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92 4. A Variational approach for the destriping issue
(FC) algorithm [Frankot and Chellappa, 1988], which provides a solution by considering
the Fourier transform of the gradient field G components. Let us denote by (ξ x ,ξ y ) the
frequency components as in [Bracewell, 1986] and recall the differentiation properties of
the Fourier transform :
F
⎧⎪ ( ∂u
( ∂x)
) = j.ξx F(u)
⎨ F ∂u
∂y
= j.ξ y F(u)
( )
F ∂ 2 u
= ξ 2 (4.73)
xF(u)
⎪ ⎩
∂ 2 x
F
(
∂ 2 u
∂ 2 y
)
= ξyF(u)
2
where j is the imaginary unit √ −1. The fourier transform of the Poisson equation (4.72)
can be written as :
(
ξ
2
x + ξy
2 )
F(u) =−jξx F(G x ) − jξ y F(G y ) (4.74)
Denoting u F , G F x and G F y the fourier transform of u, G x and G y , the Frankot-Chellapa
reconstruction algorithm gives :
u F = −jξ xG F x − jξ y G F y
ξ 2 x + ξ 2 y
(4.75)
As we will see, gradient field integration problems and their gradient-based variational
formulation offer an interesting perspective on the destriping issue. In fact, compared to
other restoration-oriented variational models, the energy functional (4.70) clearly separates
the information related to vertical gradient and horizontal gradient. Going further
in this gradient-based reasoning, we make the following remark :
If the stripe noise is additive, it mostly affects the vertical gradient of the striped image
Let us the consider the following image formation model :
I s = I + n (4.76)
where I is the stripe free true image and n is the stripe noise. The linear operator K is
considered to be the identity. Let us assume that the unidirectional signature of the stripe
noise n translates on its horizontal gradient as :
∂n(x, y)
∂x
≈ 0 (4.77)
The partial derivatives along the x and y-axis of the image formation model (4.76) :
∂I s
∂x = ∂I
∂x + ∂n
∂x
∂I s
∂y = ∂I
∂y + ∂n
∂y
(4.78)