Tamtam Proceedings - lamsin

48 Jaafar-Belaid et al.1. IntroductionThe goal of topological optimization is to find the optimal decomposition of a givendomain in two parts: the optimal design and its complementary. Similarly in image processingproblems, the goal is to split an image in several parts. We will show that in imagerestoration the detection of edges makes this operation more convenient.This paper shows that it is possible to solve the image restoration problems using topologicaloptimization tools, in fact images can be viewed as a piecewise smooth functionand edges can be considered as a set of singularities, so topological gradient approach,will be a good method to reconstruct an image where the edges are properly sealed.Let v be a given noisy image defined in a domain Ω ⊂ IR 2 and u the restored image. Werecall that a classical way to restore the image u is to solve the following PDE problem{ −div (c∇u) + u = v in Ω,(1)∂ n u = 0 on ∂Ω,where c is a small positive constant and n denotes the outward unit normal to ∂Ω. Thismethod is well known to give poor results: it smoothes noise and it blurs important structureslike edges. In order to avoid blurring of edges, nonlinear isotropic and anisotropicmethods were introduced, we can cite here the work of Perona and Malik [6], Catté, Lions,Morel and Coll [3] and more recently Weickert [7] and Aubert [2]. Our objectiveis to review this classical way of restoring a given image: we prove that it is exactly theTikhonov regularisation applied to the inversion of a compact operator which we willpresent in section 2. The topological gradient method [5] and its application to imagerestoration is presented in section 3. Some numerical experiments are discussed in section4.2. A classical approach for image restorationLet K be the canonical embedding operator defined byK : H 1 (Ω)/ IR −→ L 2 (Ω)/ IR ,u ↦−→ Ku = u(2)where H 1 (Ω)/ IR and L 2 (Ω)/ IR are the Hilbert spaces given by∫H 1 (Ω)/ IR = {u ∈ H 1 (Ω); u(x)dx = 0}, (3)Ω∫L 2 (Ω)/ IR = {v ∈ L 2 (Ω); v(x)dx = 0}, (4)ΩTAMTAM –Tunis– 2005