Univariate subdivision and multi-scale transforms: The nonlinear case

Nira Dyn, Peter Oswald

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


Over the past 25 years, fast multi-scale algorithms lead to tremendous successes in data and geometry processing, and in scientific computing in general. While linear multi-scale analysis is in a mature state, not so much is known in the nonlinear case. Nonlinearity arises naturally, e.g. in data-adaptive algorithms, in image and geometry processing, robust de-noising, or due to nonlinear constraints on the analyzed objects themselves that need to be preserved on all scales. The aim of this paper is to take the reader on a guided tour into the existing case studies for nonlinear multi-scale transforms and the emerging approaches to develop a theory. The main part of the exposition concentrates on the univariate case (multi-scale processing of scalar data and curves). It is split into a review of the basic theory of nonlinear transforms in the functional setting, and an exemplary discussion of what we call geometric subdivision schemes and multi-scale transforms. Extensions to multivariate schemes and some other recent developments will be reviewed but in less detail.

Original languageEnglish
Title of host publicationMultiscale, Nonlinear and Adaptive Approximation
Subtitle of host publicationDedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday
PublisherSpringer Berlin Heidelberg
Number of pages45
ISBN (Print)9783642034121
StatePublished - 2009


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