Visualizing hyperbolic Voronoi diagrams

We present an interactive software, HVD, that represents in-ternally the k-order hyperbolic Voronoi diagram of a finite set of sites as an equivalent clipped power diagram. HVD allows users to interactively browse the hyperbolic Voronoi diagrams and renders simultaneously the diagram in the five standard models of hyperbolic geometry: Namely, the Poincare ́ disk, the Poincare ́ upper plane, the Klein disk, the Beltrami hemisphere and the Weierstrass hyperboloid. 1

Further results on the hyperbolic Voronoi diagrams

In Euclidean geometry, it is well-known that the $k$-order Voronoi diagram in $\mathbb{R}^d$ can be computed from the vertical projection of the $k$-level of an arrangement of hyperplanes tangent to a convex potential function in $\mathbb{R}^{d+1}$: the paraboloid. Similarly, we report for the Klein ball model of hyperbolic geometry such a {\em concave} potential function: the northern hemisphere. Furthermore, we also show how to build the hyperbolic $k$-order diagrams as equivalent clipped power diagrams in $\mathbb{R}^d$. We investigate the hyperbolic Voronoi diagram in the hyperboloid model and show how it reduces to a Klein-type model using central projections.Comment: 6 pages, 2 figures (ISVD 2014

Visualization and Evolution of Software Architectures

Software systems are an integral component of our everyday life as we find them in tools and embedded in equipment all around us. In order to ensure smooth, predictable, and accurate operation of these systems, it is crucial to produce and maintain systems that are highly reliable. A well-designed and well-maintained architecture goes a long way in achieving this goal. However, due to the intangible and often complex nature of software architecture, this task can be quite complicated. The field of software architecture visualization aims to ease this task by providing tools and techniques to examine the hierarchy, relationship, evolution, and quality of architecture components. In this paper, we present a discourse on the state of the art of software architecture visualization techniques. Further, we highlight the importance of developing solutions tailored to meet the needs and requirements of the stakeholders involved in the analysis process

Bregman Voronoi diagrams

A preliminary version appeared in the 18th ACM-SIAM Symposium on Discrete Algorithms, pp. 746- 755, 2007International audienceThe 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 various variants of Voronoi diagrams depending on the class of objects, the distance function 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 one to define information-theoretic Voronoi diagrams in sta- tistical 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 connection with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation

Geodesic least squares regression on information manifolds

We present a novel regression method targeted at situations with significant uncertainty on both the dependent and independent variables or with non-Gaussian distribution models. Unlike the classic regression model, the conditional distribution of the response variable suggested by the data need not be the same as the modeled distribution. Instead they are matched by minimizing the Rao geodesic distance between them. This yields a more flexible regression method that is less constrained by the assumptions imposed through the regression model. As an example, we demonstrate the improved resistance of our method against some flawed model assumptions and we apply this to scaling laws in magnetic confinement fusion

A General Introduction To Graph Visualization Techniques

Generally, a graph is an abstract data type used to represent relations among a given set of data entities. Graphs are used in numerous applications within the field of information visualization, such as VLSI (circuit schematics), state-transition diagrams, and social networks. The size and complexity of graphs easily reach dimensions at which the task of exploring and navigating gets crucial. Moreover, additional requirements have to be met in order to provide proper visualizations. In this context, many techniques already have been introduced. This survey aims to provide an introduction on graph visualization techniques helping the reader to gain a first insight into the most fundamental techniques. Furthermore, a brief introduction about navigation and interaction tools is provided