Topologically correct surface reconstruction using alpha shapes and relations to ball-pivoting

The problem to reconstruct a surface given a finite set of boundary points is of growing interest, e.g. in the context of laser range images. While a lot of heuristic methods have been published in this context (e.g. the ball-pivoting algorithm), there exist only a few algorithms which guarantee the reconstruction to be homeomorphic to the original surface if a certain sampling density is reached. However, the sampling density mentioned is in most cases much higher than what seems to be sufficient on real data. In this paper we show how recently proved results about homology extraction from surface samples can be adopted to surface reconstruction and we significantly improve the bounds on the sampling density in case of noise-free samplings. This allows us to prove for the first time that the ball-pivoting algorithm reconstructs certain object surfaces without any topological changes and we can give bounds on the reconstruction error regarding both position and normal direction of the boundary.