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which is not unique. Indeed, the solution of (4.95) is the set {u ∈ BV (Ω)/u = f +
C, ∫ ∂C
Ω ∂x
=0} which corresponds to the set of images that differs from f on a constant
per line. Due to the variations of stripe noise along the x-direction, any solution of (4.95)
will still display residual stripes. We are then lead to the question of how to determine a
value of λ that removes all the stripe noise while maintainin the image distortion index
close to 1.
4.6.1 Tadmor-Nezzar-Vese (TNV) hierarchical decomposition
In [Tadmor et al., 2004], the authors introduce a multiscale image representation based
on a hierarchical adaptive decomposition. The ROF model is iteratively solved to generate
a sequence of solutions which sum converges to the original image. We propose a
modest adaptation of this strategy to our variational destriping model. Let us consider
the following decomposition for a striped image I s :
[u 0 ,v 0 ]= argmin
(u,v)/u+v=I s
TV x (v)+λ 0 TV y (u)
(4.96)
If the inital value λ 0 is not too small, then u 0 can be considered as a cartoon approximation
of the true stripe-free image I, while the component v 0 mainly contains stripe noise.
Following Tadmor et al.’s remark that a texture at a scale λ contains edges at a refined
scale λ 2 , the noisy component v 0 can also be decomposed using the same variational model :
[u 1 ,v 1 ]= argmin
(u,v)/u+v=v 0
TV x (v)+ λ 0
2 TV y(u) (4.97)
The previous decomposition can be seen as a dyadic refinement to the approximation u 0
and can be iterated as follows :
[u k ,v k ]= argmin
(u,v)/u+v=v k−1
TV x (v)+ λ 0
2 k TV y(u) (4.98)
leading to a simple multilayered representation of the original noisy image I s :
I s = u 0 + v 0
= u 0 + u 1 + v 1
= u 0 + u 1 + u 2 + v 2
(4.99)
= ...
= u 0 + u 1 + u 2 + u 3 + ... + u k + v k
The previous expansion translates the hierarchical decomposition of I s :
j=k
∑
lim u j = I s (4.100)
k→∞
j=0