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98 4. A Variational approach for the destriping issue Figure 4.10 – (Left) Noisy image from Terra MODIS band 30 (Center) Destriped with histogram matching (IMAPP) showing residual stripes (Right) Destriped with the UVDM illustrating complete removal of striping without any bluring or ringing artifacts Figure 4.11 – (Left) Noisy image from Terra MODIS band 33 (Center) Destriped with histogram matching (IMAPP) showing residual stripes in spite of noisy detectors being replaced with neighbors (Right) Image destriped with the UVDM showing efficient removal of radom striping without bluring or ringing artifacts these requirements dictates minimum distortion in the destriped image. The ID index in clearly dependent on the choice of the lagrange multiplier λ. The coefficient λ regulates the smoothness of the solution and its impact on the restored image can be interpreted as follows : A large value of λ may result in an oversmoothed solution, visually similar to what could be achieved with a low-pass filter, or a hard thresholding of wavelet coefficients in the lower scales. On the other hand, if λ is too small, the result might still contain noise. In fact, if we considerer the ROF model, its minimizer converges to the original noisy image f when λ → 0. In the case of the UVDM, when λ → 0, the solution converges to that of the following optimization problem : √ ∫ (∂(u ) − Is ) 2 E ɛ (u) = + ɛ ∂x 2 (4.95) Ω
99 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
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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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30 2. Remote Sensing with MODIS Mai
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38 2. Remote Sensing with MODIS
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40 3. Standard destriping technique
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Page 130 and 131: 130 BIBLIOGRAPHIE A. Chambolle. An
Page 132 and 133: 132 BIBLIOGRAPHIE E. Hochberg, S. A
Page 134 and 135: 134 BIBLIOGRAPHIE P. Perona and J.
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