Variational Image Denoising Approach with Diffusion Porous Media Flow

and Applied Analysis 3 The flow u(t) = S(t)g is a continuous semigroup of contractions in H−1(Ω) in this case or in L(Ω), in other situation, and the denoising effect of (6) is due to smoothing effect of S(t) on initial data, a property which is a characteristic to semigroups generated by the nonlinear operators A of the gradient type ∂φ, where φ is a convex and lower semicontinuous function. Since Theorem 1 applies to any initial data g ∈ H(Ω), which is a quite general space of distributions on Ω, the filter g → S(t)g can be used for smoothing very noised images g. It should be emphasized that, for filtering purpose, it suffices to compute u(t) at any time t > 0 since the smoothing effect does not increase with t. On the contrary, since for t → 0 we have [13] lim t→∞ u (t) = u ∞ in H−1 (Ω) , (11) where u ∞ is a steady-state solution to A, that is, Au ∞ = 0, one might expect that, for a large t, the filtered image could be far away from the original image g. This means that in the algorithm (10), the best filtered image can be obtained for k → ∞, kh ≈ ε, and ε arbitrarily small. If one wants to restore the image g via step descent algorithm, then (6) should be replaced by