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Image processing with partial differential equations. (English) Zbl 1065.94502
Bourlioux, Anne (ed.) et al., Modern methods in scientific computing and applications. Proceedings of the NATO Advanced Study Institute and Séminaire de Mathématiques Supérieures on modern methods in scientific computing and applications, Montréal, Québec, Canada, July 9–20, 2001. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0782-5/pbk). NATO Sci. Ser. II, Math. Phys. Chem. 75, 283-321 (2002).
Summary: In many applications computers analyse images or image sequences which are often contaminated by noise, and their quality can be poor (e.g. in medical imaging). We discuss how nonlinear partial differential equations (PDEs) can be used to automatically produce an image of much higher quality, enhance its sharpness, filter out the noise, extract shapes, etc. The models are based on the well-known Perona-Malik image selective smoothing equation and on geometrical equations of mean curvature flow type. Since the images are given on discrete grids, PDEs are discretized by variational techniques, namely by the semi-implicit finite element, finite volume and complementary volume methods in order to get fast and stable solutions. Convergence of the schemes to variational solutions of these strongly nonlinear problems and the extension of the methods to adaptive scheme strategies improving computational efficiency are discussed. Computational results with artificial and real 2D, 3D images and image sequences are presented.
For the entire collection see [Zbl 1045.65001].

MSC:
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
35A15 Variational methods applied to PDEs
68U10 Computing methodologies for image processing
Software:
ALBERT
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