Theory of Nonlinear Subdivision and Multiscale Transforms

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Over the past 25 years, fast multiscale algorithms such as pyramid transforms and subdivision methods lead to tremendous successes in data and geometry processing, and in scientific computing in general. While linear multiscale 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 denoising, or due to nonlinear constraints on the analyzed objects themselves that need to be preserved on all scales.The project aims at developing a consistent theory for the core questions of convergence, limit smoothness, and stability of such a scheme. Focus is on the stability problem which is relevant for nonlinear data and geometry compression algorithms. Both the univariate and the more complicated multivariate theory will be attacked. Work will include case studies for practically relevant schemes, such as normal multiresolution used for efficient geometry processing. Case studies are also considered important to help shape the final theory on nonlinear multiscale methods.