r-regular shape reconstruction from unorganized points

In thw paper, the problem of reconstructing a surface, given a set of scattered data points is addressed. First, a precise formulation of the reconstruction problem is proposed. The solution is mathematically defined as a particular mesh of the surface called the normalized mesh. This solution has the property to be included inside the Delaunay graph. A criterion to select boundary faces inside the Delaunay graph is proposed. This criterion is proven to provide the exact solution in 2D for points sampling a r-regular shapes with a sampling path c < 0.38r. In 3D, this results cannot be extended and the criterion cannot retrieve every faces. Some heuristics are then proposed in order to complete the surface. the object [4, 5, 6]. More complex graphs have also been introduced like crhulls and a-shapes [7, 8]. a-shapes are a generalization of the convex hull of a point set. An a-shape is a polytope surrounding the set of points. The parameter a controls the maximum “curvature” of any cavity of the polytope. Several a-shapes with different values of a are presented in figure 1. The choice of the parameter a might be tricky. O .::.. ”.... ... . . . . . . . . . . -.. . -.. “.. . . ..“