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First order algorithms in variational image processing. (English) Zbl 1372.65053

Glowinski, Roland (ed.) et al., Splitting methods in communication and imaging, science, and engineering. Cham: Springer (ISBN 978-3-319-41587-1/hbk; 978-3-319-41589-5/ebook). Scientific Computation, 345-407 (2016).
Summary: The success of non-smooth variational models in image processing is heavily based on efficient algorithms. Taking into account the specific structure of the models as sum of different convex terms, splitting algorithms are an appropriate choice. Their strength consists in the splitting of the original problem into a sequence of smaller proximal problems which are easy and fast to compute. Operator splitting methods were first applied to linear, single-valued operators for solving partial differential equations in the 60th of the last century. More than 20 years later, these methods were generalized in the convex analysis community to the solution of inclusion problems, where the linear operators have to be replaced by nonlinear, set-valued, monotone operators. Again after more than 20 years, splitting methods became popular in image processing. In particular, operator splittings in combination with (augmented) Lagrangian methods and primal-dual methods have been applied very successfully. In this chapter we give an overview of first order algorithms recently used to solve convex non-smooth variational problems in image processing. We present computational studies providing a comparison of different methods and also illustrating their success in applications.
For the entire collection see [Zbl 1362.65002].

MSC:

65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
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