Elliptic Gabriel Graph for Finding Neighbors in a Point Set

Point-based shape representation has received increased attention in recent years, mainly due to its simplicity. One of the most fundamental operations for point set processing is to find the neighbors of each point. To do this, mesh structures and neighborhood graphs are commonly used. However, though meshes are very popular in the field of computer graphics, neighbor relations encoded in a mesh are often distorted. Likewise, neighborhood graphs, such as the minimum spanning tree (MST), relative neighborhood graph (RNG), and Gabriel graph (GG), are also imperfect as they usually give too few neighbors for a given point. In this paper, we introduce a generalization of the Gabriel graph, named the EGG (Elliptic Gabriel Graph), which takes an elliptic influence region instead of the circular region in GG. In order to determine the appropriate aspect ratio α of the elliptic influence region of EGG, this paper also presents the relation between the aspect ratio α and the average valence of the resulting. Analytic and empirical test results are included.