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The Topological Gradient Method: From Optimal Design to Image Processing

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Abstract

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 Dirichlet 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.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse Université Paul Sabatier, 118 route de Narbonne, F-31062, Toulouse Cedex 9, France

    Stanislas Larnier, Jérôme Fehrenbach & Mohamed Masmoudi

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  1. Stanislas Larnier
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  2. Jérôme Fehrenbach
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  3. Mohamed Masmoudi
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Correspondence to Stanislas Larnier.

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Larnier, S., Fehrenbach, J. & Masmoudi, M. The Topological Gradient Method: From Optimal Design to Image Processing. Milan J. Math. 80, 411–441 (2012). https://doi.org/10.1007/s00032-012-0196-5

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  • DOI: https://doi.org/10.1007/s00032-012-0196-5

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