Scale-Space Properties of Nonstationary Iterative Regularization Methods

Radmoser, Esther ; Scherzer, Otmar ; Weickert, Joachim

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URN: urn:nbn:de:bsz:180-madoc-20610
Document Type: Working paper
Year of publication: 1999
The title of a journal, publication series: Technical Report / Department for Mathematics and Computer Science, University of Mannheim
Volume: 012
Place of publication: Mannheim
Publication language: English
Institution: School of Business Informatics and Mathematics > Sonstige - Fakultät für Wirtschaftsinformatik und Wirtschaftsmathematik
MADOC publication series: Veröffentlichungen der Fakultät für Mathematik und Informatik > Institut für Informatik > Technical Reports
Subject: 004 Computer science, internet
Subject headings (SWD): Regularisierung , Diffusion , Skalenraum , Ljapunov-Funktion
Keywords (English): regularization methods , diffusion filtering , scale-spaces , Lyapunov functionals
Abstract: Most scale-space concepts have been expressed as parabolic or hyperbolic partial differential equations (PDEs). In this paper we extend our work on scale-space properties of elliptic PDEs arising from regularization methods: we study linear and nonlinear regularization methods that are applied iteratively and with different regularization parameters. For these so-called nonstationary iterative regularization techniques we clarify their relations to both isotropic diffusion filters with a scalar-valued diffusivity and anisotropic diffusion filters with a diffusion tensor. We establish scale-space properties for iterative regularization methods that are in complete accordance with those for diffusion filtering. In particular, we show that nonstationary iterative regularization satisfies a causality property in terms of a maximum-minimum principle, possesses a large class of Lyapunov functionals, and converges to a constant image as the regularization parameters tend to infinity. We also establish continuous dependence of the result with respect to the sequence of regularization parameters. Numerical experiments in two and three space dimensions are presented that illustrate the scale-space behavior of regularization methods.
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