Abstract
Non-local functionals have been successfully applied in a variety of applications, such as spectroscopy or in general filtering of time-dependent data. We mention the patch-based denoising of image sequences [Boulanger et al. IEEE Transactions on Medical Imaging (2010)]. Another family of non-local functionals considered in these notes approximates total variation denoising. Thereby we rely on fundamental characteristics of Sobolev spaces and the space of functions of finite total variation (see [Bourgain et al. Journal d’Analyse Mathématique 87, 77–101 (2002)] and several follow up papers). Standard results of the calculus of variations, like for instance the relation between lower semi-continuity of the functional and convexity of the integrand, do not apply, in general, for the non-local functionals. In this paper we address the questions of the calculus of variations for non-local functionals and derive relations between lower semi-continuity of the functionals and separate convexity of the integrand. Moreover, we use the new characteristics of Sobolev spaces to derive novel approximations of the total variation energy regularisation. All the functionals are well-posed and reveal a unique minimising point. Even more, existing numerical schemes can be recovered in this general framework.
AMS 2010 Subject Classification: 49J05, 49J45, 49M25
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Acknowledgements
The work of CP and OS has been supported by the Austrian Science Fund (FWF) within the research networks NFNs Industrial Geometry, Project S09203, and Photoacoustic Imaging in Biology and Medicine, Project S10505-N20.
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Boulanger, J., Elbau, P., Pontow, C., Scherzer, O. (2011). Non-Local Functionals for Imaging. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications(), vol 49. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9569-8_8
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