Th`ese Marouan BOUALI - Sites personnels de TELECOM ParisTech

More documents

Recommendations

Info

60 3. Standard destriping techniques and application to MODIS d j+1 [k] = ∞∑ n=−∞ g[n − 2k]d j [n] =d j ⋆ g[2k] (3.39) where h[k] =h[−k] and ⋆ is the convolution symbol. The approximation coefficients a j+1 result from the convolution of approximations a j with the low-pass filter h, followed by a sub-sampling of factor 2. The sub-sampling operation consists in preserving only one coefficient out of two. Details coefficients d j+1 are deduced from the convolution of d j with the high-pass filter g. The reconstruction of the signal is obtained with the inverse wavelet transform : ∞∑ ∞∑ a j [k] = h[k − 2n]a j+1 [n]+ g[k − 2n]d j+1 [n] (3.40) n=−∞ =ă j+1 ⋆h + ˘d j+1 ⋆g n=−∞ The successive approximations a j are obtained with a convolution of signals ă j+1 and ˘d j+1 with the filters h and g. ă results from an up-sampling of factor 2 of a, where zeros are inserted between succesive samples as : ă[2n] =a[n] ă[2n + 1] = 0 (3.41) The decomposition and illustration in filter banks is illustrated in figure . Filters ¯h, ḡ, h and g are quadrature mirror filters due to the orthogonality relation between h and g : 3.7.5 2D wavelet basis g[k] = (−1) 1−n h[1 − k] (3.42) The wavelet decomposition of a bidimensional signal f ∈ L 2 (R 2 ) can be seperately computed along its dimensions, using separable wavelet basis generated from the tensor product of unidimensional orthogonal wavelet basis. In the 2D case, the wavelet decomposition is computed first on the lines and then on the columns (or inversely). The sequence {Vj 2} j∈Z defined as Vj 2 = V j ⊗ V j is a separable multiresolution approximation of L 2 (R 2 ). Similarly to the unidimensional case, the space of details Wj 2 is defined as the orthogonal complementary of Vj 2 in Vj−1 2 : Vj−1 2 = Vj 2 ⊕ Wj 2 (3.43) An orthonormal wavelet basis of L 2 (R 2 ) can be constructed from the scaling functions φ and the mother wavelet ψ. To this prupose, let us define ∀(x, y) ∈ R 2 , the following wavelets : ψ 1 (x, y) =φ(x)φ(y) ψ 2 (x, y) =ψ(x)φ(y) ψ 3 (x, y) =ψ(x)ψ(y) (3.44)
61 Figure 3.15 – (Left) Noisy image from Terra MODIS band 30 (Right) Wavelet decomposition at two resolution levels showing that (1) stripe noise is isolated in the only horizontal details d 1 j (2) Wavelet coefficients associated with stripes have a magnitude of the same order than those related to the image edges We then consider the dilated and translated versions of these wavelets defined for k =1, 2, 3 as : ψj,m,l k = 1 ( x − 2 j m 2 j ψk 2 j , y − ) 2j l 2 j (3.45) The wavelet family {ψj,m,l 1 ,ψ2 j,m,l ,ψ3 j,m,l } (m,l)∈Z 2 is an orthonormal basis of the details vector space Wj 2 and the family {ψ1 j,m,l ,ψ2 j,m,l ,ψ3 j,m,l } (j,m,l)∈Z3 is an orthonormal basis of L 2 (R 2 ). The separability of bidimensional wavelet basis is an interesting property for the multiresolution analysis of images. In fact, the wavelet family {ψj,m,l 1 ,ψ2 j,m,l ,ψ3 j,m,l } (m,l)∈Z 2 allows the extraction of details in the horizontal, vertical and diagonal directions. This particular feature is indeed the main motivation behind the use of wavelet analysis for the striping issue. Wavelet decomposition and reconstruction of a function f ∈ L 2 (R 2 ) is computed with separable bidimensional convolutions. Using the conjugate mirror filters h and g of the mother wavelet ψ, wavelet coefficients at level 2 j+1 are obtained from the approximation at level 2 j with : a j+1 [m, l] =a j ⋆ ¯h[m]¯h[l] d 1 j+1[m, l] =a j ⋆ ¯h[m]ḡ[l] d 2 j+1[m, l] =a j ⋆ ḡ[m]¯h[l] d 3 j+1[m, l] =a j ⋆ ḡ[m]ḡ[l] (3.46) The bidimensional wavelet decomposition of an image contaminated with stripe noise is illustrated in figure 3.15.
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 24 and 25: 24 2. Remote Sensing with MODIS any
Page 30 and 31: 30 2. Remote Sensing with MODIS Mai
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 110 and 111:
110 4. A Variational approach for t
Page 112 and 113:
112 5. Application : Restoration of
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
- Alaska Labrador Siberia Restorati
Page 126 and 127:
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?