Univariate subdivision and multi-scale transforms: The nonlinear case

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.