The topological gradient method for semi-linear problems and application to edge detection and noise removal.

International audienceIn this paper we propose a new variationnal method for segmenting/restoring images degraded by diverse noises and blurs. This method is based on the notion of topological gradient. First applied by [11] to restore images degraded by a Gaussian noise, we propose here to extend the segmen- tation/restoration process for possibly blurred images con- tamined either by an additive Gaussian noise, or a multi- plicative noise of gamma law or in presence of Poissonian statistics. We calculate, both for perforated and cracked do- mains, the topological gradient for each noise model. Then we present a segmentation/restoration algorithm based on this notion and we apply it to the three degradation models previously described. Then, we compare our method with the Ambrosio-Tortorelli approximation of the Mumford-Shah functional [23, 1]. We also compare our results with those given by a classical TV restoration process (see [4] for a speckle model). Many experimental results showing the ef- ficiency, the robustness and the rapidity of the approach are presented