Minimal Sampling Criteria and Method for Nonmanifold Surface Reconstruction from Unorganized Points

Surface reconstruction traditionally deals with the problem to reconstruct the outer boundary of a real world object from a set of boundary sampling points. The interior of the object is not relevant. In case of topology preserving surface reconstruction, the topological equivalence between the outcome of the method and the original object is guaranteed. The framework on topology preserving surface reconstruction answers three questions: First, what are the requirements on the shape of the real world object and how does it have to be sampled? Second, if these requirements are fulfilled, how to define an algorithm to reconstruct the surface. Third, can it be proved that the outcome of the reconstruction is topologically equivalent to the original? Our research question extends the topology preserving surface reconstruction problem by assumption that several objects of the real world are given, i.e. the interior of the real world objects can consist of several regions with common boundary, and the sampling conditions of each object and its interior regions may vary. We generalize the surface reconstruction problem to a reconstruction of non-manifold boundary of multiple regions with guaranteed preservation of topological properties. Recently a method called “(α, β)-shape reconstruction” was presented which — given a set of unorganized sample points — results in a topologically correct mesh representation of the original boundary. However the approach requires globally uniform sampling density for each point of the boundary and both the sampling density parameters and the maximum sample point deviation have to be known during the reconstruction process. In our work we first theoretically and experimentally evaluate “(α, β)-shape reconstruction” and compare the method with related work. Our main contribution is a framework for non-manifold boundary reconstruction of multi-regional real world scene with locally variable sampling conditions which provably preserves original topological properties. Founded on our new shape descriptor we define for each point of the original boundary a locally variable feature size which is nonzero even in the non-manifold and non-smooth points. Using the new feature size we define locally variable sampling conditions. Ths can be seen as an extension of the reconstruction method “Geomagic WRAP c ©”. We prove that our reconstruction method results in a refinement of the original boundary which can provably be reduced to a topologically correct reconstruction. The evaluation of our new framework consists of theoretical proofs, generalizations of sampling criteria to compare the results with related work, extensive discussion and experimental results on laser range scanner and computer tomography data. Based on our new shape descriptor we propose a method to estimate the original feature size on sampling points and use this as a criterion for data set decimation. The resulting data set is a very sparse, adaptively sampled set of points which is guaranteed to be correctly reconstructed by our algorithm, while the other related reconstruction methods fail.