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74 4. A Variational approach for the destriping issue The previous PDE, also known as the heat equation, translates an isotropic diffusion process where all directions are smoothed identically. The isotropic property of the heat equation is a major drawback for denoising applications. In fact, noise in homogeneous regions is effectively removed, however in heterogeneous areas, sharp discontinuities related to edges are processed similarly to noise. The corresponding high gradient magnitude is therefore reduced which results in a significant loss of contrast in the restored image. This limitation was first explored by [Perona and Malik, 1990] by means of anisotropic diffusion model, where the intensity of the diffusion process is dependent on the local gradient value. The degree of smoothing is inversely proportional to the gradient value so that homogeneous areas are strongly diffused while edge-discontinuities are preserved. The anisotropic diffusion PDE proposed by Perona and Malik can be formulated as : ∂u ∂t (x, y, t) =div (φ(|∇u(x, y, t)|)∇u(x, y, t)) (4.4) where div is the divergence operator, ∇ designates the spatial gradient operator and ψ is a decreasing function. The decreasing property of φ ensures that the smoothing process is weaker in regions with strong gradient values, and stronger in flat areas. Common choices for φ include exponentially decreasing functions as : φ(|∇u|) =exp(− |∇u|2 k 2 ) (4.5) If φ is a constant function, the PDE equation (4.4) of Perona and Malik is reduced to : ∂u ∂t (x, y, t) =div (∇u(x, y, t)) (4.6) which is another formulation of the isotropic heat equation. The anisotropic diffusion (4.4) also faces limitations. The presence of noise introduces strong oscillations which can be considered as edges and be subsenquently preserved. To circumvent this issue, alternative techniques were proposed independently in [Catté et al., 1992] and [Nitzberg and Shiota, 1992]. The gradient of the image is replaced by a smoothed version and the PDE (4.4) becomes : ∂u ∂t (x, y, t) =div (φ(|∇(G σ ∗ u(x, y, t))|)∇u(x, y, t)) (4.7) where G σ is gaussian operator with variance σ. Further research in the field of anisotropic diffusion introduced in [Alvarez et al., 1992] resulted in a non linear PDE : ∂u ∂t (x, y, t) =g (|∇(G σ ∗ u(x, y, t))|) |∇u(x, y, t)| div ( ∇u(x, y, t) |∇u(x, y, t)| ) (4.8) where g is a decreasing function that tends to 0 when ∇u tends to infinity. The( second ) derivative of u in the direction orthogonal to the gradent ∇u is the term |∇u|div ∇u |∇u| , and coincides with the image level lines. The intensity of the diffusion process is regulated
75 Figure 4.1 – (Left) Noisy image from Terra MODIS band 30 (Center) Application of the heat equation (ID=0.41) (Right) Application of the Perona-Malik algorithm (ID=0.84). by the term g (|∇G σ ∗ u|). In homogeneous areas, strong values of g (|∇G σ ∗ u|) induce an anisotropic diffusion in the direction orthogonal to the gradient. Along the edges of the images (strong gradient values), the weak weighting due to the descreasing property of g disables the diffusion process. In [Nordstrom, 1990], the original PDE of Perona and Malik is formulated in a way that forces the estimated solution to be close to the observed image u 0 . The resulting PDE takes the following form : ∂u ∂t (x, y, t) − div(φ(|G σ ∗∇u(x, y, t)|)∇u(x, y, t)) = β (u(x, y, t) − u 0 (x, y)) (4.9) where the coefficient β regulates the fidelity of the estimated solution to the orignal noisy sinal u 0 . In the unifying framework proposed in [Deriche and Faugeras, 1995], the authors show that image restoration based on the resolution of PDEs can be reformulated as the minimization of energy functionals. Such formulation is adequate to many ill-posed inverse problems encountered in computer vision applications such as denoising, deconvolution or inpainting. According to Hadamard’s definition, a mathematical problem can be considered as ill-posed if at least one of the following conditions is not satisfied : - A solution exists - If a solution exists, it is unique - The solution is stable ; A small perturbation in the observed data induces a small perturbation in the estimated solution. Ill-posed inverse problems are systematically regularized to impose well-posedness and are tackled under the assumption of a specific image formation model. Let us denote by K a linear transformation that accounts for the sensor multiple degradations (diffraction, defocalisation, mouvement blur). The observed image f resulting from the acquisition
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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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- Alaska Labrador Siberia Restorati
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126 5. Application : Restoration of
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128 Conclusion
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130 BIBLIOGRAPHIE A. Chambolle. An
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132 BIBLIOGRAPHIE E. Hochberg, S. A
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134 BIBLIOGRAPHIE P. Perona and J.
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