Smoothness of Nonlinear Subdivision Processes

01.03.2006 - 31.12.2006
Research funding project
Subdivision algorithms play a prominent role in Computer Grahics, in Geometry Modeling, and in connection with wavelts. Their study poses a number of mathematical challenges. Convergence and smoothness analysis of linear subdivision schemes in the one-dimensional case can be considered complete, and also in the higher-dimensional setting (subdivision of polyhedra) the smoothness problem has been solved in 1995. In view of the wealth of applications it is not surprising that subdivision processes have been generalized to nonlinear geometries like surfaces, Riemannian manifolds, Euclidean space minus obstacles, and Lie groups. Also in the univariate case there are signal processing applications where some properties of wavelet transforms are incompatible with linearity. Recently a systematic theory of proximity of subdivison schemes has emerged, and meanwhile a rich class of nonlinear analogues of linear one-dimensional subdivision schemes has been investigated with regard to convergence, approximation properties, and smoothness. Despite these successes, great parts of the theory are still missing: the systematic development of the higher-dimensional case, especially smoothness analysis at extraordinary vertices; finer smoothness analysis (Hoelder regularity), and nonlinear energy minimizing subdivision algorithms. It is the aim of the proposed research project to continue research in these directions.

People

Project leader

Project personnel

Institute

Grant funds

  • FWF - Österr. Wissenschaftsfonds (National) Austrian Science Fund (FWF)

Research focus

  • Mathematical and Algorithmic Foundations: 100%

Keywords

GermanEnglish
Glattheitsmoothness
Nichtlinearitaetnonlinearity
Unterteilungsalgorithmensubdivision algorithms

Publications