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Meyer’s work can then be used in hope of isolating the stripe noise from other structures
present in the true scene.
4.3.1 Yves Meyer’s model for oscillatory functions
The total variation regularization used in the previous section as a denoising technique
can be seen from a different perspective as an image decomposition model, where
the observed image f is approximated by a sketchy version u that lies in the BV space
and the component v = f − u contains small sace details such as noise and/or texture.
Many variational methods have the explicit goal of extracting an image u composed of
homogeneous areas separated by sharp discontinuities but do not retain the component
v as it is considered to be noise. This can be problematic for images containing texture
because noise and texture are both oscillatory functions, processed equally with the TV
regularization. From an image decomposition perspective, ROF model can be formulated
as the following minimization :
inf
(u,v)∈BV (Ω)×L 2 (Ω)/u+v=f
(
)
TV(u)+λ‖v‖ 2 L 2 (Ω)
(4.43)
In his investigation of the standard ROF model [Meyer, 2002], Yves Meyer pointed out
that small values of λ can remove fine details related to texture. To overcome this issue,
he proposes a different decomposition, where the classical L 2 norm associated with the
residual v = f − u is replaced by a weaker-norm, more sensitive to oscillatory functions.
Yves Meyer suggests finding a component v in the space G defined as the Banach space
composed of all distributions v which can be written as :
v(x, y) =∂ x g 1 (x, y)+∂ y g 2 (x, y) (4.44)
where g 1 and g 2 both belong to the space L ∞ (R 2 ). The space G is endowed with the norm
‖v‖ G defined as the lower bound of all L ∞ norms of the functions |⃗g| where ⃗g =(g 1 ,g 2 )
and |⃗g(x, y)| = √ g 1 (x, y) 2 + g 2 (x, y) 2 . Additionnaly to G, which can be viewed as the
dual space of BV , Meyer introduces the spaces E and F also suited to model texture.
The space E (dual of Ḃ 1,1
1 ) is defined similarly to G except that g 1,g 2 are in the space
of bounded mean oscillations functions denoted by BMO(R 2 ). For the space F (dual of
H 1 ), g 1 ,g 2 belong to Besov space B∞
−1,∞ (R 2 ). When the texture component v is assumed
to lie in the space G, Yves Meyer proposes the following decomposition model :
(
TV(u)+λ‖v‖G(Ω ))
2 (4.45)
inf
(u,v)∈BV (Ω 2 )×G(Ω 2 )/u+v=f
The ‖.‖ G -norm can efficiently capture oscillating patterns because it is weaker than the
‖.‖ 2 -norm (L 2 (Ω) ⊂ G(Ω) ). However, due to its mathematical form, the Euler-Lagrange
equation of (4.45) cannot be expressed explicitly and several u + v decomposition models
have been later introduced as an approximation to Yves Meyer model.