Bregman Voronoi Diagrams: Properties, Algorithms and Applications

The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending on the class of objects, the distance functions and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g. k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connexion with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation. Finally, we give some applications of Bregman Voronoi diagrams which are of interest in the context of computational geometry and machine learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures

Succinct representation of triangulations with a boundary

We consider the problem of designing succinct geometric data structures while maintaining efficient navigation operations. A data structure is said succinct if the asymptotic amount of space it uses matches the entropy of the class of structures represented. For the case of planar triangulations with a boundary we propose a succinct representation of the combinatorial information that improves to 2.175 bits per triangle the asymptotic amount of space required and that supports the navigation between adjacent triangles in constant time (as well as other standard operations). For triangulations with $m$ faces of a surface with genus g, our representation requires asymptotically an extra amount of 36(g-1)lg m bits (which is negligible as long as g << m/lg m)

Evaluating Signs of Determinants Using Single-Precision Arithmetic

We propose a method of evaluating signs of 2×2 and 3×3 determinants with b-bit integer entries using only b and (b + 1)-bit arithmetic, respectively. This algorithm has numerous applications in geometric computation and provides a general and practical approach to robustness. The algorithm has been implemented and compared with other exact computation methods

Globally Optimal Spatio-temporal Reconstruction from Cluttered Videos

International audienceWe propose a method for multi-view reconstruction from videos adapted to dynamic cluttered scenes under uncontrolled imaging conditions. Taking visibility into account, and being based on a global optimization of a true spatio-temporal energy, it oilers several desirable properties: no need for silhouettes, robustness to noise, independent from any initialization, no heuristic force, reduced flickering results, etc. Results on real-world data proves the potential of what is, to our knowledge, the only globally optimal spatio-temporal multi-view reconstruction method

Dispelling the N^3 myth for the Kt jet-finder

At high-energy colliders, jets of hadrons are the observable counterparts of the perturbative concepts of quarks and gluons. Good procedures for identifying jets are central to experimental analyses and comparisons with theory. The Kt family of successive recombination jet finders has been widely advocated because of its conceptual simplicity and flexibility and its unique ability to approximately reconstruct the partonic branching sequence in an event. Until now however, it had been believed that for an ensemble of N particles the algorithmic complexity of the Kt jet finder scaled as N^3, a severe issue in the high multiplicity environments of LHC and heavy-ion colliders. We here show that the computationally complex part of Kt jet-clustering can be reduced to two-dimensional nearest neighbour location for a dynamic set of points. Borrowing techniques developed for this extensively studied problem in computational geometry, Kt jet-finding can then be performed in N ln N time. Code based on these ideas is found to run faster than all other jet finders in current use.Comment: 11 pages, 3 figures; v2, to appear in Phys.Lett.B, includes an extra section briefly discussing the issues of jet areas and pileup subtraction, and also the Cambridge/Aachen jet finde

Pore-scale Modeling of Viscous Flow and Induced Forces in Dense Sphere Packings

We propose a method for effectively upscaling incompressible viscous flow in large random polydispersed sphere packings: the emphasis of this method is on the determination of the forces applied on the solid particles by the fluid. Pore bodies and their connections are defined locally through a regular Delaunay triangulation of the packings. Viscous flow equations are upscaled at the pore level, and approximated with a finite volume numerical scheme. We compare numerical simulations of the proposed method to detailed finite element (FEM) simulations of the Stokes equations for assemblies of 8 to 200 spheres. A good agreement is found both in terms of forces exerted on the solid particles and effective permeability coefficients

Comparative validation of single-shot optical techniques for laparoscopic 3-D surface reconstruction

Intra-operative imaging techniques for obtaining the shape and morphology of soft-tissue surfaces in vivo are a key enabling technology for advanced surgical systems. Different optical techniques for 3-D surface reconstruction in laparoscopy have been proposed, however, so far no quantitative and comparative validation has been performed. Furthermore, robustness of the methods to clinically important factors like smoke or bleeding has not yet been assessed. To address these issues, we have formed a joint international initiative with the aim of validating different state-of-the-art passive and active reconstruction methods in a comparative manner. In this comprehensive in vitro study, we investigated reconstruction accuracy using different organs with various shape and texture and also tested reconstruction robustness with respect to a number of factors like the pose of the endoscope as well as the amount of blood or smoke present in the scene. The study suggests complementary advantages of the different techniques with respect to accuracy, robustness, point density, hardware complexity and computation time. While reconstruction accuracy under ideal conditions was generally high, robustness is a remaining issue to be addressed. Future work should include sensor fusion and in vivo validation studies in a specific clinical context. To trigger further research in surface reconstruction, stereoscopic data of the study will be made publically available at www.open-CAS.com upon publication of the paper

FastJet user manual

FastJet is a C++ package that provides a broad range of jet finding and analysis tools. It includes efficient native implementations of all widely used 2-to-1 sequential recombination jet algorithms for pp and e+e- collisions, as well as access to 3rd party jet algorithms through a plugin mechanism, including all currently used cone algorithms. FastJet also provides means to facilitate the manipulation of jet substructure, including some common boosted heavy-object taggers, as well as tools for estimation of pileup and underlying-event noise levels, determination of jet areas and subtraction or suppression of noise in jets.Comment: 69 pages. FastJet 3 is available from http://fastjet.fr