In project C05 “Computational and structural aspects of point set surfaces”, we develop discrete differential geometric representations for point set surfaces and effective computational algorithms. Instead of first reconstructing a triangle based mesh, our operators act directly on the point set data. The concepts have contact to meshless methods and ansatz spaces of radial basis functions. As proof of concept of our theoretical investigations we transfer and implement key algorithms from surface processing, for example, for surface parametrization and for feature aware mesh filtering on point set surfaces.

Point set surfaces have a more than 15 year long history in geometry processing and computer graphics as they naturally arise in 3D-data acquisition processes. A guiding principle of these algorithms is the direct processing of raw scanning data without prior meshing – a principle that has a long-established history in classical numerical computations. However, their usage mostly restricts to full dimensional domains embedded in R² or R³ and a thorough investigation of a differential geometric representation of point set surfaces and their properties is not available.

Inspired by the notion of manifolds, we develop new concepts for meshless charts and atlases. These are used to implement higher order differential operators including curvature descriptors. On this solid basis of meshless differential operators, we develop novel algorithms for important geometry processing tasks, such as feature recognition, filtering operations, and surface parameterization.

Project C05 is a MATHEON associated project.