Nonlinear Subdivision Schemes and Multi-Scale Transforms

Over the past 25 years, fast multi-scale algorithms such as wavelet-type pyramid transforms for hierarchical data representation, multi-grid solvers for the numerical solution of operator equations, and subdivision methods in computer-aided geometric design 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 [10, 18, 26, 15, 12, 23], 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. For illustration, and to guide our further discussions, let us introduce three univariate examples of nonlinear multi-scale transforms that have played a central role in the development of the emerging theory.