Th`ese Marouan BOUALI - Sites personnels de TELECOM ParisTech

More documents

Recommendations

Info

82 4. A Variational approach for the destriping issue Figure 4.3 – (Left) Image from Terra MODIS band 30 denoised with TV regularization (ID=0.5750) (Right) Denoised with the variational model () (ID=0.7768) If we discard the presence of gaussian noise n or assume it is implicitly accounted for in the gain and offset parameters A and B, the covariance matrix K reduces to the identity matrix and can be discarded. Then, a destriping approach similar to [Shen et al., 2008] can be achieved by minimizing : E(u) =λ‖(f − Au − B)‖ 2 + TV(u) (4.42) Destriping via the minimization of (4.42) relies on the observational model (4.32)) where A and B are assumed to be known. This is however not the case in pratice and a preprocessing stage is required to estimate the values of A and B. In their work, [Shen et al., 2008] used the moment matching technique to evaluate the values of A and B from the image mean value and variance. The limitations of such approach are directly attached to its hybrid aspect. In fact, the TV or Huber-Markov regularization acts as a post-processing stage, smoothing residual stripes that moment matching or any other technique based on linear adjustment (OFOV method for exemple) fails to remove. Nonetheless, the destriped results are much less distorted than those obtained with a direct application of the ROF model (figure 4.3). 4.3 Striping as a texture The main characteristic of stripping effect lies on it’s unidirectional aspect. To this extent, stripe noise can be considered as a structured texture with sharp fluctuations along a single axis of the image. This is particularly the case on MODIS spectral bands where striping is periodic. Texture discriminating variational models inspired from Yves
83 Meyer’s work can then be used in hope of isolating the stripe noise from other structures present in the true scene. 4.3.1 Yves Meyer’s model for oscillatory functions The total variation regularization used in the previous section as a denoising technique can be seen from a different perspective as an image decomposition model, where the observed image f is approximated by a sketchy version u that lies in the BV space and the component v = f − u contains small sace details such as noise and/or texture. Many variational methods have the explicit goal of extracting an image u composed of homogeneous areas separated by sharp discontinuities but do not retain the component v as it is considered to be noise. This can be problematic for images containing texture because noise and texture are both oscillatory functions, processed equally with the TV regularization. From an image decomposition perspective, ROF model can be formulated as the following minimization : inf (u,v)∈BV (Ω)×L 2 (Ω)/u+v=f ( ) TV(u)+λ‖v‖ 2 L 2 (Ω) (4.43) In his investigation of the standard ROF model [Meyer, 2002], Yves Meyer pointed out that small values of λ can remove fine details related to texture. To overcome this issue, he proposes a different decomposition, where the classical L 2 norm associated with the residual v = f − u is replaced by a weaker-norm, more sensitive to oscillatory functions. Yves Meyer suggests finding a component v in the space G defined as the Banach space composed of all distributions v which can be written as : v(x, y) =∂ x g 1 (x, y)+∂ y g 2 (x, y) (4.44) where g 1 and g 2 both belong to the space L ∞ (R 2 ). The space G is endowed with the norm ‖v‖ G defined as the lower bound of all L ∞ norms of the functions |⃗g| where ⃗g =(g 1 ,g 2 ) and |⃗g(x, y)| = √ g 1 (x, y) 2 + g 2 (x, y) 2 . Additionnaly to G, which can be viewed as the dual space of BV , Meyer introduces the spaces E and F also suited to model texture. The space E (dual of Ḃ 1,1 1 ) is defined similarly to G except that g 1,g 2 are in the space of bounded mean oscillations functions denoted by BMO(R 2 ). For the space F (dual of H 1 ), g 1 ,g 2 belong to Besov space B∞ −1,∞ (R 2 ). When the texture component v is assumed to lie in the space G, Yves Meyer proposes the following decomposition model : ( TV(u)+λ‖v‖G(Ω )) 2 (4.45) inf (u,v)∈BV (Ω 2 )×G(Ω 2 )/u+v=f The ‖.‖ G -norm can efficiently capture oscillating patterns because it is weaker than the ‖.‖ 2 -norm (L 2 (Ω) ⊂ G(Ω) ). However, due to its mathematical form, the Euler-Lagrange equation of (4.45) cannot be expressed explicitly and several u + v decomposition models have been later introduced as an approximation to Yves Meyer model.
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: 32 2. Remote Sensing with MODIS Fig
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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?