Retrieval of Geographic Data using Ellipsoidal Quadtrees

Transcription

1 Retrieval of Geographic Data using Ellipsoidal Quadtrees Patrik Ottoson Department of Geodesy and Photogrammetry Royal Institute of Technology, KTH S , Stockholm, Sweden Abstract. Geographic visualisation systems require methods for efficient data access. Retrieval of geographic data from large databases, of tens to thousands of Gbytes, needs optimisation using spatial indexing mechanisms. This paper describes how the indexing mechanism based on Ellipsoidal Quadtrees, EQT, can be implemented in software, e.g. for real-time or Internet visualisation. EQT fulfils the so-called equal-area criterion, which means that all index cells can be constructed having equal area. For flexible usage, the data structures in the implementation phase are based on how the Lisp programming language handles hierarchical lists. The concerns of handling levels of details, LOD, and optimisation of data flow are presented. The primary objective to handle LOD in the way described in the paper is to manage and structure geographic data for different scales. This structure can reduce the data flow and the amount of data to be rendered, which means that applications of geographic information can be optimised. 1 Introduction Visualisation frequently demands optimisation, which can lead to special database solutions. These solutions usually offer fast and flexible access to data, which is obtained by using standard formats and viewers. Unfortunately, these viewers are usually not optimised for the requirements of geographic visualisation. Most viewers lack the possibility to create large scenes with variable resolution. In a standard viewer, for example, the horizon and objects far away can be rendered as textures. This may look nice, but it is just a cheap trick, which is not a sustainable and robust database solution. We will show how an indexing mechanism that considers the shape of the earth can be the basis for a robust database solution. The existing methods for global indexing originate from polyhedral tessellation, adaptive sub-divisions, and map projection techniques. The polyhedral tessellation is based on sub-dividing a tetrahedron, a hexahedron, an octahedron [1], [2], [9], a dodecahedron [15] or an icosahedron [4]. Adaptive sub-division means that tessellation of the earth (spherical approximation) is carried out using Voronoi polygons [6]. The most interesting map projection technique is based on UTM subdivision of the earth [7]. This is achieved by dividing the UTM-zones and sub-zones

2 into a regular grid of patches. The patches can be indexed with a linear quadtree technique. The UTM sub-division is inflexible, because the division has to follow the UTM zones and the area of the patches varies. In this paper, we will introduce the equalarea criterion, which means that all patches or index cells should be constructed having equal area. This criterion makes it possible to have approximately the same number of objects per index cell, which contributes to stabilising the indexing mechanism. A stable mechanism is expected to have approximately the same access time for any object in a database. Ottoson and Hauska [12] presented a technique that fulfils the equal-area criterion and considers the true shape of the earth, i.e. the ellipsoidal shape. This method is called Ellipsoidal Quadtree, EQT. The method is flexible, because the size of the top-level quadtree can be set arbitrarily. The quadtree technique introduced by Klinger and Dyer [5] and developed by Samet [13] can be explained as a tree-like representation of data that forks data into four new areas for each level. The tree can recursively be divided into smaller regions, where every region is represented by a spatial index, see fig. 1. Level 1 Level 2 Level 3 Level 4 Fig. 1. Quadtree representation of a polygon network sub-divided into four levels. This paper will present the implementation of the indexing mechanism based on Ellipsoidal Quadtrees, EQT. The main objective for this indexing mechanism is to gain fast access to geographic data for visualisation. Spatial indexing, based on EQT, can be used for any kind of visualisation, e.g. Internet visualisation of global or national web maps, and Virtual Reality applications. 2 Indexing mechanism Indexing mechanisms are used to speed up the retrieval of data under certain search conditions [3]. The mechanisms are usually based on either structuring data (e.g. sorted lists, clustering indices, and binary trees) or mathematical algorithms (e.g. hashing). The indices are used to create primary and secondary keys. Normally, at least one primary key is used for each table. This gives faster access to data and makes it possible to build relations between different tables. Geographic objects differ from ordinary database attributes. A common query for geographic objects is to find certain objects within an area (defined as a rectangle, a cube, a polygon, or a volume). Such a query implies that an additional indexing mechanism has to consider spatial extent.

3 2.1 Ellipsoidal quadtrees, EQT The ellipsoidal quadtree, EQT, was introduced elsewhere [12]. Here, we will shortly describe some basic facts about EQT. The surface of the earth ellipsoid can be divided into quadrangular facets. We call these facets ellipsoidal quadrangles (analogous to spherical triangles). Four normal vectors span the surface of each facet (fig. 2a). The sides of each facet are parallel to the meridians (north/south) and the parallels (east/west) (fig. 2b). The geometric model is based on the geodetic datum WGS84 and the reference ellipsoid GRS80. z λ ϕ 1 ϕ 2 x 2 x 1 Fig. 2. a) Definition of the facets along the earth ellipsoid. b) The surface of the earth divided with meridians and parallels. The facets spanned by the normal vectors can be used as a base for indexing and addressing. EQT has a hierarchical structure similar to ordinary quadtrees. The facets can be constructed in such a way that they have equal areas at the same level in the quadtree. The surface of the earth is divided into bands of facets parallel to the meridians and to the parallels. The division creates slices in the north/south direction and bands in the east/west direction, which results in a grid-like pattern. To fulfil the equal-area criterion either λ or ϕ has to vary. With constant λ can be used to compute ϕ 1 and ϕ 2, which is identical to ϕ. The size of a facet can be chosen arbitrarily, but λ can be set to reach the same dimensions for λ and ϕ, in metres, at the principal latitude. At a latitude in for example central Sweden, one degree in longitude has approximately the same length, in metres, as one half degree in latitude. It is possible to obtain suitable values for λ in order to get squared quadrangles at this latitude. The distribution of data is probably equal in latitude and longitude directions, which means that squared quadrangles are desirable. This helps to stabilise the indexing mechanism. Computation of the index key requires known and computed parameters: the longitude difference, λ, the principal latitude, ϕ p, the number of facets with the width of λ for one band, q λ, the area of each facet, slice, S. The computation of expressed as q λ, S, and S f, and the total area of a S f is described in [12]. The index key, k, is ( λ / λ) + int( S / S qλ k = int f ). (1)

4 Sub-division of the top-level, from the EQT division, can be carried out with quadtree decomposition technique. Thereby, each quadrangle is sub-divided into four new quadrangles. In geographic visualisation, the quadtree structure can be constructed to hold information in all nodes, not only in the terminal ones. This method is useful, because the hierarchical structure can be used for rendering multiple levels of detail, LOD. 3 Tree-search and shuffle mechanism Data retrieval is carried out by tree-search. In the implementation phase, use of the quadtree structure may be inconvenient. We will present a more flexible data structure, but still it is equivalent to the quadtree structure. In order to optimise the usage of data and reduce data flow, a shuffle mechanism can be implemented. This mechanism is used to avoid data retrieval already loaded. 3.1 Structuring of LOD Landscape visualisation normally implies large data sets. This raises the importance of optimising the visualisation process, which can be carried out using multiple levels of details, LOD. There are two options implementing LOD in the visualisation process: applying LOD either to the entire scene or to some parts of the scene. We will show how LOD can be used in a part of the scene (around a midpoint). The midpoint is defined as the virtual point of observation. The resolution and LOD can be reduced with increasing distance away from the midpoint. In the construction of the scene around a midpoint, we have to take into account that the facets are quadrangles. The quadtree structure of the database has to be considered as well, because in the quadtree structure data is retrieved block-wise which sets some restrictions. The careful reader may intuitively think that 3x3 quadrangles as in fig. 3b would be a good choice or a rounded shape as in fig. 3a. If only the distance to the midpoint was considered, the rounded shape would be preferable. Unfortunately, we have to follow the restrictions set by the quadtree structure and consequently exclude the shapes described in fig. 3a-b. Fig. 3. a) Rounded shape that grows by the distance to a midpoint. b) Quadratic shape, which grows in each direction, is based on 3x3 quadrangles. c) Quadratic shape, which follows the internal structure of the quadtree, is based on 4x4 quadrangles. The construction of fig. 3a-b is based on retrieving and rendering quadrangles that are not along the same branch of the quadtree. The only way to find a specific quadrangle is to search the tree from the top. Handling of data along the same branch

5 is more convenient, because subsequent smaller quadrangles are retrieved by following the branch. It is also convenient to handle whole entities, which consists of 2x2 quadrangles (a super-quadrangle). The larger quadrangles in fig. 3a-b are not identical to any stored quadrangle in the tree. To solve this problem, the quadrangles have to be modified or re-structured as shown in fig. 3c. Using the construction of fig. 3c means that only whole entities are used. The centre of viewing (the midpoint) should be placed as centrally as possible, which implies that the centre should be placed as far away as possible from the border to the next larger quadrangle. A structure that handles whole entities and centralises the centre of viewing can be constructed using 3x3 super-quadrangles, see fig. 4a. If the centre is placed within any of the four most centralised quadrangles, there are always at least two steps (quadrangles) to the border. This method can be inherited to all levels; i.e. the next level is also defined by 3x3 super-quadrangles (fig. 4b). Fig. 4. a) The centre is placed in the most centralised set of quadrangles. A super-quadrangle consists of 2x2 quadrangles. b) The shape of sub-divided quadrangles for three LODs. 3.2 Implementing the tree-search The nodes are found by tree-search to the desired level. Each level in the quadtree is equal to one level in the LOD. The process to retrieve the quadrangles for fig. 4b can be described as: 1. Use the co-ordinates of the midpoint to get the first entity (four quadrangles). 2. Place seed points around the retrieved entity to get the eight adjacent entities. 3. Place new seed points around each 3x3 super-quadrangles to get adjacent entities for each desired LOD. In the database, the nodes of the quadtree structure for fig. 4b appear as in fig. 5. Fig. 5. The quadtree structure of the scene in fig. 4b. The quadtree provides a comprehensive view of the data structure, but it is impractical in the implementation phase. It could be necessary to deviate from the quaternary division and to decompose some nodes into more or fewer nodes than

6 four. Flexible and robust implementation of the a quadtree structure can be carried out using the programming language Lisp's list handler, called car and cdr. The car of a list is the first item in the list and cdr is the rest of the list [8]. This means that no more than two pointers are used for each node, one for car and one for cdr, i.e. the car pointer goes down in the tree and the cdr pointer goes to the right in the tree. The usage of car and cdr is very flexible, because each branch may hold an arbitrarily number of nodes. The car and cdr structure can be used directly to handle the structure in fig. 5 without loss of meaning, see fig. 6. LOD 1 LOD 2 LOD j LOD j+1 LOD j+2 Fig. 6. The car and cdr structure of the scene in fig. 4b. LOD j+3 LOD j The shuffle mechanism Structuring data as shown in fig. 4b means that it is easy to implement a good shuffle mechanism, because whole entities are used and the resolution of data is unchanged for adjacent areas. When the midpoint moves to either of the adjacent quadtree entities, new data have to be loaded. In the shuffle mechanism, data already loaded are kept unchanged, see fig. 7. This means that only 5/9 or 3/9 of the total data set has to be loaded. The shuffle mechanism is not beneficial, if the midpoint is moved outside the adjacent quadrangles, because all data have to be loaded. Steps and translations to adjacent areas are the most common moves in visualisation. Fig. 7. The pictures show eight possible movements of the midpoint. The light grey and white areas represent data that are already loaded. The light grey areas are kept when the midpoint is moved. The dark grey areas represent data to be loaded.

7 4 Visualisation of EQT In this example, EQT is implemented for visualisation in order to show the index cells. In the normal case, the index cells are not visualised, but only used to store pointers to data. Land classification data from USGS [14] were used as the test data set, where the land classes were given pseudo colours. In fig. 8, we can see how the earth is divided into ellipsoidal quadrangles. The indexed data set is seamless and without borders all around the earth. Gaps do not exist at the top-level, because of the regularity of the quadrangles and the fact that the quadrangles have the same size. Fig. 8. The top-level of the ellipsoidal quadtree structure of the earth with North America turned to the observer. In this case, the area of each quadrangle is very large, about 1900 km 2. Normally, the area should be set to some few square kilometres, in order to get a manageable amount of data. At the National Land Survey of Sweden, we stored about objects per index cell. The amount of data per index cell may be set differently for various computers. Before implementing and building an index, the performance of the indexing mechanism should be tested and calibrated, in order to determine a suitable number of objects per index cell. The number of index cells should not be too large either, because many accesses to index cells increase data retrieval time. Fig. 8 is based on approximately squared quadrangles at the latitude 62 degrees north. The possibility to create squared index cells is practical for national databases, because data are supposed to have equal distribution all directions. Non-squared cells handle this condition poorly. In the example, the squared cells are created for an imagined database over Sweden. The quadrangles are stumpy near the equator and narrow near the poles. The area of the quadrangles is the same everywhere on the earth. The sub-division mechanism of EQT is used to obtain a structure where the resolution decreases away from a midpoint. Each quadtree level is divided into four

8 new quadrangles for each higher level, see fig. 9a-b. The figures also show the result of the shuffle mechanism when the midpoint has been moved in the Southeast direction. Fig. 9. a-b) Sub-division of the earth with EQT for two different midpoints. In the visualisation process, the quadrangles are rendered as two planar triangles. Rendering of quadtree models (here the index cells) requires some considerations. Since the curvature of the earth is the base for construction of the ellipsoidal quadrangles, the border between the quadrangles with higher and lower resolution results in gaps, see fig. 10. The gaps can be eliminated by lifting the smaller quadrangles up to the interpolated value between two corners of the adjacent larger quadrangles. The shared points are not moved, because the co-ordinates are the same. The procedure to lift and change the shape is a controlled procedure. Re-construction of the outer quadrangles will never result in worse accuracy than adjacent larger quadrangles. In fig. 10, each quadrangle is divided into two triangles, because triangles are the geometric primitives used in the visualisation process. Gap Fig. 10. The gap between quadrangles with higher and lower resolution. 5 Conclusion and discussion The background of EQT was described by Ottoson and Hauska [12]. EQT is a mechanism that can be used in geographic indexing to address three-dimensional geographic data. In this paper, it has been shown that the mechanism can be implemented in software. Ottoson and Hauska [12] have shown two methods to divide the surface of the earth, either with varying λ and constant ϕ or varying ϕ and constant λ. In general, it is preferable to use varying ϕ and constant λ, because it creates a regular grid (fig. 8). This regularity makes it possible to

9 handle LOD around a midpoint and to create the shuffle mechanism, which is shown in the paper. Equation (1) implies that the index key can be computed. This is the case for the nodes of the top-level, while the nodes for the subsequent levels are reached with treesearch. Ottoson and Hauska [12] showed that it is possible to find the subsequent nodes via an equation. By implementing such an equation, access time can be reduced, because finding the desired node only requires one file access. Finding the desired node with a tree-search may need several accesses, depending on the numbers of LODs. The nodes of the quadtree are given identity numbers incrementally. It would have been possible to construct a quadtree for direct index search. Each level in the quadtree would require a pre-defined number of pointers. For static databases, where data are stable and all nodes contain data, this would not cause any problem. For dynamic databases, where new nodes are created and decomposition is carried out continuously, it is not practical and difficult. For large databases with many nodes, we recommend that index keys should be eight-byte integers. An eight-byte integer makes it possible to use index cells for the whole earth as small as one square centimetre. In this case, the quadtree will have 28 levels requiring on the average 56 accesses to find a desired index cell. Here, we have discussed how EQT can be used to retrieve data from databases. The quadtree construction depends on the purpose of the database. Static or dynamic databases may set different requirements. EQT can be used for decomposition of data where tables of data pointers can be found in the end nodes. In visualisation, every node can have data for each LOD. Handling of LOD and the EQT structure can directly be connected to each other. The quadrangles can be approximately squared at some latitudes. The quadtree in itself can be used to search data as well as an indexing mechanism. All this shows that EQT is very flexible and it could be used for several problems. We solved our problem as described in the paper, but it is obvious that other applications may need other implementation strategies. The purpose of a LOD structure centred on a midpoint (fig. 4) is to reduce the amount of data needed for the entire scene. This structure makes it possible to create an efficient shuffle mechanism. The shuffle mechanism combined with geographic indexing speed up data access, but this alone is not sufficient for real-time visualisation. A strategy which includes adaptive data models, reduction of data, effective LOD handling, enhanced rendering, etc is required to optimise the visualisation process. This paper describes the mechanisms of data retrieval from geographic databases. The next step should be to test those mechanisms on a database containing geographic objects and demonstrate its usefulness and performance. Ottoson [10], [11] showed how quadtrees could be used in data fusion of a DEM and land use data. It could be interesting to find out how data fusion should be carried out using EQT, when considering the curvature of the earth.

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