## Nonlinear diffusion filtering of images using the topological gradient approach to edges detection

• In this thesis, the problem of nonlinear diffusion filtering of gray-scale images is theoretically and numerically investigated. In the first part of the thesis, we derive the topological asymptotic expansion of the Mumford-Shah like functional. We show that the dominant term of this expansion can be regarded as a criterion to edges detection in an image. In the numerical part, we propose the finite volume discretization for the Catté et al. and the Weickert diffusion filter models. The proposed discretization is based on the integro-interpolation method introduced by Samarskii. The numerical schemes are derived for the case of uniform and nonuniform cell-centered grids of the computational domain $$\Omega \subset \mathbb{R}^2$$. In order to generate a nonuniform grid, the adaptive coarsening technique is proposed.

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Verfasserangaben: Monika Muszkieta urn:nbn:de:hbz:386-kluedo-21730 Helmut Neunzert Dissertation Englisch 2007 2007 Technische Universität Kaiserslautern Technische Universität Kaiserslautern 30.08.2007 17.03.2008 image analysis; nonlinear diffusion filtering; topological asymptotic expansion Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Jxx Elliptic equations and systems [See also 58J10, 58J20] / 35J20 Variational methods for second-order elliptic equations 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Jxx Elliptic equations and systems [See also 58J10, 58J20] / 35J60 Nonlinear elliptic equations 80-XX CLASSICAL THERMODYNAMICS, HEAT TRANSFER (For thermodynamics of solids, see 74A15) / 80Mxx Basic methods / 80M35 Asymptotic analysis Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011