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100 4. A Variational approach for the destriping issue
Figure 4.12 – Optimal destriping using Tadmor et al. hierarchical decomposition approach.
From left to right and top to bottom, noisy image from Terra band 30 and successive
destriped results for iterations 1 to 7. The value λ 0 = 10 have been selected to ensure
initial oversmoothing
This iterated process provides a multiscale representation of I s where the successive terms
u k extract details of the noise-free image I related to the scale λ 0 2 k , while sharper details
associated with stripe and included in the finer scale λ 0 2 k+1 are isolated in the term v k . The
original TNV hierachical decomposition was initially proposed as a multiscale framework
in the context of ROF regularization but does not aim at removing noise from the images.
In the later case, a stopping criteria is required so that the term ∑ j=k
j=0 u j does not contain
stripe details. This is discussed in section 4.6.3. TNV hierarchical decomposition is applied
to a striped image extracted from Terra MODIS band 30 and illustrated in figure 4.12.
The stripe noise extracted at each iteration is displayed in figure 4.13
4.6.2 Osher et al. iterative regularization method
In the orignal ROF model, the term f − u can still contain fine edges and textures if
the lagrange multiplier λ is not choosen carefully. To minimize the removal of structures
from the estimate solution, [Osher et al., 2005] introduced an iterative refinement of the
ROF model and proposed a generalization for other variational models based on the use of
Bregman distances. In the case of ROF regularization, the iterative methodology follows
three steps :
Step 1 : Solve the original ROF model :
{∫ ∫
u 1 =
inf
u∈BV (Ω)
|∇u| + λ
Ω
Ω
(f − u) 2 }
(4.101)