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88 4. A Variational approach for the destriping issue
We also refer to the variational model initially proposed for 1D signals in [] and later
explored by [Nikolova, 2004]. The proposed functional replaces the L 2 norm of the original
ROF model by the L 1 norm as :
inf
(u,v)∈BV (Ω) 1 (Ω)
(
)
TV(u)+λ‖f − (u + v)‖ 1 L 1 (Ω)
(4.68)
4.3.5 Experimental results and discussion
The original ROF model, Vese-Osher (VO) and Osher-Solé-Vese (OSV) decomposition
models have been applied to Terra MODIS images from band 30 and 33, in an attempt to
extract stripe noise in the texture component v. The u + v decompositions are illustrated
in figures 4.4 and 4.5. Traditionally, the lagrange multiplier λ is selected so that the L 2
norm of v is the same for every decomposition model. This strategy aims at visually
evaluating the ability of decomposition models to discriminate texture and noise from the
image main content and is well suited when no requirements are imposed on the cartoon
component u. In our case however, the primary goal is to isolate the striping on v while
preserving an acceptable distortion in u as it constitute an estimate of the destriped image.
Consequently, we proceed by selecting a value of λ that ensures approximatively the same
ID index for u for every decomposition model.
Figures 4.4 and 4.5 underscore a better ability of (VO) and (OSV) models compared to the
ROF model in terms of texture discrimation. In fact, it can be seen that the v component
of the ROF model also contains smooth structures related to ocean and clouds, and not
visible with VO and OSV decompositions. Nevertheless, despite a strong regularization
(ID=0.7), striping is still visible in the u-component of all three decomposition models.
Remarquably, we point out that the u-component derived from ROF model contains less
stripes than that obtained with (VO) and (OSV) models. This observation is in agreement
with the conlusion drawn from the wavelet analysis of section 3.7. Indeed, as a result of its
high intensity (in terms of gradient values), striping tends to be considered as an image
discontinuity and is therefore better preserved in the cartoon component u of texture
discriminating variational models.
For all three models, the presence of residual stripes despite oversmoothing, is a limitation
related to the contradictive compromise between the terms of the energy functionals. In
fact, in addition to the true image u being attached simply to the noisy image f (regardless
the norm used), the commonly used TV-norm reinforces the anisotropic preservation of
structures and does not distinguish strong gradient values related to striping from those
corresponding to edges. We recall that the initial motivation behind the use of variational
decomposition models is the textured aspect of striping due to its unidirectionality. This
feature however, is not accounted for neither in the fidelity term nor in the regularizing
term.
We shall see in the following sections how the introduction of directional information in
variational models offers a new perpective for the removal of stripe noise.