Stanislav Harizanov

• Subdivision is a process of recursively refining discrete data using a set of subdivision rules to generate limits (curves, surfaces, height fields) with desirable properties such as continuity, smoothness, reproduction of shape features, and many more. The wide range of applications as well as the necessity of improving the performance of the existing algorithms lead to the invention of a great variety of subdivision schemes. In many cases, such as preserving the data shape, using normal meshes for better compression rates, removing heavily-tailed noise, working with manifold-valued data, linear multi-scale transforms give unsatisfactory results or cannot be applied at all, and nonlinear alternatives are necessary. There are still very few results about Lipschitz stability and Hölder regularity of nonlinear subdivision schemes and the associated multi-scale transforms, which is a very active research field with many open problems, that is driven by both theory and applications. In this thesis we develop a general stability analysis of both univariate schemes and their associated multi-scale transforms in the nonlinear functional setting. We show that, unlike the linear setting, convergence and stability analysis are no longer equivalent, and we derive numerical criteria for the verification of each of them. We extend the univariate convergence and stability results to the multivariate regular setting via local approximation techniques. We establish a general theory for normal multi-scale transforms for curves, based on approximating prediction operators. We propose a globally-convergent normal multi-scale transform, and build an adaptive algorithm based on it that defines a well-posed transform with smooth limits and high detail decay rates. We investigate several extensions of the classical setup for normal multi-scale transforms, namely we use another subdivision operator to generate the normal directions, the combined action of two different subdivision operators for the prediction step, and nonlinear geometry-based predictors, respectively, and show that the properties of the normal multi-scale transforms improve when such extensions are considered.