Geometrical Degeneracy Removal by Virtual Disturbances - An Application to Surface Reconstruction from Point Slice Samples

In surface reconstruction from slice samples (typical in medical imaging, coordinate measurement machines, stereolithography, etc.) the available methods attack the geometrical and topological aspects or a combinationof these. Topological methods classify the events occurred in the 2-manifold between two consecutive slices. Geometrical methods synthesize the surface based on local proximity of contours in consecutive slices. Many of these methods work with modifications of Voronoi Delaunay (VD) techniques, applied on slices i and i + 1. Superimposed 2D Voronoi Diagrams V Di and V Di+1 (used in surface reconstruction) present topological problems if, for example, a site of V Di lies on an site or an edge of V Di+1. The usual treatment of this problem in literature is to apply a geometrical disturbance to either V Di or V Di+1, thus eliminating the degeneracy. Recent works seek to quantify the amount of the disturbance applied in relation to the probability distribution of the event “change in the topology of VD”. In this article, in contrast, virtual disturbances are proposed and implemented, which allow for the application of subsequent steps of the algorithm at hand (in this case, tetrahedra construction for surface reconstruction) regardless of to the geometrical exception. Tetrahedra (or any other downstream constructs) can then be instantiated as per non-degenerate conditions. Although this method is applied for surface reconstruction, it gives insight as to how to circumvent degeneracies in procedures based on VD methods.