# American Institute of Mathematical Sciences

February  2016, 10(1): 51-86. doi: 10.3934/ipi.2016.10.51

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

 1 Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, France, France

Received  December 2014 Revised  July 2015 Published  February 2016

The goal of this paper is to apply the topological gradient method to edge detection and noise removal for images degraded by various noises and blurs. First applied to edge detection for images degraded by a Gaussian noise, we propose here to extend the method to blurred images contaminated either by a multiplicative noise of gamma law or to blurred Poissonian images. We compute, both for perforated and cracked domains, the topological gradient for each noise model. Then we present an edge detection/restoration algorithm based on this notion and we apply it to the two degradation models previously described. We compare our method with other classical variational approaches such that the Mumford-Shah and TV restoration models and with the classical Canny edge detector. Some experimental results showing the efficiency, the robustness and the rapidity of the method are presented.
Citation: Audric Drogoul, Gilles Aubert. The topological gradient method for semi-linear problems and application to edge detection and noise removal. Inverse Problems & Imaging, 2016, 10 (1) : 51-86. doi: 10.3934/ipi.2016.10.51
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