Perturbations and Vertex Removal in a 3D Delaunay Triangulation

Though Delaunay triangulations are very well known geometric data structures, the problem of the robust removal of a vertex in a three-dimensional Delaunay triangulation is still a problem in practice. We propose a simple method that allows to remove any vertex even when the points are in very degenerate configurations. The solution is available in \cgal\footnote{\texttt- {http://www.cgal.org} (releases 2.3 and 2.4)}

Efficient moving point handling for incremental 3D manifold reconstruction

As incremental Structure from Motion algorithms become effective, a good sparse point cloud representing the map of the scene becomes available frame-by-frame. From the 3D Delaunay triangulation of these points, state-of-the-art algorithms build a manifold rough model of the scene. These algorithms integrate incrementally new points to the 3D reconstruction only if their position estimate does not change. Indeed, whenever a point moves in a 3D Delaunay triangulation, for instance because its estimation gets refined, a set of tetrahedra have to be removed and replaced with new ones to maintain the Delaunay property; the management of the manifold reconstruction becomes thus complex and it entails a potentially big overhead. In this paper we investigate different approaches and we propose an efficient policy to deal with moving points in the manifold estimation process. We tested our approach with four sequences of the KITTI dataset and we show the effectiveness of our proposal in comparison with state-of-the-art approaches.Comment: Accepted in International Conference on Image Analysis and Processing (ICIAP 2015

3D boundary recovery by constrained Delaunay tetrahedralization

Three-dimensional boundary recovery is a fundamental problem in mesh generation. In this paper, we propose a practical algorithm for solving this problem. Our algorithm is based on the construction of a {\it constrained Delaunay tetrahedralization} (CDT) for a set of constraints (segments and facets). The algorithm adds additional points (so-called Steiner points) on segments only. The Steiner points are chosen in such a way that the resulting subsegments are Delaunay and their lengths are not unnecessarily short. It is theoretically guaranteed that the facets can be recovered without using Steiner points. The complexity of this algorithm is analyzed. The proposed algorithm has been implemented. Its performance is reported through various application examples

Preferred directions for resolving the non-uniqueness of Delaunay triangulations

AbstractThis note proposes a simple rule to determine a unique triangulation among all Delaunay triangulations of a planar point set, based on two preferred directions. We show that the triangulation can be generated by extending Lawson's edge-swapping algorithm and that point deletion is a local procedure. The rule can be implemented exactly when the points have integer coordinates and can be used to improve image compression methods

Constrained Delaunay tetrahedral mesh generation and refinement

A {\it constrained Delaunay tetrahedralization} of a domain in $\mathbb{R}^3$ is a tetrahedralization such that it respects the boundaries of this domain, and it has properties similar to those of a Delaunay tetrahedralization. Such objects have various applications such as finite element analysis, computer graphics rendering, geometric modeling, and shape analysis. This article is devoted to presenting recent developments on constrained Delaunay tetrahedralizations of piecewise linear domains. The focus is for the application of numerically solving partial differential equations using finite element or finite volume methods. We survey various related results and detail two core algorithms that have provable guarantees and are amenable to practical implementation. We end this article by listing a set of open questions