Delaunay triangulation

About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.

Delaunay Tessellation of Proteins: Four Body Nearest Neighbor Propensities of Amino Acid Residues

Abstract: Delaunay tessellation is applied for the first time in the analysis of protein structure. By representing amino acid residues in protein chains by Cα atoms, the protein is described as a set of points in three-dimensional space. Delaunay tessellation of a protein structure generates an aggregate of space-filling irregular tetrahedra, or Delaunay simplices. The vertices of each simplex define objectively four nearest neighbor Cα atoms, i.e., four nearest-neighbor residues. A simplex classification scheme is introduced in which simplices are divided into five classes based on the relative positions of vertex residues in protein primary sequence. Statistical analysis of the residue composition of Delaunay simplices reveals nonrandom preferences for certain quadruplets of amino acids to be clustered together. This nonrandom preference may be used to develop a four-body potential that can be used in evaluating sequence–structure compatibililty for the purpose of inverted structure prediction.

On Bregman Voronoi diagrams

Abstract: The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a well-shaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a by-product, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other by convex duality or embedding. Moreover, we can always compute them indirectly as power diagrams in primal or dual spaces, or directly after linearization in an extra-dimensional space as the projection of a Euclidean polytope. Finally, our paper proposes to generalize Bregman divergences to higher-order terms, called κ-jet Bregman divergences, and touch upon their Voronoi diagrams.

Methods, apparatus and computer program products that reconstruct surfaces from data point sets

Abstract: Methods, apparatus and computer program products provide efficient techniques for reconstructing surfaces from data point sets. These techniques include reconstructing surfaces from sets of scanned data points that have preferably undergone preprocessing operations to improve their quality by, for example, reducing noise and removing outliers. These techniques include reconstructing a dense and locally two-dimensionally distributed 3D point set (e.g., point cloud) by merging stars in two-dimensional weighted Delaunay triangulations within estimated tangent planes. The techniques include determining a plurality of stars from a plurality of points p i in a 3D point set S that at least partially describes the 3D surface, by projecting the plurality of points p i onto planes T i that are each estimated to be tangent about a respective one of the plurality of points p i . The plurality of stars are then merged into a digital model of the 3D surface.