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58 3. Standard destriping techniques and application to MODIS
one eventually converges to the original fourier transform where temporal information is
completely lost.
3.7.2 Multiresolution analysis
Multiresolution analysis, and more precisely wavelet transform constitutes a powerful
alternative to fourier transform, exploited in numerous signal processing applications.
Multiresolution analysis allows the decomposition of a function f ∈ L 2 (R 2 ), as a sum of
approximations associated with different resolution levels. In [Mallat, 2000], Mallat defines
an multiresolution approximation as a set of closed vector sub-spaces {V j } j∈Z that verify
several properties among which :
∀j ∈ Z, ⊂ V j+1 ⊂ V j ⊂ L 2 (R) (3.27)
∀j ∈ Z,f(t) ∈ V j ⇔ f( t 2 ) ∈ V j+1 (3.28)
The embedding property (3.27) of vector spaces {Vj} ensures that the approximation of
the function f at resolution 2 j is obtained from an approximation at a higher resolution
2 j+1 . The property (3.28) garantees that the projection of f on the space V j contains
twice as much details as the projection on the space V j+1 . Denoting φ, a function ∈ L 2 (R)
which translations {φ(t − k)} k∈Z form an orthonormal basis of the space V 0 , it is shown
that the family of functions {φ j,k } k∈Z obtained as dilatations and translations of φ is an
orthonormal basis of V j where :
φ j,k = 1 √
2 j φ ( t
2 j − k )
(3.29)
The function φ, known as scale function or father wavelet, is dilated and translated to
form an orthonormal basis of the space V j , where the orthorgonal projection of a function
f defines its approximation at the resolution 2 −j .
3.7.3 Wavelet basis
Going from a resolution 2 j to a lower resolution 2 j+1 , leads to a loss of information.
Details visible in the approximation of resolution 2 j are lost at the resolution 2 j+1 . The
embedding property of multiresolution approximations and the inclusion of vector space
V j in V j−1 can be used to define the space of details W j as the orthogonal complementary
of V j in V j−1 :
V j−1 = V j ⊕ W j (3.30)
Similarly to the vector space of approximations V 0 , it is shown that the space of details W 0
is generated from an orthonormal basis composed of translated version {ψ(t − k)} k∈Z of a