Ivana Kolingerová

A cell-based point-in-polygon algorithm suitable for large sets of points

Abstract The paper describes a new algorithm for solving the point-in-polygon problem. It is especially suitable when it is necessary to check whether many points are placed inside or outside a polygon. The algorithm works in two steps. First, a grid of cells equal in size is generated, and the polygon is laid on that grid. A heuristic approach is proposed for cell dimensioning. The cells of the grid are marked as being inside, outside, or on the polygon border. A modified flood-fill algorithm is applied for cell classification. In the second step, points are tested individually. If the tested point falls into an inner or an outer cell, the result is returned without any additional calculations. If the cell contains the polygon border, it is possible to determine the local point position. The analysis of time complexity shows that the initialization is finished in O(n n ) time, while the expected time complexity for checking an individual point is O( n ) , where n represents the number of polygon edges. The algorithm works with O ( n ) space complexity. The paper also gives practical results using artificial and real polygons from a GIS environment.

An incremental construction algorithm for Delaunay triangulation using the nearest-point paradigm

This paper introduces a new algorithm for constructing a 2D Delaunay triangulation. It belongs to the class of incremental insertion algorithms, which are known as less demanding from the implementation point of view. The most time consuming step of the incremental insertion algorithms is locating the triangle containing the next point to be inserted. In this paper, this task is transformed to the nearest point problem, which is solved by a two-level uniform subdivision acceleration technique. Dependencies on the distribution of the input points are reduced using this technique. The algorithm is compared with other popular triangulation algorithms: two variants of Guibas, Knuth, and Sharirs incremental insertion algorithm, two different implementations of Mückes algorithm, Fortunes sweep-line algorithm, and Lee and Schachters divide and conquer algorithm. The following point distributions are used for tests: uniform, regular, Gaussian, points arranged in clusters, and real data sets from a GIS database. Among all tested algorithms, the divide and conquer approach turns out to be the best. The proposed algorithm is the second fastest except for input points with highly non-uniform distribution. As implementation of the algorithm is simple, it represents an attractive alternative to other Delaunay triangulation algorithms used in practice.

Improvements to randomized incremental Delaunay insertion

Abstract Delaunay triangulation construction is one of the fundamental problems we are facing in computer graphics and computational geometry. As a result, many solutions have been developed, incremental insertion being one of the most popular algorithms. Although it is not worst-case optimal, it is simple, robust and behaves well in expected time. This paper suggests two improvements to the algorithm. The first one speeds up the computation without increasing memory requirements. The second refinement decreases memory requirements, trading space for small slow down. Both improvements are easy to implement and can be used either side-by-side or each of them independently.

Parallel Delaunay triangulation in E2 and E3 for computers with shared memory

This paper presents several parallel algorithms for the construction of the Delaunay triangulation in E^2 and E^3-one of the fundamental problems in computer graphics. The proposed algorithms are designed for parallel systems with shared memory and several processors. Such a hardware configuration (especially the case with two-processors) became widely spread in the last few years in the computer graphics area. Some of the proposed algorithms are easy to be implemented but not very efficient, while some of them prove opposite characteristics. Some of them are usable in E^2 only, other work in E^3 as well. The algorithms themselves were already published in computer graphics where the computer graphics criteria were highlighted. This paper concentrates on parallel and systematic point of view and gives detailed information about the parallelization of a computational geometry application to parallel and distributed computation oriented community.

Reconstructing domain boundaries within a given set of points, using Delaunay triangulation

Given an input set of planar points, which occupy a non-convex polygon area, possibly with holes, we reconstruct the shape of its boundary domain, without previous knowledge of which points or edges belong to the boundary. Our approach is based on different qualities of the Delaunay triangles inside and outside the domain. This method is heuristic and does not ensure success in all cases but it is very simple and there is no other method for this problem known to us. The method was derived on real GIS data but experiments show that it could also be used for mechanical engineering data, with positive results.

Multicriteria-optimized triangulations

Triangulation of a given set of points in a plane is one of the most commonly solved problems in computer graphics and computational geometry. Because they are useful in many applications, triangulations must provide well-shaped triangles. Many criteria have been developed to provide such meshes, namely weight and angular criteria. Each criterion has its pros and cons, some of them are difficult to compute, and sometimes even the polynomial algorithm is not known. By any of the existing deterministic methods, it is not possible to compute a triangulation which satisfies more than one criterion or which contains parts developed according to several criteria. We explain how such a mixture can be generated using genetic optimization.

Parallel Delaunay triangulation in E3: make it simple

The randomized incremental insertion algorithm of Delaunay triangulation in E3 is very popular due to its simplicity and stability. This paper describes a new parallel algorithm based on this approach. The goals of the proposed parallel solution are not only to make it efficient but also to make it simple. The algorithm is intended for computer architectures with several processors and shared memory. Several versions of the proposed method were tested on workstations with up to eight processors and on datasets of up to 200000 points with favorable results.

Fast Discovery of Voronoi Vertices in the Construction of Voronoi Diagram of 3D Balls

Solving geometrical problems on a set of 3D balls is a challenging task in computational geometry. They can be solved effectively when the Voronoi diagram for the set is available. The diagram is usually constructed by the edge-tracing or similar algorithms based on finding Voronoi vertices along edges. However, its expected quadratic time complexity makes it impractical. This can be improved significantly by our new approach. Whenever a vertex needs to be found, Delaunay triangulation of ball centers is searched through to find one specific ball. The search is kept inside a spatial filter, which can be reduced in size during the search. The improvement is demonstrated on protein data (a set of balls represents atoms in a molecule), because this is our intended application.

Technical Section: Comparison of triangle strips algorithms

Triangle surface models belong to the most popular type of geometric objects description in computer graphics. Therefore, the problem of fast visualization of this type of data is often solved. One popular approach is stripification, i.e., a conversion of a triangulated object surface into strips of triangles. This enables a reduction of the rendering time by reducing the data size which avoids redundant lighting and transformation computations. The problem of finding an optimal decomposition of triangle surface models to a set of strips is NP-hard and there exist a lot of different heuristic stripification techniques. This paper should help to orient in the jungle of stripification algorithms. We present an overview of existing stripification methods and detailed description of several important stripification methods for fully triangulated meshes. As different authors usually use different data sets and different architectures, it is nearly impossible to compare the quality of stripification methods. For this reason we also present a set of tests of these methods to give the reader a better possibility to compare these methods.

Parallel Delaunay triangulation based on circum-circle criterion

This paper describes a newly proposed simple and efficient parallel algorithm for the construction of the Delaunay triangulation (DT) in E2 by randomized incremental insertion. The construction of the DT is one of the fundamental problems in computer graphics. The proposed algorithm is designed for parallel systems with shared memory and several processors. Such hardware (especially with two-processors) became available in the last few years thanks to low prices and at present, there is still a lack of parallel algorithms that are simple to implement and efficient enough to be an attractive alternative to long existing serial algorithms. The designed algorithm incorporates new method for synchronization among PEs based on the simple geometric test (i.e. if no other points lie in the circum-circle of accessed triangle, this triangle can be modified independently on others PEs). We implemented the algorithm in C++ and tested it on workstations up to four processors where we reached relatively good speed-up to our serial implementation. When only two processors were used we reached even super-linear speed-up.