The Topological Gradient Method: From Optimal Design to Image Processing

Stanislas Larnier (1) , Jérôme Fehrenbach (1) , Mohamed Masmoudi (1)

1 MIP - Mathématiques pour l'Industrie et la Physique

The aim of this article is to review and extend the applications of the topological gradient to major image processing problems. We briefly review the topological gradient, and then present its application to the crack localization problem, which can be solved using the Dirich-let to Neumann approach. A very natural application of this technique in image processing is the inpainting problem, which can be solved by identifying the optimal location of the missing edges. Edge detection is of extreme importance, as edges convey essential information in a picture. A second natural application is then the image reconstruction. A class of image reconstruction problems is considered that includes restoration , demosaicing, segmentation and super-resolution. These problems are studied using a unified theoretical framework which is based on the topological gradient method. This tool is able to find the localization and orientation of the edges for blurred, low sampled, partially masked, noisy images. We review existing algorithms and propose new ones. The performance of our approach is compared with conventional image reconstruction processes.

Mots clés: Edge detection; Topological gradient; Anisotropic diffusion; Image denoising; Image inpainting; Image segmentation; Super-resolution; Demosaicing

Domaines: Optimisation et contrôle [math.OC]; Traitement des images [eess.IV]

Stanislas Larnier, Jérôme Fehrenbach, Mohamed Masmoudi. The Topological Gradient Method: From Optimal Design to Image Processing. Milan Journal of Mathematics, 2012, 80 (2), pp.411-441. ⟨10.1007/s00032-012-0196-5⟩. ⟨hal-01333752⟩