7.
The destriping variational model (4.80) is applied to images extracted from Terra
MODIS band 27, 30 and 33 and the results are illustrated in figures 4.8 and 4.9. In the case
of homogeneous data, as in band 27, we observe a complete removal of the striping effect
without any visible blur in the destriped result. We point out that the overall blur, i.e the
image distortion index defined in section 3.8, depends on the lagrange multiplier λ, whch
choice is discussed further in section 4.6. Although the results on band 27 are satisfactory,
the destriping model does not perform as well on images containing strong discontinuities.
Figure 4.9, clearly shows local blurring artifacts along sharp edges at ocean-land and oceanclouds
transitions. This major limitation is, analogically to Tikhonov regularization, due
to the L 2 norm used in (4.80) which is not able to preserve sharp structures.
4.5 A unidirectional variational destriping model
From a variational perspective, we first tackled the striping issue with ROF total variation
regularization model. We also explored image decomposition models for their ability