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74 4. A Variational approach for the destriping issue
The previous PDE, also known as the heat equation, translates an isotropic diffusion process
where all directions are smoothed identically. The isotropic property of the heat
equation is a major drawback for denoising applications. In fact, noise in homogeneous
regions is effectively removed, however in heterogeneous areas, sharp discontinuities related
to edges are processed similarly to noise. The corresponding high gradient magnitude
is therefore reduced which results in a significant loss of contrast in the restored image.
This limitation was first explored by [Perona and Malik, 1990] by means of anisotropic
diffusion model, where the intensity of the diffusion process is dependent on the local
gradient value. The degree of smoothing is inversely proportional to the gradient value
so that homogeneous areas are strongly diffused while edge-discontinuities are preserved.
The anisotropic diffusion PDE proposed by Perona and Malik can be formulated as :
∂u
∂t (x, y, t) =div (φ(|∇u(x, y, t)|)∇u(x, y, t)) (4.4)
where div is the divergence operator, ∇ designates the spatial gradient operator and ψ is
a decreasing function. The decreasing property of φ ensures that the smoothing process is
weaker in regions with strong gradient values, and stronger in flat areas. Common choices
for φ include exponentially decreasing functions as :
φ(|∇u|) =exp(− |∇u|2
k 2 ) (4.5)
If φ is a constant function, the PDE equation (4.4) of Perona and Malik is reduced to :
∂u
∂t (x, y, t) =div (∇u(x, y, t)) (4.6)
which is another formulation of the isotropic heat equation. The anisotropic diffusion (4.4)
also faces limitations. The presence of noise introduces strong oscillations which can be
considered as edges and be subsenquently preserved. To circumvent this issue, alternative
techniques were proposed independently in [Catté et al., 1992] and [Nitzberg and Shiota,
1992]. The gradient of the image is replaced by a smoothed version and the PDE (4.4)
becomes :
∂u
∂t (x, y, t) =div (φ(|∇(G σ ∗ u(x, y, t))|)∇u(x, y, t)) (4.7)
where G σ is gaussian operator with variance σ. Further research in the field of anisotropic
diffusion introduced in [Alvarez et al., 1992] resulted in a non linear PDE :
∂u
∂t (x, y, t) =g (|∇(G σ ∗ u(x, y, t))|) |∇u(x, y, t)| div
( ∇u(x, y, t)
|∇u(x, y, t)|
)
(4.8)
where g is a decreasing function that tends to 0 when ∇u tends to infinity. The( second )
derivative of u in the direction orthogonal to the gradent ∇u is the term |∇u|div ∇u
|∇u|
,
and coincides with the image level lines. The intensity of the diffusion process is regulated