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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)