Borut Žalik

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.

A universal trapezoidation algorithm for planar polygons

Abstract A new algorithm for decomposing non-monotone planar polygons, which may contain holes, into trapezoids is described. The holes may be nested and may have common edges. In the first part of the paper, the main idea is explained for non-monotone polygons without holes, and the algorithm is then extended to polygons containing holes; the holes can also be decomposed into trapezoids, if desired. Finally, it is shown that the algorithm performs trapezoidation of general polygons in O( n 2 log 2 n ) time, where n is the common number of polygon vertices.

Surface interrogation methods

Abstract This paper deals with an overview and analysis of methods for inspecting surface quality. The overview is done by a new classification based on common mathematical concept yielding four groups of methods. First, techniques based on curvature, such as colour-coded map, lines of curvature, isolines of curvature, and focal surfaces are considered. They are followed by methods based on illumination models like isophotes, lv-curves, reflection lines, highlight lines, and shaded-image rendering. Then, methods using special linear transformations are included: polar and orthonomic surfaces. The final methods studied are contour lines. At the end, the results of a comparative analysis of all considered methods, such as their suitability for determining continuities, inflection points, and convex/concave regions, their speed, sensitivity to surface changes, and invariance to geometric transformations, which could be very interesting and useful for a user, are presented.

An almost distribution-independent incremental Delaunay triangulation algorithm

This paper presents a new incremental insertion algorithm for constructing a Delaunay triangulation. Firstly, the nearest point is found in order to speed up the location of a triangle containing a currently inserted point. A hash table and 1–3 deterministic skip lists, combined with a walking strategy, are used for this task. The obtained algorithm is compared with the most popular Delaunay triangulation algorithms. The algorithm has the following attractive features: it is fast and practically independent of the distribution of input points, it is not memory demanding, and it is numerically stable and easy to implement.

Construction of a non-symmetric geometric buffer from a set of line segments

The paper presents an algorithm for constructing the geometric outlines of a given set of line segments, using a sweep-line approach. The algorithm works in four steps: creating the basic geometric outlines, identifying the intersection points between them, constructing cycles, and determining the spatial relationships amongst the cycles. In the worst case, when each basic geometric buffer intersects all the other, the time complexity of the algorithm is O(n2 log n), where n is the number of input line segments. However, in practice, the expected time complexity is much lower, and this has been confirmed by experiments--it has been shown that O(n log n) time complexity is a good estimation in most situations. The algorithm has been used in GIS applications over a period and has proved to be successful.

Merging a set of polygons

Abstract The paper presents a new algorithm for merging a set of polygons based on a sweep-line technique. The algorithm operates on simple polygons and a limited set of non-simple polygons considered as legal polygons. At first, the algorithm separates loops and rings of the polygons located at the odd levels of hierarchy. The sweeping process is then started separately for the loops and the rings. During sweeping the loops, the inner polygons are identified. They participate at filling the holes what is done in the second step. The main task of the sweeping process is to determine common vertices and overlapping edges. After that, merging of the polygons is realized by a walk-about of the edges. The expected time complexity of the algorithm is O(k log k), k=m+I where m represents the total number of vertices of all polygons in the input set, and I is the number of touching edges among polygons. At the end the efficiency of the algorithm is illustrated using data from geodesy.

A universal chain code compression method

A novel universal algorithm for various chain code compressions is presented.New chain code binarization scheme is proposed.The compression is based on RLE and variation of LZ77.The method achieves on average higher compression than state-of-the-art methods. This paper introduces a new approach for lossless chain code compression. Firstly, the chain codes are converted into the binary stream, independent on the input chain code. Then, the compression is done using three modes: RLE 0 , LZ 77 0 and COPY. RLE 0 compresses the runs of the 0-bits, LZ 77 0 is a simplified version of LZ 77 and handles the repetitions within the bit stream, whilst COPY is an escape mode used, when the other two methods are unsuccessful. This method has been tested on the Freeman chain code in eight and four directions, the Vertex chain code, the Three OrThogonal chain code, and the Normalized angle difference chain code. The experiments confirmed better compression ratios on various benchmark datasets in comparison to the state-of-the-art lossless chain code compression methods.

Polygon trapezoidation by sets of open trapezoids

Abstract A new efficient algorithm is described for the simple trapezoidation of polygons based on a sweep-line paradigm. As the sweep-line glides over the plane, a set of so-called open trapezoids is generated and maintained. It is shown that a boundary case (more polygon vertices are located on the sweep-line) can be solved safely and does not slow down the algorithm. If desired, the polygon holes can be trapezoidated simultaneously. This proposed algorithm when compared with the fastest known algorithm developed by Seidel resulted in more efficiency for different classes of polygons.