11411902
Kinetic connectivity for unit disks
We describe a kinetic data structure (KDS) that maintains the connected components of the union of a set of unit-radius disks moving in the plane. We assume that the motion of each disk can be specified by a low-degree algebraic trajectory; this trajectory, however, can be modified in an on-line fashion. While the disks move continuously, their connectivity changes at discrete times. Our main result is an O(n) space data structure that takes O(log n\slash \kern -1pt log log n) time per connectivity query of the form ``are disks A and B in the same connected component?’’ A straightforward approach based on dynamically maintaining the overlap graph requires Ω (n 2 ) space. Our data structure requires only linear space and must deal with O(n 2 + e ) updates in the worst case, each requiring O(log 2 n) amortized time, for any e>0 . This number of updates is close to optimal, since a set of n moving unit disks can undergo Ω (n 2 ) connectivity changes.
9035359
Visibility Queries in Simple Polygons and Applications
In this paper we explore some novel aspects of visibility for stationary and moving points inside a simple polygon P. We provide a mechanism for expressing the visibility polygon from a point as the disjoint union of logarithmically many canonical pieces using a quadraticspace data structure. This allows us to report visibility polygons in time proportional to their size, but without the cubic space overhead of earlier methods. The same canonical decomposition can be used to determine visibility within a frustum, or to compute various attributes of the visibility polygon efficiently. By exploring the connection between visibility polygons and shortest path trees, we obtain a kinetic algorithm that can track the visibility polygon as the viewpoint moves along polygonal paths inside P, at a polylogarithmic cost per combinatorial change in the visibility. The combination of the static and kinetic algorithms leads to a space query-time tradeoff for the visibility from a point problem and an output-sensitive algorithm for the weak visibility from a segment problem.
15556637
Proximity problems on moving points
A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dlmensions: the first structure maintains the closest pair of a set of continuously moving points, and is provably efficient. The second structure maintains a spanning tree of the moving points whose cost remains within some prescribed factor of the minimum spanning tree.
591702
Compact Voronoi Diagrams for Moving Convex Polygons
We describe a kinetic data structure for maintaining a compact Voronoi-like diagram of convex polygons moving around in the plane. We use a compact diagram for the polygons, dual to the Voronoi, first presented in [MKS96]. A key feature of this diagram is that its size is only a function of the number of polygons and not of their complexity. We demonstrate a local certifying property of that diagram, akin to that of Delaunay triangulations of points. We then obtain a method for maintaining this diagram that is output-sensitive and costs O(log n) per update. Furthermore, we show that for a set of k polygons with a total of n vertices moving along bounded degree algebraic motions, this dual diagram, and thus their compact Voronoi diagram, changes combinatorially Ω(n2) and O(kn2β(k)β(n)) times, where β(ċ) is an extremely slowly growing function. This compact Voronoi diagram can be used for collision detection or retraction motion planning among the moving polygons.
5430292
A practical evaluation of kinetic data structures
In many applications of computational geometry to modeling objects and processes in the physical world, the participating objects are in a state of continuous change. Motion is the most ubiquitous kind of continuous transformation but others, such as shape deformation, are also possible. In a recent paper, Baech, Guibas, and Hershberger [BGH97] proposed the framework of kinetic data structures (KDSS) as a way to maintain, in a completely on-line fashion, desirable information about the state of a geometric system in continuous motion or change. They gave examples of kinetic data structures for the maximum of a set of (changing) numbers, and for the convex hull and closest pair of a set of (moving) points in the plane. The KDS frameworkallowseach object to change its motion at will according to interactions with other moving objects, the environment, etc. We implemented the KDSSdescribed in [BGH97],es well as came alternative methods serving the same purpose, as a way to validate the kinetic data structures framework in practice. In this note, we report some preliminary results on the maintenance of the convex hull, describe the experimental setup, compare three alternative methods, discuss the value of the measures of quality for KDSS proposed by [BGH97],and highlight some important numerical issues.
9735559
Probabilistic analysis for combinatorial functions of moving points
We initiate a probabilistic study of configuration functions of moving points. In our probabilistic model, a particle is given an initiaf position and a velocity drawn independently at random from the same distribution D. We show that if n particles are drawn independently at random from the uniform distribution on the square, their convex hull undergoes El(logz n) combinatorial changes in expectation, their Voronoi diagram undergoes e(n312 ) combinatorial changes, and their closest pair undergoes El(n) combinatorial changes.
6311626
Combining strengths of circuit-based and CNF-based algorithms for a high-performance SAT solver
We propose Satisfiability Checking (SAT) techniques that lead to a consistent performance improvement of up to 3x over state-of-the-art SAT solvers like Chaff on important problem domains in VLSI CAD. We observe that in circuit oriented applications like ATPG and verification, different software engineering techniques are required for the portions of the formula corresponding to learnt clauses compared to the original formula. We demonstrate that by employing the same innovations as in advanced CNF-based SAT solvers, but in a hybrid approach where these two portions of the formula are represented differently and processed separately, it is possible to obtain the consistently highest performing SAT solver for circuit oriented problem domains. We also present controlled experiments to highlight where these gains come from. Once it is established that the hybrid approach is faster, it becomes possible to apply low overhead circuit-based heuristics that would be unavailable in the CNF domain for greater speedup.
7408744
Projection defocus analysis for scene capture and image display
In order to produce bright images, projectors have large apertures and hence narrow depths of field. In this paper, we present methods for robust scene capture and enhanced image display based on projection defocus analysis. We model a projector's defocus using a linear system. This model is used to develop a novel temporal defocus analysis method to recover depth at each camera pixel by estimating the parameters of its projection defocus kemel in frequency domain. Compared to most depth recovery methods, our approach is more accurate near depth discontinuities. Furthermore, by using a coaxial projector-camera system, we ensure that depth is computed at all camera pixels, without any missing parts. We show that the recovered scene geometry can be used for refocus synthesis and for depth-based image composition. Using the same projector defocus model and estimation technique, we also propose a defocus compensation method that filters a projection image in a spatially-varying, depth-dependent manner to minimize its defocus blur after it is projected onto the scene. This method effectively increases the depth of field of a projector without modifying its optics. Finally, we present an algorithm that exploits projector defocus to reduce the strong pixelation artifacts produced by digital projectors, while preserving the quality of the projected image. We have experimentally verified each of our methods using real scenes.
16021441
Hybrid inter- and intra-wavelet scale image restoration
This paper exploits both the inter- and intra-scale interdependencies that exist in wavelet coefficients to improve image restoration from noise-corrupted data. Using an over-complete wavelet expansion, we group the wavelet coefficients with the same spatial orientation at several scales. We then apply the linear minimum mean squared-error estimation to smooth noise. This scheme exploits the inter-scale correlation information of wavelet coefficients. To exploit the intra-scale dependencies, we calculate the co-variance matrix of each vector locally using a centered square-shaped window. Experiments show that the proposed hybrid scheme significantly outperforms methods exploiting only the intra- or inter-scale dependencies. The performance of noise removal also depends on wavelet filters. In our experiments a biorthogonal wavelet, which best characterizes the image inter-scale dependencies, achieves the best results.
627898
Fault-tolerant data delivery for multicast overlay networks
Overlay networks represent an emerging technology for rapid deployment of novel network services and applications. However, since public overlay networks are built out of loosely coupled end-hosts, individual nodes are less trustworthy than Internet routers in carrying out the data forwarding function. Here we describe a set of mechanisms designed to detect and repair errors in the data stream. Utilizing the highly redundant connectivity in overlay networks, our design splits each data stream to multiple sub-streams which are delivered over disjoint paths. Each sub-stream carries additional information that enables receivers to detect damaged or lost packets. Furthermore, each node can verify the validity of data by periodically exchanging Bloom filters, the digests of recently received packets, with other nodes in the overlay. We have evaluated our design through both simulations and experiments over a network testbed. The results show that most nodes can effectively detect corrupted data streams even in the presence of multiple tampering nodes.
32889588
Enhanced visualization of time series through higher fourier harmonics
Visualization techniques can enable the exploration and detection of patterns and relationships in a complex data set by presenting the data in a graphical format in which the key characteristics become more apparent. A new visualization technique, the first Fourier harmonic projection (FFHP) was introduced to translate the multi-dimensional data into a two dimensional scatter plot where the spatial relationship of the points reflects the structure of the original data set. FFHP has been shown capable of visualizing various gene expression data sets. However, dimension arrangement is crucial for the effectiveness of FFHP and certain data, such as time series, prevents dimensions (time points) from being freely rearranged. In this paper, we present an alternative approach through higher Fourier harmonic projections to enhance the visualization. Our algorithm takes advantage of the theoretical meaning of the Fourier harmonics and "reshuffles" the dimensions of the data set without physically rearranging them. The experimental results demonstrated significant improvement of the visualizations.