FAST ROBUST ARITHMETICS FOR GEOMETRIC ALGORITHMS AND APPLICATIONS TO GIS

Geometric predicates are used in many GIS algorithms, such as the construction of Delaunay Triangulations for Triangulated Irregular Networks (TIN) or geospatial predicates. With floating-point arithmetic, these computations can incur roundoff errors that may lead to incorrect results and inconsistencies, causing computations to fail. This issue has been addressed using a combination of exact arithmetics for robustness and floating-point filters to mitigate the computational cost of exact computations. The implementation of exact computations and floating-point filters can be a difficult task, and code generation tools have been proposed to address this. We present a new C++ meta-programming framework for the generation of fast, robust predicates for arbitrary geometric predicates based on polynomial expressions. We show examples of how this approach produces correct results for GIS data sets that could lead to incorrect predicate results for naive implementations. We also show benchmark results that demonstrate that our implementation can compete with state-of-the-art solutions.

Recognizing Visibility Graphs of Triangulated Irregular Networks

A Triangulated Irregular Network (TIN) is a data structure that is usually used for representing and storing monotone geographic surfaces, approximately. In this representation, the surface is approximated by a set of triangular faces whose projection on the XY-plane is a triangulation. The visibility graph of a TIN is a graph whose vertices correspond to the vertices of the TIN and there is an edge between two vertices if their corresponding vertices on TIN see each other, i.e. the segment that connects these vertices completely lies above the TIN. Computing the visibility graph of a TIN and its properties have been considered thoroughly in the literature. In this paper, we consider this problem in reverse: Given a graph G, is there a TIN with the same visibility graph as G? We show that this problem is ∃ℝ-Complete.

3D Simplification Methods and Large Scale Terrain Tiling

This paper tackles the problem of generating world-scale multi-resolution triangulated irregular networks optimized for web-based visualization. Starting with a large-scale high-resolution regularly gridded terrain, we create a pyramid of triangulated irregular networks representing distinct levels of detail, where each level of detail is composed of small tiles of a fixed size. The main contribution of this paper is to redefine three different state-of-the-art 3D simplification methods to efficiently work at the tile level, thus rendering the process highly parallelizable. These modifications focus on the restriction of maintaining the vertices on the border edges of a tile that is coincident with its neighbors, at the same level of detail. We define these restrictions on the three different types of simplification algorithms (greedy insertion, edge-collapse simplification, and point set simplification); each of which imposes different assumptions on the input data. We implement at least one representative method of each type and compare both qualitatively and quantitatively on a large-scale dataset covering the European area at a resolution of 1/16 of an arc minute in the context of the European Marine Observations Data network (EMODnet) Bathymetry project. The results show that, although the simplification method designed for elevation data attains the best results in terms of mean error with respect to the original terrain, the other, more generic state-of-the-art 3D simplification techniques create a comparable error while providing different complexities for the triangle meshes.

A Modified Methodology for Generating Indoor Navigation Models

Automatic methods for constructing navigation routes do not fully meet all requirements. The aim of this study was to modify the methodology for generating indoor navigation models based on the Medial Axis Transformation (MAT) algorithm. The simplified method for generating corridor axes relies on the Node-Relation Structure (NRS) methodology. The axis of the modeled structure (corridor) is then determined based on the points of the middle lines intersecting the structure (polygon). The proposed solution involves a modified approach to the segmentation of corridor space. Traditional approaches rely on algorithms for generating Triangulated Irregular Networks (TINs) by Delaunay triangulation or algorithms for generating Thiessen polygons known as Voronoi diagrams (VDs). In this study, both algorithms were used in the segmentation process. The edges of TINs intersected structures. Selected midpoints on TIN edges, which were located in the central part of the structure, were used to generate VDs. Corridor structures were segmented by polygon VDs. The identifiers or structure nodes were the midpoints on the TIN edges rather than the calculated centroids. The generated routes were not zigzag lines, and they approximated natural paths. The main advantage of the proposed solution is its simplicity, which can be attributed to the use of standard tools for processing spatial data in a geographic information system.

A Modified Methodology for Generating Indoor Navigation Models

Automatic methods for constructing navigation routes do not fully meet all requirements. The aim of this study was to modify the methodology for generating indoor navigation models based on the Medial Axis Transformation (MAT) algorithm. The simplified method for generating corridor axes relies on the Node-Relation Structure (NRS) methodology. The axis of the modeled structure (corridor) is determined based the points of the middle lines intersecting the structure (polygon). The proposed solution involves a modified approach to the segmentation of corridor space. Traditional approaches rely on algorithms to construct Triangulated Irregular Networks (TINs) by Delaunay triangulation or algorithms for generating Thiessen polygons known as Voronoi diagrams (VDs). In this study, both algorithms were used in the segmentation process. The edges of TINs intersect structures. Selected midpoints on TIN edges, which are located in the central part of the structure, are used to generate VDs. Polygon VDs segment corridor structures. The identifiers or structure nodes are the midpoints on TIN edges rather than the calculated centroids. The generated routes are not zigzag lines, and they approximate natural paths. The main advantage of the proposed solution is its simplicity which can be attributed to the use of standard tools for processing spatial data in a geographic information system.