Th`ese Marouan BOUALI - Sites personnels de TELECOM ParisTech

More documents

Recommendations

Info

90 4. A Variational approach for the destriping issue Figure 4.5 – Decomposition of the image from Terra MODIS band 33 as a cartoon u (Left) + v (Right). λ is selected so that ID(u) is the same for the three models. From top to bottom, TV model (ID=0.447), VO model (ID=0.472) and OSV model (ID=0.461)
91 4.4 Destriping via gradient field integration This far, we introduced variational methods based on PDE’s and energy functional minimization, with image denoising applications in mind. These techniques are actually used extensively in computer vision/graphics, to solve over-constrained geometric problems. It is typically the case of Photometric Stereo Methods (PSM) and Shape From Shading (SFS) applications where the goal is to recover a depth map (or an image) by integrating a gradient field with discrete values. Let us denote by G =(G x ,G y ) a bidimensional gradient vector defined in a subspace Ω of R 2 . The problem of gradient field integration consists in determining a function u whose gradient ∇u is close to G. Two classes of techniques can be used to tackle this ill-posed problem and estimate the true image u. Local integration techniques [Coleman et al., 1982], [Healey and Jain, 1984] and [Wu and Li, 1988] rely on a curve integration : ∫ u(x, y) =u(x 0 ,y 0 )+ G x dx + G y dy (4.69) where γ is the integration path from a starting point (x 0 ,y 0 ) to pixel (x, y) ∈ Ω. Local integration techniques recover an image u starting with an initial height and then propagating height values according to the neighborhood gradient values. Local integration techniques are dependent on the data accuracy and can propagate error values when dealing with noisy gradient fields. Global integration methods [Horn and Brooks, 1986], [], [Frankot and Chellappa, 1988] and [Horn, 1990] can be formulated in a variational framework as the minimization of an energy functional : ∫ E(u) = Ω γ ( ) ∂u 2 ( ) ∂u 2 ∂x − G x + ∂y − G y dx dy (4.70) Unlike denoising and decomposition variational models described this far, both terms in the energy functional (4.70) are fidelity terms that measure the L 2 norm difference between the gradient field components of u and the observed gradient field G. For instance, the energy functional does not include any lagrange multiplier λ. The Euler-Lagrange equations of the previous functionals leads to : ∂ ∂x ( ∂u ∂x − G x which can be simplified into the following Poisson equation : ) + ∂ ( ) ∂u ∂x ∂y − G y = 0 (4.71) ∇ 2 u = ∇.G (4.72) The previous equation can be solved using a fast marching method [Ho et al., 2006] or the streaming multigrid method proposed in [Kazhdan and Hoppe, 2008]. Among several other techniques, we draw a particular attention to the well-known Frankot-Chellapa
Page 1:
École Doctorale d’Informatique,
Page 5:
5 Résumé Nou abordons dans cette
Page 8 and 9:
8 TABLE DES MATIÈRES 3.5 Frequency
Page 10 and 11:
10 TABLE DES MATIÈRES
Page 12 and 13:
12 1. Introduction A common issue f
Page 14 and 15:
14 1. Introduction
Page 16 and 17:
16 2. Remote Sensing with MODIS In
Page 18 and 19:
18 2. Remote Sensing with MODIS Fig
Page 20 and 21:
Page 22 and 23:
Page 24 and 25:
24 2. Remote Sensing with MODIS any
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
30 2. Remote Sensing with MODIS Mai
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Page 38 and 39:
38 2. Remote Sensing with MODIS
Page 40 and 41: 40 3. Standard destriping technique
Page 72 and 73: Table 3.4 - Noise Reduction (NR), I
Page 74 and 75: 74 4. A Variational approach for th
Page 100 and 101: 100 4. A Variational approach for t
Page 112 and 113: 112 5. Application : Restoration of
Page 124 and 125: - Alaska Labrador Siberia Restorati
Page 128 and 129: 128 Conclusion
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.
show all

Th`ese Marouan BOUALI - Sites personnels de TELECOM ParisTech

Create successful ePaper yourself

Delete template?

Save as template?