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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.
Figure 4.4 – Decomposition of the image from Terra MODIS band 30 as u (Left) + v (Right). λ is selected so that ID(u) is the same for the three models. From top to bottom, TV model (NR=4.56, ID=0.739), VO model (NR=4.23, ID=0.735) and OSV model (NR=2.47, ID=0.735) 89
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École Doctorale d’Informatique,
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5 Résumé Nou abordons dans cette
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8 TABLE DES MATIÈRES 3.5 Frequency
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10 TABLE DES MATIÈRES
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12 1. Introduction A common issue f
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14 1. Introduction
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16 2. Remote Sensing with MODIS In
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18 2. Remote Sensing with MODIS Fig
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24 2. Remote Sensing with MODIS any
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