Tamtam Proceedings - lamsin

Image restoration and edge detection 514. Numerical ApplicationsThe algorithm consists in inserting small heterogeneities in regions where the topologicalgradient given by Theorem 3.1 is smaller than a given threshold α < 0. Theseregions represent the edges ω ρ of the image . The final algorithm is as followsAlgorithm- Initialization : c = c 0 .- Calculation of u 0 and v 0 : solutions of the direct (1) and adjoint (14) problems.- Computation of the 2 × 2 matrix M and its lowest eigenvalue λ min at each point of thedomain.- Set{ ε if x ∈ Ω such that λmin < α < 0, ε > 0c 1 =(19)elsewhere.c 0- Calculation of u 1 solution to problem (1) with c = c 1 .From the numerical point of view, it is more convenient to simulate the cracks by a smallvalue of c.Figure 1 shows the restored Self-Portrait Van Gogh using this algorithm: the originalimage, the perturbed image which is obtained with an additive noise (random from MAT-LAB) of 20%, the restored image and the isovalues of the topological gradient.Figure 2 illustrates the importance of the topological gradient for detecting edges. Itshows the original image, the perturbed image, the restored image using the same algorithmand the isovalues of the topological gradient.For solving (1) and (14), we compared the Preconditioned Conjugate Gradient (PGC)and the Gauss Elimination (GE) methods. We used as a preconditioner a discrete cosinetransform. The computation times, on a PC Pentium4, 256 Mo DDR, are given in table 1.Size of the image 100× 100 250 × 250 450 × 450Time of computation/GE 1,76s 106,41s 1,54e+003sTime of computation/PCG 2,26s 11s 28sTable1: Computation times5. ConclusionWe have presented in this work, a new method for image restoration with edge detection.To make this method relevant for real life applications, we considered the Discretecosine transform as a preconditioner for PCG, we have compared different algorithmsto solve linear systems associated to (1) and (14). The extension of this method to otherproblems in image processing like segmentation and classification are under development.TAMTAM –Tunis– 2005