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Register a new userBayesian Image Reconstruction Based on Voronoi Diagrams
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G. F. Cabrera
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We present a Bayesian Voronoi image reconstruction technique (VIR) for interferometric data. Bayesian analysis applied to the inverse problem allows us to derive the a-posteriori probability of a novel parameterization of interferometric images. We use a variable Voronoi diagram as our model in place of the usual fixed pixel grid. A quantization of the intensity field allows us to calculate the likelihood function and a-priori probabilities. The Voronoi image is optimized including the number of polygons as free parameters. We apply our algorithm to deconvolve simulated interferometric data. Residuals, restored images and chi^2 values are used to compare our reconstructions with fixed grid models. VIR has the advantage of modeling the image with few parameters, obtaining a better image from a Bayesian point of view.
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We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram $mathcal{V}(S)$ (and several variants thereof) of a set $S$ of $n$ sites in the plane as sites are added. We define a general update operation for planar graphs modeling the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in $mathbb{R}^3$. We show that the amortized number of edge insertions and removals needed to add a new site is $O(sqrt{n})$. A matching $Omega(sqrt{n})$ combinatorial lower bound is shown, even in the case where the graph of the diagram is a tree. This contrasts with the $O(log{n})$ upper bound of Aronov et al. (2006) for farthest-point Voronoi diagrams when the points are inserted in order along their convex hull. We present a semi-dynamic data structure that maintains the Voronoi diagram of a set $S$ of $n$ sites in convex position. This structure supports the insertion of a new site $p$ and finds the asymptotically minimal number $K$ of edge insertions and removals needed to obtain the diagram of $S cup {p}$ from the diagram of $S$, in time $O(K,mathrm{polylog} n)$ worst case, which is $O(sqrt{n};mathrm{polylog} n)$ amortized by the aforementioned result. The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained at all times and can be traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in $O(log n)$ time, or to determine whether two given sites are neighbors in the Delaunay triangulation.
The Voronoi diagrams are an important tool having theoretical and practical applications in a large number of fields. We present a new procedure, implemented as a set of CUDA kernels, which detects, in a general and efficient way, topological changes in case of dynamic Voronoi diagrams whose generating points move in time. The solution that we provide has been originally developed to identify plastic events during simulations of soft-glassy materials based on a Lattice Boltzmann model with frustrated-short range attractive and mid/long-range repulsive-interactions. Along with the description of our approach, we present also some preliminary physics results.
In this paper we initiate the study of tropical Voronoi diagrams. We start out with investigating bisectors of finitely many points with respect to arbitrary polyhedral norms. For this more general scenario we show that bisectors of three points are homeomorphic to a non-empty open subset of Euclidean space, provided that certain degenerate cases are excluded. Specializing our results to tropical bisectors then yields structural results and algorithms for tropical Voronoi diagrams.
Torsional modes within a complex molecule containing various functional groups are often strongly coupled so that the harmonic approximation and one-dimensional torsional treatment are inaccurate to evaluate their partition functions. A family of multi-structural approximation methods have been proposed and applied in recent years to deal with the torsional anharmonicity.However, these methods approximate the exact almost periodic potential energy as a summation of local periodic functions with symmetric barrier positions and heights. In the present theoretical study, we illustrated that the approximation is inaccurate when torsional modes present non-uniformly distributed local minima. Thereby, we proposed an improved method to reconstruct approximate potential to replace the periodic potential by using information of the local minima and their Voronoi tessellation. First, we established asymmetric barrier heights by introducing two periodicity parameters and assuming that the exact barrier positions are at the boundaries of Voronoi cells. Second, we used multiplicatively weighted Voronoi tessellation to refine the barrier heights and positions by defining a structure-related distance metric. The proposed method has been tested for a few higher-dimensional cases, all of which show promising improved accuracy.
Given a tesselation of the plane, defined by a planar straight-line graph $G$, we want to find a minimal set $S$ of points in the plane, such that the Voronoi diagram associated with $S$ fits $G$. This is the Generalized Inverse Voronoi Problem (GIVP), defined in cite{Trin07} and rediscovered recently in cite{Baner12}. Here we give an algorithm that solves this problem with a number of points that is linear in the size of $G$, assuming that the smallest angle in $G$ is constant.
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