Covering point sets with two disjoint disks or squares

Open archive-ElsevierWe study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks CR and CB with disjoint interiors such that the number of red points covered by CR plus the number of blue points covered by CB is maximized. We give an algorithm to solve this problem in O(n8/3 log2 n) time, where n denotes the total number of points. We also show that the analogous problem of finding two axis-aligned unit squares SR and SB instead of unit disks can be solved in O(nlog n) time, which is optimal. If we do not restrict ourselves to axis-aligned squares, but require that both squares have a common orientation, we give a solution using O(n3 log n) time

Thinning-free Polygonal Approximation of Thick Digital Curves Using Cellular Envelope

Since the inception of successful rasterization of curves and objects in the digital space, several algorithms have been proposed for approximating a given digital curve. All these algorithms, however, resort to thinning as preprocessing before approximating a digital curve with changing thickness. Described in this paper is a novel thinning-free algorithm for polygonal approximation of an arbitrarily thick digital curve, using the concept of "cellular envelope", which is newly introduced in this paper. The cellular envelope, defined as the smallest set of cells containing the given curve, and hence bounded by two tightest (inner and outer) isothetic polygons, is constructed using a combinatorial technique. This envelope, in turn, is analyzed to determine a polygonal approximation of the curve as a sequence of cells using certain attributes of digital straightness. Since a real-world curve=curve-shaped object with varying thickness, unexpected disconnectedness, noisy information, etc., is unsuitable for the existing algorithms on polygonal approximation, the curve is encapsulated by the cellular envelope to enable the polygonal approximation. Owing to the implicit Euclidean-free metrics and combinatorial properties prevailing in the cellular plane, implementation of the proposed algorithm involves primitive integer operations only, leading to fast execution of the algorithm. Experimental results that include output polygons for different values of the approximation parameter corresponding to several real-world digital curves, a couple of measures on the quality of approximation, comparative results related with two other well-referred algorithms, and CPU times, have been presented to demonstrate the elegance and efficacy of the proposed algorithm

Identification of Change in a Dynamic Dot Pattern and its use in the Maintenance of Footprints

Examples of spatio-temporal data that can be represented as sets of points (called dot patterns) are pervasive in many applications, for example when tracking herds of migrating animals, ships in busy shipping channels and crowds of people in everyday life. The use of this type of data extends beyond the standard remit of Geographic Information Science (GISc), as classification and optimisation problems can often be visualised in the same manner. A common task within these fields is the assignment of a region (called a footprint) that is representative of the underlying pattern. The ways in which this footprint can be generated has been the subject of much research with many algorithms having been produced. Much of this research has focused on the dot patterns and footprints as static entities, however for many of the applications the data is prone to change. This thesis proposes that the footprint need not necessarily be updated each time the dot pattern changes; that the footprint can remain an appropriate representation of the pattern if the amount of change is slight. To ascertain the appropriate times at which to update the footprint, and when to leave it as it is, this thesis introduces the concept of change identifiers as simple measures of change between two dot patterns. Underlying the change identifiers is an in-depth examination of the data inherent in the dot pattern and the creation of descriptors that represent this data. The experimentation performed by this thesis shows that change identifiers are able to distinguish between different types of change across dot patterns from different sources. In doing so the change identifiers reduce the number of updates of the footprint while maintaining a measurably good representation of the dot pattern

Voronoi diagrams in the max-norm: algorithms, implementation, and applications

Voronoi diagrams and their numerous variants are well-established objects in computational geometry. They have proven to be extremely useful to tackle geometric problems in various domains such as VLSI CAD, Computer Graphics, Pattern Recognition, Information Retrieval, etc. In this dissertation, we study generalized Voronoi diagram of line segments as motivated by applications in VLSI Computer Aided Design. Our work has three directions: algorithms, implementation, and applications of the line-segment Voronoi diagrams. Our results are as follows: (1) Algorithms for the farthest Voronoi diagram of line segments in the Lp metric, 1 ≤ p ≤ ∞. Our main interest is the L2 (Euclidean) and the L∞ metric. We first introduce the farthest line-segment hull and its Gaussian map to characterize the regions of the farthest line-segment Voronoi diagram at infinity. We then adapt well-known techniques for the construction of a convex hull to compute the farthest line-segment hull, and therefore, the farthest segment Voronoi diagram. Our approach unifies techniques to compute farthest Voronoi diagrams for points and line segments. (2) The implementation of the L∞ Voronoi diagram of line segments in the Computational Geometry Algorithms Library (CGAL). Our software (approximately 17K lines of C++ code) is built on top of the existing CGAL package on the L2 (Euclidean) Voronoi diagram of line segments. It is accepted and integrated in the upcoming version of the library CGAL-4.7 and will be released in september 2015. We performed the implementation in the L∞ metric because we target applications in VLSI design, where shapes are predominantly rectilinear, and the L∞ segment Voronoi diagram is computationally simpler. (3) The application of our Voronoi software to tackle proximity-related problems in VLSI pattern analysis. In particular, we use the Voronoi diagram to identify critical locations in patterns of VLSI layout, which can be faulty during the printing process of a VLSI chip. We present experiments involving layout pieces that were provided by IBM Research, Zurich. Our Voronoi-based method was able to find all problematic locations in the provided layout pieces, very fast, and without any manual intervention

Interdependence of Flow and Shape Morphological Dynamics For Flow-Induced Erosion of Bluff Bodies

Flow-induced erosion encompasses all processes in which fluid-solid interactions result in the removal and transport of material from the solid. The removed material may change its physical state and/or chemical composition and may be redeposited onto the solid body or advected away by the fluid and deposited elsewhere. Common to all flow induced erosion processes is that they involve an eroding surface, and eroding agent, and a fluid flow which delivers the eroding agent to the eroding surface. Consequently, the study of erosion is difficult as it requires detailed knowledge of the material, mechanical, and/or thermophysical properties of the eroding surface; the transport mechanisms that deliver the eroding agent to the eroding surface; and the transport mechanisms that entrain and advect the eroded material into and within the fluid flow. This difficulty is compounded by the fact that that there is a feedback coupling between the eroding surface and the fluid dynamics that control the transport mechanisms important to erosion. Specifically, during erosion, surface morphological changes to the eroding surface will alter the flow field thereby increasing or decreasing the rate at which the eroding agent is delivered to the eroding surface. This in turn alters the surface morphology. Thus a complex feedback cycle exists between the fluid and surface dynamics. The study of this feedback cycle has received little attention in the fluid mechanics community. This relative neglect is understandable due to its non-equilibrium nature, yet surprising when one considers how much erosion by the action of a flow is an integral part of major scientific and engineering fields, for example geophysics, environmental, manufacturing, and aerospace. The underlying research objective of this dissertation is to better understand the two-way coupling between an eroding body and the surface flux of the eroding agent by evaluating the shape dynamics of eroding bluff bodies through the erosion process. The problem is challenging since, as described above, the surface flux of the eroding agent will vary as the surface morphology of the eroding body evolves. In order to investigate the complex interdependence between the flow and surface morphology of an eroding body during flow-induced erosion, physical ablation and dissolution experiments will be performed and existing numerical datasets will be analyzed to: (i) re-evaluate existing scaling laws regarding geometric properties (cross-sectional area, wetted perimeter, and curvature) of bluff bodies undergoing erosion in (a) uniform, unidirectional flow, (b) in spatially and temporally varying flow, and (c) in convectively driven flow; (ii) identify a shape parameter of the eroding surface that is well-correlated with local evolutional changes to the eroding agent surface flux; and (iii) develop a simple feedback erosion model that bypasses the fluid dynamics and adjusts the local eroding agent surface flux based on the evaluation of the identified shape parameter. The focus on the erosion of bluff bodies was chosen because, in principle, it is more amenable to the study of the erosion feedback cycle as the evolution of the shape dynamics and morphological changes to the surface of the eroding bluff body are a direct result of the, unknown, instantaneous magnitude of the local eroding agent surface flux. Since the evolution of the local eroding agent surface flux is a direct consequence of the feedback from the eroding surface on the flow dynamics, an improved understanding of the erosion feedback cycle is possible by evaluating only the morphological changes to the surface of the eroding bluff body

Abstracts for the twentyfirst European workshop on Computational geometry, Technische Universiteit Eindhoven, The Netherlands, March 9-11, 2005

This volume contains abstracts of the papers presented at the 21st European Workshop on Computational Geometry, held at TU Eindhoven (the Netherlands) on March 9–11, 2005. There were 53 papers presented at the Workshop, covering a wide range of topics. This record number shows that the field of computational geometry is very much alive in Europe. We wish to thank all the authors who submitted papers and presented their work at the workshop. We believe that this has lead to a collection of very interesting abstracts that are both enjoyable and informative for the reader. Finally, we are grateful to TU Eindhoven for their support in organizing the workshop and to the Netherlands Organisation for Scientific Research (NWO) for sponsoring the workshop