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The goal is to find the true image u from the noisy observation f assuming n to be white
gaussian noise with a known variance σ 2 . This ill-posed problem can be regularized by
assuming that the true image u belongs to the space of functions of bounded variations,
denoted BV and defined as :
BV (Ω) =
{
u ∈ L 1 (Ω)|sup ϕ∈C 1 c ,|ϕ|≤1
(∫
) }
u divϕ < ∞
(4.15)
The total variation of a signal u ∈ BV (Ω) is equivalent to the L 1 norm of its gradient
norm and is given by :
{∫
}
TV(u) =sup u divϕ, ϕ ∈Cc 1 , |ϕ| ≤ 1
(4.16)
Hereafter, we consider u to be in BV (Ω) ∩C 1 , and the total variation simplifies into :
∫
TV(u) = |∇u|dΩ (4.17)
To retrieve u from f, Rudin, Osher and Fatemi propose to solve a constrained optimisation
problem :
minimize
TV(u)
subject to ∫ Ω f = ∫ Ω u, and ‖u − f‖2 = σ 2 (4.18)
In the previous formulation the equality constraint is not convex and can be replaced by
an inequality constraint as :
Ω
minimize
TV(u)
subject to ∫ Ω f = ∫ Ω u, and ‖u − f‖2 ≤ σ 2 (4.19)
Chambolle and Lions have shown in [Chambolle and Lions, 1997] that if ‖f− ∫ Ω fdΩ‖2 >σ 2
then problems (4.18) and (4.19) are equivalent. (4.19) can then be reformulated as the
minimization of :
E(u) =λ‖u − f‖ 2 + TV(u) (4.20)
where the lagrangian multiplier λ> 0 regulates the compromise between the fidelity term
and the regularizing term. For a given value of λ, Karush-Kuhn-Tucker conditions ensure
the equivalence between (4.19) and (4.20). The energy functional (4.20) will be refered to
hereafter as the ROF model. The minimization of ROF model is obtained via its Euler-
Lagrange equation :
⎛
⎛
−2λ(u − f)+ ∂
∂x
⎜
⎝
√ ( ∂u
∂x
∂u
∂x
) 2
+
(
∂u
∂y
⎞
⎟
) 2 ⎠ + ∂ ⎜
∂y ⎝
√ ( ∂u
∂x
∂u
∂x
) 2
+
(
∂u
∂y
⎞
⎟
) 2 ⎠ =0 (4.21)