An overview is presented of the use of spatial data structures in spatial databases. The focus is on hierarchical data structures, including a number of variants of quadtrees, which sort the data with respect to the space occupied by it. Suchtechniques are known as spatial indexing methods. Hierarchical data structures are based on the principle of recursive decomposition. They are attractive because they are compact and depending on the nature of the data they save space as well as time and also facilitate operations such as search. Examples are given of the use of these data structures in the representation of different data types such as regions, points, rectangles, lines, and volumes.

A tutorial survey is presented of hierarchical data structures for representing collections of small rectangles. Rectangles are often used as an approximation of shapes for which they serve as the minimum rectilinear enclosing object. They arise in applications in cartography as well as very large-scale integration (VLSI) design rule checking. The different data structures are discussed in terms of how they support the execution of queries involving proximity relations. The focus is on intersection and subset queries. Several types of representations are described. Some are designed for use with the plane-sweep paradigm, which works well for static collections of rectangles. Others are oriented toward dynamic collections. In this case, one representation reduces each rectangle to a point in a higher multidimensional space and treats the problem as one involving point data. The other representation is area based-that is, it depends on the physical extent of each rectangle.

Techniques are presented for navigating between adjacent triangles of greater or equal size in a hierarchical triangle mesh where the triangles are obtained by a recursive quadtree-like subdivision of the underlying space into four equilateral triangles. These techniques are useful in a number of applications including finite element analysis, ray tracing, and the modeling of spherical data. The operations are implemented in a manner analogous to that used in a quadtree representation of data on the two-dimensional plane where the underlying space is tessellated into a square mesh. A new technique is described for labeling the triangles which is useful in implementing the quadtree triangle mesh as a linear quadtree (i.e., a pointer-less quadtree); the navigation can then take place in this linear quadtree. When the neighbors are of equal size, the algorithms take constant time. The algorithms are very efficient, as they make use of just a few bit manipulation operations and can be impl...

Abstract--A ray tracing implementation is described that is based on an octree representation of a scene. Rays are traced through the scene by calculating the blocks through which they pass. This calculation is performed in a bottom-up manner through the use of neighbor finding. The octrees are assumed to be implemented by a pointer representation. The most basic operation in computer graphics is the conversion of an internal model of a three-dimensional scene into a two-dimensional scene that lies on the viewplane. The purpose is to generate an image of the

Algorithms are presented for moving between adjacent blocks in an octree representation of an image. Motion is possible in the direction of a face, edge, and a vertex, and between blocks of arbitrary size. The algorithms are based on a generalization and simplification of techniques developed earlier for two dimensions (i.e., in quadtrees). They are also applicable to quadtrees. The difference lies in the graph-theoretical classification of adjacencies-i.e., in terms of vertices, edges, and faces. Algorithms are given for octrees that are implemented with pointers and with pointerless representations such as the linear octree. 0 1989 Academic PRSS. ITIC. 1.

A number of algorithms are presented for obtaining~a~i&?ter representation for an image given its quadtree. The algorithms are given in an evolutionary manner starting with the straightforward top-down approach that visits each run in a row in succession starting at the root of the tree. The remaining algorithms proceed in a manner akin to an inorder tree traversal. All of the algorithms are analyzed and an indication is given as to when each is preferable. The execution time of all of the algorithms is shown to be proportional to the sum of the heights of the blocks comprising the image. 1.

The extraordinary format of spatial data and the fact that there is no straightforward mapping of spatial objects from the multidimensional space to the 1-dimensional space, stimulated various researchers during the past two decades to develop multidimensional access methods that facilitate efficient indexing of spatial objects in large databases. This survey paper tries a classification of existing multidimensional access methods, according to the types of data they are most suitable for (points or objects with spatial extent), their structure (hierarchical or flat), and their performance over spatial queries. Most of this work is based on an excellent survey paper [Gaed97]

Abstract. The use of bounding structures in the form of orthogonal polygons (also known as rectilinear polygons) with a varying number of vertices in contrast with a minimum bounding rectangle (an orthogonal polygon with just 4 vertices in two dimensions) as an object approximation method is presented. Orthogonal polygons can be used to improve the performance of the re ne step in the lterre ne query processing strategy employed in spatial databases. The orthogonal polygons are represented using the vertex representation implemented as a vertex list. The advantage of the vertex representation implemented as a vertex list is that it can be used to represent orthogonal polygons in arbitrary dimensions using just their vertices. This is in contrast to conventional methods such as the chain code which only work in two dimensions and cannot be extended to deal with higher dimensional data. Algorithms are given for varying the number of vertices used to represent the objects. It is shown that the use of non-trivial orthogonal polygons (i.e., with more than four vertices) is of bene t when a spatial index is used in the lter step for processing spatial queries such as point-in-object and windowing. If no spatial index is used, then all objects must be examined. In this case, many of the objects are small thereby not bene ting from the variation in the number of vertices that they have as the simple bounding box is adequate. 1

The size of five trie-based methods of sorting large collections of line segments in a spatial database is investigated analytically using a random lines image model and geometric probability techniques. The methods are based on sorting the line segments with respect to the space that they occupy. Since the space is two-dimensional, the trie is formed by interleaving the bits corresponding to the binary representation of the x and y coordinates of the underlying space and then testing two bits at each iteration. The result of this formulation yields a class of representations that are referred to as quadtrie variants, although they have been traditionally referred to as quadtree variants. The analysis differs from prior work in that it uses a detailed explicit model of the image instead of relying on modeling the branching process represented by the tree and leaving the underlying image unspecified. The analysis provides analytic expressions and bounds on the expected size of these quadtree variants. This enables the prediction of storage required by the representations and of the associated performance of algorithms that rely on them. The results are useful in two ways: 1. They reveal the properties of the various representations and permit their comparison using analytic, non-experimental, criteria. Some of the results confirm previous analyses (e.g., that the storage requirement of the MX quadtree is proportional to the total lengths of the line segments). An important new result is that for a PMR and Bucket PMR quadtree with sufficiently high values of the splitting threshold (i.e., # 4) the number of nodes is proportional to the numberof line segments and is independent of the maximum depth of the tree. This provides a theoretical justification for ...

www.cs.umd.edu/˜hjs Techniques are presented to progressively approximate and compress in a lossless manner two-colored (i.e. binary) 3D objects (as well as objects of arbitrary dimensionality). The objects are represented by a region octree implemented using a pointerless representation based on locational codes. Approximation is achieved through the use of a forest. This method labels the internal nodes of the octree as GB or GW, depending on the number of children being of type GB or GW. In addition, all BLACK nodes are labeled GB, while all WHITE nodes are labeled GW. A number of different image approximation methods are discussed that make use of a forest. The advantage of these methods is that they are progressive which means that as more of the object is transmitted, the better is the approximation. This makes these methods attractive for use on the worldwide web. Progressive transmission has the drawback that there is an overhead in requiring extra storage. Aprogressive forest-based approximation and transmission method is presented where the total amount of data that is transmitted is not larger than MIN(B,W), where B and W are the number of BLACK and WHITE blocks, respectively, in the region octree of the set of objects. 1