Structure Detection from a 3D Set of Points with Anisotropic alpha-Shapes

We present a method to extract polyhedral structures from a three-dimensional set of points, even if these structures are embedded in a perturbed background. The method is based on a family of affine diagrams which is an extension of the Voronoi diagram. These diagrams, namely anisotropic diagrams, are defined by using a parameterized distance whose unit ball is an ellipsoidal one. The parameters, upon which depends this distance, control the elongation and the orientation of the associated ellipsoidal ball. The triangulations, dual to the anisotropic diagrams, have the property to connect points that are not neighbors in the Voronoi diagram. Based on these triangulations, we define a family of three-dimensional anisotropic α-shapes. Unlike Euclidean α-shapes, anisotropic ones allow us to detect linear and planar structures in a given direction. The detection of more general polyhedral structures is obtained by merging several anisotropic α-shapes computed for different orientations.