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102 4. A Variational approach for the destriping issue
the iterative procedure can be written as :
u k+1 = inf
u
{
TV x (u − f)+λT V y (u) −
∫
Ω
}
u∂J(u k )
(4.106)
In practice, we consider a smoothed version of TV y , and the subdifferential ∂J can be
replaced by the gradient of J(u) with respect to u. Equation (4.106) then becomes :
⎛ ⎛
⎞⎞
∫
u k+1 = inf TV x (u − f)+λT V y (u) − u ⎝λ ∂ ∂u k
⎝
∂y
√ ⎠⎠ (4.107)
u
Ω ∂y
( ∂u k
∂y )2 + ɛ
4.6.3 Stopping criteria
Whether using TVN decomposition or Osher et al. methodology, there exists an iteration
k, where the estimate û k is the closest to the true sripe free image. In both cases,
if k →∞, û k converges to the noisy image I s . In [Osher et al., 2005] the discrepancy
principle is used as a stopping rule. Assuming that the noise level δ is known, the iterative
procedure is stopped as soon as the residual term ‖û k − I s ‖ reaches a value of the same
order as δ. In our case, the stripe noise level is not known and we have to relie on another
approach. The unidirectionality of stripe noise can be exploited, again, to define a reliable
stopping criteria. The variational model (4.84), was designed in order to constrain the
regularization only to the direction of striping. Nevertheless, if the lagrange multiplier λ is
exessively high, the estimated solution will also be smoothed in the horizontal direction.
If we recall that the unidirectionality of striping translates as the horizontal gradients of
the noisy image I s and the true image I being of the same order, λ has to be chosen so
that :
∫
∂(û − f)
∣ ∂x ∣ ≤ ɛ (4.108)
Ω
where ɛ is a tolerance parameter that regulates the amount of distortion introduced in
û. In pratice, the difficulties related to the determination of ɛ can be easily overcome. In
fact, an optimal destriping is expected to preserve the ensemble averaged power spectrum
down the lines of the noisy image because striping only affects one direction. This means
that the spectral distribution of the information averaged accross the swath should be
approximatively the same for the striped image and the estimated true scene. Conveniently,
such measure is provided by the Image distortion index (ID) ; We recall that the ID reflects
a spectral fidelity between the destriped and original signals in the direction orthogonal to
striping. As TVN decomposition and Osher et al. iterative procedures result in a sequence
of solutions {u k } that converges to an image û = I s + C with TV x (C) = 0, we have :
lim
k→∞ ID(u k)=1 (4.109)
A stopping criteria can then be established using a threshold value, ID thres = 0.95.
This threshold was determined heuristically to ensure simultaneously complete removal