Plane Sweep Research Articles

Skeleton computation of orthogonal polyhedra

AbstractSkeletons are powerful geometric abstractions that provide useful representations for a number of geometric operations. The straight skeleton has a lower combinatorial complexity compared with the medial axis. Moreover, while the medial axis of a polyhedron is composed of quadric surfaces the straight skeleton just consist of planar faces. Although there exist several methods to compute the straight skeleton of a polygon, the straight skeleton of polyhedra has been paid much less attention. We require to compute the skeleton of very large datasets storing orthogonal polyhedra. Furthermore, we need to treat geometric degeneracies that usually arise when dealing with orthogonal polyhedra. We present a new approach so as to robustly compute the straight skeleton of orthogonal polyhedra. We follow a geometric technique that works directly with the boundary of an orthogonal polyhedron. Our approach is output sensitive with respect to the number of vertices of the skeleton and solves geometric degeneracies. Unlike the existing straight skeleton algorithms that shrink the object boundary to obtain the skeleton, our algorithm relies on the plane sweep paradigm. The resulting skeleton is only composed of axis‐aligned and 45° rotated planar faces and edges.

Image analysis of cracks in concrete: methodology, opportunities and pitfalls

Damage in concrete is mostly visualised in sections or at the surface of specimens subjected to internal and/or external loading. It continuously develops by growth and coalescence of tiny cracks into a spatial network structure. This structure can be seen as the fi nger-print of the material refl ecting its history of loadings under given environmental conditions. The methodology of contrast improvement as an essential link in visualising damage is touched upon. However, a major focus of the paper is on describing damage by submitting images of crack patterns consisting of lineal features in the plane to a sweeping test line system and counting intersections. To obtain three-dimensional damage information in an economic way, the damage structure is assumed in the most general case revealing a partial orientation of mixed lineal and planar nature (the so-called ‘Stroevenconcept’). The practical cases are elaborated of prevailing compressive and tensile stresses. This reduces the number of unknown crack portions to two. As a consequence, quantitatively analyzing the image patterns can be restricted to vertical sections only. This involves a dramatic reduction of sawing efforts and simplifi es the image analysis stage as well, of course. Only two orthogonal intersection counting operations are required for the assessment of specifi c crack surface area and of the degree and direction of preferred crack orientation. When observations would have been obtained in more directions, so-called roses of intersections (or intersection densities) can be determined. For very large images this would be circles. For random cracks in the image plane a circle around the origin, for oriented cracks, a circle through the origin is found. This concept, in addition to mathematical formulations is employed to demonstrate that automation of quantitative image analysis generally yields biased information, unless the analysis is executed under conditions discussed herein.

Overlay Algorithm for Simple Subdivision of Arbitrary Polygon in Plane

The overlay algorithm is an important branch in computational geometry field, it is an important process for computing exact Minkowski sum of two convex polyhedrons. By improving the existing plane sweep algorithm, the overlay algorithm for simple subdivision of arbitrary polygon in plane is given. The algorithm can be used to overlay arbitrary polygon after subdivision into simple polygon in the plane. It has lower time complexity than the existing overlay algorithm. The whole algorithm consists of three steps: line segment intersection, reconstructing topology and constructing the DCEL for overlay graph. The results show that the algorithm can compute the overlay of two planar subdivisions in linear time.

Image Plane Sweep Volume Illumination

In recent years, many volumetric illumination models have been proposed, which have the potential to simulate advanced lighting effects and thus support improved image comprehension. Although volume ray-casting is widely accepted as the volume rendering technique which achieves the highest image quality, so far no volumetric illumination algorithm has been designed to be directly incorporated into the ray-casting process. In this paper we propose image plane sweep volume illumination (IPSVI), which allows the integration of advanced illumination effects into a GPU-based volume ray-caster by exploiting the plane sweep paradigm. Thus, we are able to reduce the problem complexity and achieve interactive frame rates, while supporting scattering as well as shadowing. Since all illumination computations are performed directly within a single rendering pass, IPSVI does not require any preprocessing nor does it need to store intermediate results within an illumination volume. It therefore has a significantly lower memory footprint than other techniques. This makes IPSVI directly applicable to large data sets. Furthermore, the integration into a GPU-based ray-caster allows for high image quality as well as improved rendering performance by exploiting early ray termination. This paper discusses the theory behind IPSVI, describes its implementation, demonstrates its visual results and provides performance measurements.

Variants of the Quantized Visibility Graph for Efficient Path Planning

We propose variants of the quantized visibility graph (QVG) for efficient path planning. Conventional visibility graphs have been used for path planning when the obstacles are polygonal. The QVG extends its usability to arbitrarily-shaped objects by representing the obstacles as polygons. We propose QVG variants which represent all combinations of three factors, each with two alternatives: (i) quantization level (fixed-level or multiple-level), (ii) object representation method (inner and boundary cells together or boundary cells only), and (iii) methods used to check whether pairs of points are mutually visible (rotational plane sweep algorithm or sign inequality discrimination (SID) algorithm). In the verification of the efficiency of the proposed QVGs, (i) all QVGs produced the same best path, which was shorter than the convectional algorithms, (ii) computational cost to find the shortest path is lower when using QVGs than when using the convectional algorithms and (iii) the QVG that uses multi-level quantization, partial obstacle representation and SID visibility checking provides the shortest best path and has lower computational cost than all other methods.

Aspects of the Facet‐Edge Data Structure for the Construction of Voronoi Cells

AbstractThe construction of Voronoi diagrams has two main aspects: The construction algorithm and the data structure. For the construction of planar Voronoi diagrams e.g. the well known plane sweep algorithm can be applied. Another efficient method is the incremental construction of the Delaunay triangulation by using the quad‐edge data structure. Within this data structure the Delaunay triangulation is treated as a planar graph. Incremental construction implies that manipulations of the triangulation are allowed. Its Voronoi diagram is obtained simply by accessing the triangulation's dual graph. We extend this method to tree dimensions, by implementing a 3D Delaunay triangulation on the facet‐edge data structure. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Spatial Modelling for Mobile Robot’s Vision-based Navigation

We present an algorithm to model 3D workspace and to understand test scene for mobile robot's navigation or human computer interaction. This has done by line-based modeling and recognition algorithm. Line-based recognition using 3D lines has been tried by many researchers however its reliability still needs improvement due to ambiguity of 3D line feature information from original images. To improve the outcome, we approach firstly to find real planes using given 3D lines and then to implement recognition process. The methods we use are principle component analysis (PCA), plane sweep, occlusion query, and iterative closest point (ICP). During the implementation, we also use 3D map information for localization. We apply this algorithm to real test scene images and find out our result can be useful to identify doors or walls in indoor environment with better efficiency.

Arrangements on Parametric Surfaces I: General Framework and Infrastructure

We introduce a framework for the construction, maintenance, and manipulation of arrangements of curves embedded on certain two-dimensional orientable parametric surfaces in three-dimensional space. The framework applies to planes, cylinders, spheres, tori, and surfaces homeomorphic to them. We reduce the effort needed to generalize existing algorithms, such as the sweep line and zone traversal algorithms, originally designed for arrangements of bounded curves in the plane, by extensive reuse of code. We have realized our approach as the Cgal package Arrangement_on_surface_2. We define a compact and modular interface for our framework; for a given application a required small subset of the interface can be identified. Then, only this subset must be implemented. A companion paper describes concretizations for several types of surfaces and curves embedded on them, and applications. This is the first implementation of a generic algorithm that can handle arrangements on a large class of parametric surfaces.

A cell-based algorithm for evaluating directional distances in GIS

Directional distance is commonly used in geographical information systems as a measure of openness. In previous works, the sweep line method and the interval tree method have been employed to evaluate the directional distances on vector maps. Both methods require rotating original maps and study points in every direction of interest. In this article, we propose a cell-based algorithm that pre-processes a map only once; that is, it subdivides the map into a group of uniform-sized cells and records each borderline of the map into the cells traversed by its corresponding line segment. Based on the pre-processing result, the neighbouring borderlines of a study point can be directly obtained through the neighbouring cells of the point, and the borderlines in a definite direction can be simply acquired through the cells traversed by the half line as well. As a result, the processing step does not need to enumerate all the borderlines of the map when determining whether a point is on a borderline or finding the nearest intersection between a half line and the borderlines. Furthermore, we implement the algorithm for determining fetch length in coastal environment. Once the pre-processing is done, the algorithm can work in a complex archipelago environment such as to calculate the fetch lengths in multiple directions, to determine the inclusion property of a point, and to deal with the singularity of a study point on a borderline.

Sub-region data structure and its composition algorithm in pattern generator

To realize the software functional module in pattern generator,this paper gave a new data structure,doubly-connected edge list,and a well-designed composite algorithm of the sub-regions by using the plane sweep algorithm in computational geometry. In that algorithm,a general approach was given to process any situation appearing in sub-regions composition. To find the internal-hollowed polygon,a discriminate method and a search method were proposed based on graph reconstruction. The time complexity in this algorithm has a logarithmic item and linear space complexity,better than the quadratic one. The data structure is easy to realize in PCs because it is mainly composed of pointers and linked-lists.

격자화된 공간상에서 4중-나무 구조를 이용한 가시성 검사를 바탕으로 한 새로운 경로 계획 알고리즘과 그 개선 방안들

In this paper, we introduce a new path planning algorithm which combines the merits of a visibility graph algorithm and an adaptive cell decomposition. We quantize a given map with empty cells, blocked cells, and mixed cells, then find the optimal path on the quantized map using a visibility graph algorithm. For reducing the number of the quantized cells we use the quad-tree technique which is used in an adaptive cell decomposition, and for improving the performance of the visibility checking in making a visibility graph we propose a new visibility checking method which uses the property of the quad-tree instead of the well-known rotational sweep-line algorithm. For the more efficient visibility checking, we propose two additional improvements for our suggested method. Both of them are used for reducing the visited cells in the quad-tree. The experiments for a performance comparison of our algorithm with other well-known algorithms show that our proposed method is superior to others.

Low complexity algorithms for optimal consumer push-pull partial covering in the plane

This paper considers a model for locating a consumer within a bounded region in the plane with respect to a set of n existing pull-push suppliers. The objective is to maximize the difference of total profits and costs incurred due to the partial covering of the consumer by the suppliers pull and push influence areas. We develop efficient polynomial time algorithms for the resulting problems in the rectilinear and the Euclidean planes where the bounded region is either a rectangle or a constant size polygon, respectively. Based on these solutions, we develop algorithms for evaluating efficiently the objective function at any possible location of the consumer inside the bounded region. We also employ the algorithms for the Euclidean optimization problem and the rectilinear query computation to solve efficiently their corresponding dynamic versions, where an appearance of a new supplier or an absence of an existing one occurs. Being easy to implement due to the extensive use of simple data structures, such as the balanced and binary segment tree, and the employment of standard mechanisms, such as the sweep line, the Voronoi diagram and the circular ray shooting, our solutions potentially have wide usability.

Visibility-based modelling and control for network-based robotics

We present an algorithm to model 3D workspace and to understand test scene for navigation or human computer interaction in network-based mobile robot. This was done by line-based modelling and recognition algorithm. Line-based recognition using 3D lines has been tried by many researchers however its reliability still needs improvement due to ambiguity of 3D line feature information from original images. To improve the outcome, we approach firstly to find real planes using the given 3D lines and then to implement recognition process. The methods we use are principle component analysis (PCA), plane sweep, visibility test, and iterative closest point (ICP). During the implementation, we also use 3D map information for localization. We apply this algorithm to real test scene images and to find out our result can be useful to identify doors or walls in indoor environment with better efficiency.

Determining directional distances between points and shorelines using sweep line technique

Distances from points to closest shorelines in a given direction are used, for example, in some models for estimating wave exposure. Such distances, also called fetch lengths, can be determined using standard geographic information systems. However, performance may be a problem if these distances are required for a great number of study points. Two new algorithms for determining fetch lengths for study points in the same directions are presented in this paper. It is assumed that the two‐dimensional map is stored in vector format, i.e. shorelines of islands and mainland are stored as polygons. The first algorithm works on a set of undirected line segments derived from the shoreline polygons. The other works on a raster representation of the map. The algorithm saves memory by postponing the rasterisation until necessary. Both of the new algorithms have superior efficiency to a previously reported algorithm when the number of study points is large.

Computing visibility on terrains in external memory

Given an arbitrary viewpoint v and a terrain, the visibility map or viewshed of v is the set of points in the terrain that are visible from v . In this article we consider the problem of computing the viewshed of a point on a very large grid terrain in external memory. We describe algorithms for this problem in the cache-aware and cache-oblivious models, together with an implementation and an experimental evaluation. Our algorithms are a novel application of the distribution sweeping technique and use O ( sort ( n )) I/Os, where sort ( n ) is the complexity of sorting n items of data in the I/O-model. The experimental results demonstrate that our algorithm scales up and performs significantly better than the traditional internal-memory plane sweep algorithm and can compute visibility for terrains of 1.1 billion points in less than 4 hours on a low-cost machine compared to more than 32 hours with the internal-memory algorithm.

Performance Analysis of a Collision Detection Algorithm of Spheres Based on Slab Partitioning

In this paper, we consider a collision detection problem of spheres which asks to detect all pairs of colliding spheres in a set of n spheres located in d-dimensional space. We propose a collision detection algorithm for spheres based on slab partitioning technique and a plane sweep method. We derive a theoretical upper bound on the time complexity of the algorithm. Our bound tells that if both the dimension and the maximum ratio of radii of two spheres are bounded, then our algorithm runs in O(n log n + K) time with O(n + K) space, where K denotes the number of pairs of colliding spheres.

Computational Quantification of Peptides from LC-MS Data

Liquid chromatography coupled to mass spectrometry (LC-MS) has become a major tool for the study of biological processes. High-throughput LC-MS experiments are frequently conducted in modern laboratories, generating an enormous amount of data per day. A manual inspection is therefore no longer a feasible task. Consequently, there is a need for computational tools that can rapidly provide information about mass, elution time, and abundance of the compounds in a LC-MS sample. We present an algorithm for the detection and quantification of peptides in LC-MS data. Our approach is flexible and independent of the MS technology in use. It is based on a combination of the sweep line paradigm with a novel wavelet function tailored to detect isotopic patterns of peptides. We propose a simple voting schema to use the redundant information in consecutive scans for an accurate determination of monoisotopic masses and charge states. By explicitly modeling the instrument inaccuracy, we are also able to cope with data sets of different quality and resolution. We evaluate our technique on data from different instruments and show that we can rapidly estimate mass, centroid of retention time, and abundance of peptides in a sound algorithmic framework. Finally, we compare the performance of our method to several other techniques on three data sets of varying complexity.

A sweep-line algorithm for spatial clustering

This paper presents an agglomerative hierarchical clustering algorithm for spatial data. It discovers clusters of arbitrary shapes which may be nested. The algorithm uses a sweeping approach consisting of three phases: sorting is done during the preprocessing phase, determination of clusters is performed during the sweeping phase, and clusters are adjusted during the post processing phase. The properties of the algorithm are demonstrated by examples. The algorithm is also adapted to the streaming algorithm for clustering large spatial datasets.

Topological sweep of the complete graph

We present a novel, simple and easily implementable algorithm to report all intersections in an embedding of a complete graph. For graphs with N vertices and complexity K measured as the number of segments of the embedding, the running time of the algorithm is Θ ( K + N M ) , where M is the maximum number of edges cut by any vertical line. Our algorithm handles degeneracies, such as vertical edges or multiply intersecting edges, without requiring numerical perturbations to achieve general position. The algorithm is based on the sweep line technique, one of the most fundamental techniques in computational geometry, where an imaginary line passes through a given set of geometric objects, usually from left to right. The algorithm sweeps the graph using a topological line, borrowing the concept of horizon trees from the topological sweep method [H. Edelsbrunner, L.J. Guibas, Topologically sweeping an arrangement, J. Comput. Syst. Sci. 38 (1989) 165–194; J. Comput. Syst. Sci. 42 (1991) 249–251 (corrigendum)]. The novelty in our approach is to control the topological line through the use of the moving wall that separates at any time the graph into two regions: the region of known structure, in front of the moving wall, and the region that may contain intersections generated by edges–that have not yet been registered in the sweep process–behind the wall. Our method has applications to graph drawing and for depth-based statistical analysis, for computing the simplicial depth median for a set of N data points [G. Aloupis, S. Langerman, M. Soss, G. Toussaint, Algorithms for bivariate medians and a Fermat-Torricelli problem for lines, Comp. Geom. Theory Appl. 26 (1) (2003) 69–79]. We present the algorithm, its analysis, experimental results and extension of the method to general graphs.

Determining the structural layout of orthogonal framed buildings

When designing framed commercial buildings, achieving a satisfactory column layout is one of the primary objectives. Design systems to help designers with conceptual building design have, over the last decade, been developed using evolutionary algorithms. However, one limiting feature of the methods which have been developed is that the representation is limited to rectangular buildings. Most buildings are not rectangular and so this is a significant deficiency. A method of determining column layouts for orthogonal buildings is developed using the sweep line algorithm coupled to an adjacency graph. The method is tested on various examples and is shown to work well unless the building shape results in a large number of partitions.