Martijn Meijers

Topologically consistent 3D city models obtained by extrusion

One of the simplest methods to construct a 3D city model is to extrude building footprints to obtain ‘block-shaped’ polyhedra representing buildings. Although the method is well known and easy to implement, if the 2D topological relationships between the footprints are not taken into account, the resulting 3D city models will not necessarily be topologically consistent (i.e. primitives shared by 3D buildings will be duplicated and/or intersect each others). As a result, the model will be of little use for most applications, besides visualization that is. In this article, we present a new extrusion procedure to construct topologically consistent 3D city models. It is based on the use of a constrained triangulation, is conceptually simple and offers great flexibility to create city models in different formats (e.g. CityGML or a surface-based representation). We have implemented the procedure, tested it with real-world data sets and validated it.

Vario-scale data structures supporting smooth zoom and progressive transfer of 2D and 3D data

This paper introduces the concept of the smooth topological Generalized Area Partitioning (tGAP) structure represented by a space-scale partition, which we term the space-scale cube. We take the view of ‘map generalization as extrusion of data into an additional dimension’. For 2D objects the resulting vario-scale representation is a 3D structure, while for 3D objects the result is a 4D structure. This paper provides insights in: (1) creating valid data for the cube and proof that this is always possible for the implemented 2D tGAP generalization operators (line simplification, merge and split/collapse), (2) obtaining a valid 2D polygonal map representation at arbitrary scale from the cube, (3) using the vario-scale structure to provide smooth zoom and progressive transfer between server and client, (4) exploring which other possibilities the cube brings for obtaining maps having non-homogenous scales over their domain (which we term mixed-scale maps), and (5) using the same principles also for higher dimensional data; illustrated with 3D input data represented in a 4D hypercube. The proposed new structure has very significant advantages over existing multi-scale/multi-representation solutions (in addition to being truly vario-scale): (1) due to tight integration of space and scale, there is guaranteed consistency between scales, (2) it is relatively easy to implement smooth zoom, and (3) compact, object-oriented encoding is provided for a complete scale range.

A storage and transfer efficient data structure for variable scale vector data

This paper deals with efficient data handling of variable scale vector data. Instead of pre-building a collection of data sets on different scales, we create an index structure on the base data set (largest scale data) that enables us to extract a map at exactly the right scale the moment we need it. We present both the classic version of the tGAP (topological Generalized Area Partitioning) data structure for storing our variable scale map, as well as an ameliorated version, both based on topological concepts. We prove that the classic structure needs in a worst case scenario O(e 2) edges (with e the number of edges at largest scale). In practice we observed up to a factor 15 more edges in the variable scale data structure. The tGAP structure has been optimized to reduce geometric redundancy, but the explosion of additional edges is due to the changing topological references. Our main achievement finds its roots in the reduction of the number of edge rows to be stored for the ‘lean’ version (by removing the topological referential redundancy of the classic tGAP), which is beneficial both for storage and transfer. We show that storage space for the data set, plus the index, is less than twice the size of the original data set. The ‘lean’ tGAP, as the classic tGAP, offers true variable scale access to the data and has also improved performance, mainly due to less data communication between server and client.

Simultaneous & topologically-safe line simplification for a variable-scale planar partition

We employ a batch generalization process for obtaining a variable- scale planar partition. We describe an algorithm to simplify the boundary lines after a map generalization operation (either a merge or a split operation) has been applied on a polygonal area and its neighbours. The simplification is performed simultaneously on the resulting boundaries of the new polygonal areas that replace the areas that were processed. As the simplification strategy has to keep the planar partition valid, we define what we consider to be a valid planar partition (among other requirements, no zero-sized areas and no unwanted intersections in the boundary polylines). Furthermore, we analyse the effects of the line simplification for the content of the data structures in which the planar partition is stored.

A triangulation-based approach to automatically repair GIS polygons

Although the validation of a single GIS polygon can be considered as a solved issue, the repair of an invalid polygon has not received much attention and is still in practice a semi-manual and time-consuming task. We investigate in this paper algorithms to automatically repair a single polygon. Automated repair algorithms can be considered as interpreting ambiguous or ill-defined polygons and returning a coherent and clearly defined output (the definition of the international standards in our case). We present a novel approach, based on the use of a constrained triangulation, to automatically repair invalid polygons. Our approach is conceptually simple and easy to implement as it is mostly based on labelling triangles. It is also flexible: it permits us to implement different repair paradigms (we describe two in the paper). We have implemented our algorithms, and we report on experiments made with large real-world polygons that are often used by practitioners in different disciplines. We show that our approach is faster and more scalable than alternative tools. HighlightsThe repair of invalid polygons is still in practice a manual and time-consuming task.We investigate methods to automatically repair invalid polygons (according to the international standards).We propose a constrained triangulation-based approach, and show that in practice, with typical real-world polygons used by practitioners in disciplines related to the geosciences, it is fast, stable and scales to massive polygons.Our implementation is open-source and freely available under a GPL licence.

Continuous Road Network Generalization throughout All Scales

Until now, road network generalization has mainly been applied to the task of generalizing from one fixed source scale to another fixed target scale. These actions result in large differences in content and representation, e.g., a sudden change of the representation of road segments from areas to lines, which may confuse users. Therefore, we aim at the continuous generalization of a road network for the whole range, from the large scale, where roads are represented as areas, to mid- and small scales, where roads are represented progressively more frequently as lines. As a consequence of this process, there is an intermediate scale range where at the same time some roads will be represented as areas, while others will be represented as lines. We propose a new data model together with a specific data structure where for all map objects, a range of valid map scales is stored. This model is based on the integrated and explicit representation of: (1) a planar area partition; and (2) a linear road network. This enables the generalization process to include the knowledge and understanding of a linear network. This paper further discusses the actual generalization options and algorithms for populating this data structure with high quality vario-scale cartographic content.

Data Structures for Continuous Generalisation: tGAP and SSC

Spatial zoom and thematic navigation are indispensable functionalities for digital web and mobile maps. Therefore, recent map generalisation research has introduced the first truly smooth vario-scale structure (after several near vario-scale representations), which supports continuous or smooth zooming. In the implementation, the vario-scale representation of 2D geo-information can be stored as a single 3D (2D+scale) data structure. A single uniform scale map in 2D is then derived by computing a horizontal slice through the structure.

An Area Merge Operation for Smooth Zooming

When zooming a digital map it is often necessary that two or more area features must be merged. If this is done abruptly, it leads to big changes in geometry, perceived by the user as a “jump” on the screen. To obtain a gradual merge of two input objects to one output object this chapter presents three algorithms to construct a corresponding 3D geometry that may be used for the user’s smooth zooming operations. This is based on the assumption that every feature in the map is represented in 3D, where the 2D coordinates are the original representation, and 1D represents the scale as a Z value. Smooth zooming in or out is thus equivalent to the vertical movement of a horizontal slice plane (downwards or upwards).

Representing Three-Dimensional Topography in a DBMS With a Star-Based Data Structure

For storing and modelling three-dimensional topographic objects (e.g. buildings, roads, dykes and the terrain), tetrahedralisations have been proposed as an alternative to boundary representations. While in theory they have several advantages, current implementations are either not space efficient or do not store topological relationships (which makes spatial analysis and updating slow, or require the use of a costly 3D spatial index). We discuss in this paper an alternative data structure for storing tetrahedralisations in a DBMS. It is based on the idea of storing only the vertices and stars of edges; triangles and tetrahedra are represented implicitly. It has been used previously in main memory, but not in a DBMS—we describe how to modify it to obtain an efficient implementation in a DBMS. As we demonstrate with one real-world example, the structure is around 20 % compacter than implemented alternatives, it permits us to store attributes for any primitives, and has the added benefit of being topological. The structure can be easily implemented in most DBMS (we describe our implementation in PostgreSQL) and we present some of the engineering choices we made for the implementation.

The Potential of the 3D Dual Half-Edge (DHE) Data Structure for Integrated 2D-Space and Scale Modelling: A Review

Scaling factor is one of the most crucial aspect in 2D and 3D models especially in computer graphics, CAD, GIS, and games. Different user or/and application need different scale models during various stages of the use of data, including visualization and interaction. There are some arisen issues on 3D data model especially to meet GIS requirements while minimize the redundancy of the datasets. In GIS modelling, various data structures and data models have been proposed to support variety of applications and dimensionalities, but only a few in scale dimension. Some of them have succeeded in modelling scale such as in Space-Scale Cube (SSC) model. The recently implemented Dual Half-Edge (DHE) data structure within the PostgreSQL database is suitable for any valid 3D spatial model; not yet being explored for other dimensional such as scale environment. Using the same vario-scale approach, the DHE data model is also capable to implement a variable Level of Detail (LoD) representation such as SSC model. Some advantages of the DHE are described in this paper such as the dynamic property (valid updates based on Euler operations) and topology approach in comparison with other existing data structures. The last section of this paper describes capability of the DHE data structure to provide a better platform for GIS integrated space-scale data model.