A modified binary space partitioning tree for geographic information systems

Spatial data structures form a core ingredient of many geometric algorithms, both in theory and in practice. Many of these data structures, especially the ones used in practice, are based on partitioning the underlying space (examples are binary space partitions and decompositions of polygons) or partitioning the set of objects (examples are bounding-volume hierarchies). The efficiency of such data structures---and, hence, of the algorithms that use them---depends on certain characteristics of the partitioning. For example the performance of many algorithms that use binary space partitions (BSPs) depends on the size of the BSPs. Similarly, the performance of answering range queries using bounding-volume hierarchies (BVHs) depends on the so-called crossing number that can be associated with the partitioning on which the BVH is based. Much research has been done on the problem of computing partitioning whose characteristics are good in the worst case. In this thesis, we studied the problem from a different point of view, namely instance-optimality. In particular, we considered the following question: given a class of geometric partitioning structures---like BSPs, simplicial partitions, polygon triangulations, …---and a cost function---like size or crossing number---can we design an algorithm that computes a structure whose cost is optimal or close to optimal for any input instance (rather than only worst-case optimal). We studied the problem of finding optimal data structures for some of the most important spatial data structures. As an example having a set of n points and an input parameter r, It has been proved that there are input sets for which any simplicial partitions has crossing number ¿(vr). It has also been shown that for any set of n input points and the parameter r one can make a simplicial partition with stabbing number O(vr). However, there are input point sets for which one can make simplicial partition with lower stabbing number. As an example when the points are on a diagonal, one can always make a simplicial partition with stabbing number 1. We started our research by studying BSPs for line segments in the plane, where the cost function is the size of the BSPs. A popular type of BSPs for line segments are the so-called auto-partitions. We proved that finding an optimal auto-partition is NP-hard. In fact, finding out if a set of input segments admits an auto-partition without any cuts is already NP-hard. We also studied the relation between two other types of BSPs, called free and restricted BSPs, and showed that the number of cuts of an optimal restricted BSP for a set of segments in R2 is at most twice the number of cuts of an optimal free BSP for that set. The details are being represented in Chapter 1 of the thesis. Then we turned our attention to so-called rectilinear r-partitions for planar point sets, with the crossing number as cost function. A rectilinear r-partition of a point set P is a partitioning of P into r subsets, each having roughly |P|/r points. The crossing number of the partition is defined using the bounding boxes of the subsets; in particular, it is the maximum number of bounding boxes that can be intersected by any horizontal or vertical line. We performed some theoretical as well as experimental studies on rectilinear r-partitions. On the theoretical side, we proved that computing a rectilinear r-partition with optimal stabbing number for a given set of points and parameter r is NP-hard. We also proposed an exact algorithm for finding optimal rectilinear r-partitions whose running time is polynomial when r is a constant, and a faster 2-approximation algorithm. Our last theoretical result showed that considering only partitions whose bounding boxes are disjoint is not sufficient for finding optimal rectilinear r-partitions. On the experimental side, we performed a comparison between four different heuristics for constructing rectilinear r-partitions. The so-called windmill KD-tree gave the best results. Chapter 2 of the thesis describes all the details of our research on rectilinear r-partitions. We studied another spatial data structure in Chapter 3 of the thesis. Decomposition of the interior of polygons is one of the fundamental problems in computational geometry. In case of a simple polygon one usually wants to make a Steiner triangulation of it, and when we have a rectilinear polygon at hand, one typically wants to make a rectilinear decomposition for it. Due to this reason there are algorithms which make Steiner triangulations and rectangular decompositions with low stabbing number. These algorithms are worst-case optimal. However, similar to the two previous data structures, there are polygons for which one can make decompositions with lower stabbing numbers. In 3 we proposed a 3-approximation for finding an optimal rectangular decomposition of a rectilinear polygon. We also proposed an O(1)-approximation for finding optimal Steiner triangulation of a simple polygon. Finally, in Chapter 4 of the thesis, we considered another optimization problem, namely how to approximate a piecewise-linear function F: R ¿R with another piecewise-linear function with fewer pieces. Here one can distinguish two versions of the problem. The first one is called the min-k problem; the goal is then to approximate the function within a given error e such that the resulting function has the minimum number of links. The second one is called the min-e problem; here the goal is to find an approximation with at most k links (for a given k) such that the error is minimized. These problems have already been studied before. Our contribution is to consider the problem for so-called uncertain functions, where the value of the input function F at its vertices is given as a discrete set of different values, each with an associated probability. We show how to compute an approximation that minimizes the expected error.

This paper discusses an approach to automating CAD model acquisition by creating binary space partitioning (BSP) trees from laser range data. Distinct views of the object, each represented by a BSP tree, are then merged using set operations. BSP trees have proven their utility in 3D modeling, graphics and image processing, and their tree structure allows efficient algorithms to be developed that are compact and numerically robust. These properties are of primary importance when considering an intermediate representation between raw sensor data and existing CAD models.

Non-revisiting stochastic search revisited: Results, perspectives, and future directions

An image coding technique based on symmetry extraction and Binary Space Partitioning (BSP) tree representation for still pictures is presented. Axes of symmetry, detected through a principal axis of inertia approach and a coefficient of symmetry measure, are used to divide recursively an input image into a finite number of convex regions. This recursive partitioning results in the BSP tree representation of the image data. The iterative partition occurs whenever the current left/right node of the tree cannot be represented `symmetrically' by its counterpart, i.e., the right/left node. This splitting process may also end whenever the region associated with a given node has homogeneous characteristics or its size falls below a certain threshold. Given a BSP tree partition for a given input image, and the `seed' leaf nodes (i.e., those that cannot be generated by mirroring their counterparts), the remaining leaf nodes of the tree are reconstructed using a predictive scheme with respect to the `seed' leaf nodes.

The research aims to address one of the complex problems that may be encountered when generating computer images for three-dimensional (3D) objects, which is the hidden surface detection and elimination. Several methods have been devised, through the last three decades, to solve this problem. However, the most popular methods used widely nowadays are depth or Z-Buffer (ZB) and Binary Space Partition Tree (BSPT). The first method (ZB) is very simple and more general but requires additional memory to store depth values in addition to intensity. On the other hand, (BSPT) method is more complex to implement but requires memory for pixel intensity only. Modelling graphical database in a binary tree make the dealing with parts of the database feasible. This is so important for clipping a part or more of the database when outside the viewing zone. The focus of this paper is to study these two methods, design an algorithm to implement each of them. After that a reasonable procedure is to found and applied for testing the performance of each of the two methods using exactly the same graphic load. A step by step increase of this load should be possible so that the behavior of the execution time at different complexity level and on average bases can be reported. The outcome results of the two methods should be compared and a recommendation to use which one of them is to be concluded [1]. Since one of the most popular graphics library which is being used during the last ten years and nowadays is the Open Graphic Library( OpenGL) from(Silicon Graphic Incorporation (SGI)[2], the algorithms of this work are implemented and the related software is carried out using this library with visual C platform.

An orthogonal BSP (binary space partitioning) tree is a commonly used spatial subdivision data structure for ray-tracing acceleration. While the construction of a BSP tree takes a relatively short time, the efficiency of a traversal algorithm significantly influences the overall rendering time. We propose a new fast traversal algorithm based on statistical evaluation of all possible cases occuring during the traversal of a BSP tree. More frequent cases are handled simply, while less frequent ones are more computationally expensive. The proposed traversal algorithm handles all singularities correctly, and saves between 30% and 50% of traversal time compared with the commonly-known Sung and Arvo algorithms.

This paper proposes a new steganography algorithm for 3D models using a spatial subdivision technique. Our algorithm first decomposes the bounding volume of the cover model into voxels based on a Binary Space Partitioning (BSP) tree. The voxels are then further categorized into eight subspaces, each of which is numbered and represented as three-digit binary characters. In the embedding process, we first traverse the BSP tree, locating a leaf voxel; then we embed every three bits of the payload message into the vertex inside the leaf voxel. This is realized by translating a vertex’s current position to the corresponding numbered subspace. This technique is a substitutive blind extraction scheme, where messages embedded can be extracted without the aid of the original cover model. This technique achieves high data capacity, equivalent to at least three times the number of the embedded vertices in the cover model. In addition, the stego model has insignificant visual distortion. Finally, this scheme is robust against similarity transformation attacks.

Geographic information systems (GISs) are widely used for representation, management, and analysis of spatial data in many disciplines. In particular, geoscientists increasingly use these tools for data integration and management purposes in many environmental applications, ranging from water resources management to the study of global warming. Beyond these capabilities, geoscientists need to model and simulate three-dimensional (3D) dynamic fields and readily integrate those results with other relevant spatial information in order to have a better understanding of the environmental problems. However, GISs are very limited for the modeling and simulation of spatial fields, which are mostly 3D and dynamic. These limitations are mainly related to the existing GIS spatial data structures that are static and limited to 2D space. In order to overcome these limitations, we develop and implement a new kinetic 3D spatial data structure based on Delaunay tetrahedralization and a 3D Voronoi diagram to support a 3D dynamic field simulation within GISs. In this article, we describe in detail the different steps from discretization of a 3D continuous field to its numerical integration, based on an event-driven method. For validation of the proposed spatial data structure itself and its potential for the simulation of a dynamic field, two case studies are presented in the article. According to our observations, during the simulation process, the data structure is maintained and the 3D spatial information is managed adequately. Furthermore, the results obtained from both experiments are very satisfactory and are comparable with the results obtained from other existing methods for the simulation of the same dynamic field. To conclude, we discuss the current challenges related to the development of the 3D kinetic data structure itself and its adaptation to 3D dynamic field simulation and suggest some solutions for its improvement.

Collision detection is a critical problem in multi-axis NC machining with complex machining environments. Using a combination of machining environment culling and a two-phase collision detection strategy, a rapid and robust collision detection approach is implemented. With this approach, the machining environment is initially subdivided and organized hierarchically using a binary space partitioning (BSP) tree structure. Before carrying out precise collision detection, the machining environment is culled conservatively to remove most of the irrelevant geometries from further checking steps which are relatively time consuming. For the later stage of precise collision detection, which is the most complicated and time-consuming task in many existing approaches, a strategy of using two-phase collision detection is developed in order to make the approach not only efficient in computation and but also maintain a high detection precision at the same time. Another important benefit of this approach is that it takes advantage of the rapid performance of graphics hardware to generate visibility information very efficiently. We show how this information could be used in assisting collision detection. An example of five-axis machining in a complex environment is adopted as the case study to demonstrate the proposed approach. It has shown that the collision detection task can be achieved with a near real-time performance.

We present a volume and complexity bounded solid simplification of models represented by Binary Space Partition (BSP). Depending on the compact and robust representation of a solid model in BSP-tree, the boundary surface of a simplified model is guaranteed to be watertight and self-intersection free. Two techniques are investigated in this paper. The volume bounded convex simplification can collapse parts with small volumes on the model into a simple convex volume enclosing the volumetric cells on the input model. The selection of which region to simplify is based on a volume-difference metric, with the help of which the volume difference between the given model and the simplified one is minimized. Another technique is a plane collapse method which reduces the depth of the BSP-tree. These two techniques are integrated into our solid simplification algorithm to give satisfactory results.

Generating piecewise linear approximations of digitized or “densely sampled” curves is an important problem in many areas. Here, we consider how to approximate an arbitrary digitized 3-D space curve, made of n+1 points, with m line segments. We present an O(n 3 log m) time, O(n 2 log m) space, dynamic programming algorithm which finds an optimal approximation. We then introduce an iterative heuristic algorithm, based upon the notions of curve length and spherical image, which quickly computes a good approximation of a space curve in O(N iter n) time and O(n) space. We apply this fast heuristic algorithm to display space curve segments and implicit surface patches, and to linearly approximate curved 3D objects, made by rotational sweeping, by binary space partitioning trees that are well-balanced.

Association rules are useful for determining correlations between attributes of a relation and have applications in the marketing, financial, and retail sectors. Furthermore, optimized association rules are an effective way to focus on the most interesting characteristics involving certain attributes. Optimized association rules are permitted to contain uninstantiated attributes and the problem is to determine instantiations such that either the support, confidence, or gain of the rule is maximized. In this paper, we generalize the optimized gain association rule problem by permitting rules to contain disjunctions over uninstantiated numeric attributes. Our generalized association rules enable us to extract more useful information about seasonal and local patterns involving the uninstantiated attribute. For rules containing a single numeric attribute, we present an algorithm with linear complexity for computing optimized gain rules. Furthermore, we propose a bucketing technique that can result in a significant reduction in input size by coalescing contiguous values without sacrificing optimality. We also present an approximation algorithm based on dynamic programming for two numeric attributes. Using recent results on binary space partitioning trees, we show that the approximations are within a constant factor of the optimal optimized gain rules. Our experimental results with synthetic data sets for a single numeric attribute demonstrate that our algorithm scales up linearly with the attribute's domain size as well as the number of disjunctions. In addition, we show that applying our optimized rule framework to a population survey real-life data set enables us to discover interesting underlying correlations among the attributes.

In this paper the exploitation of PolSAR temporal series datasets is presentedin the context of change detection and characterization. A Binary Partition Tree (BPT) data structure is employed in order to extract homogeneous regions of the image containing pixels that are following a similar polarimetric temporal evolution. Then the temporal dimension of the data is analyzed firstly to quantify the significance of the polarimetric temporal variation, making possible the detection of scene changes, and secondly to analyze and characterize those changes. Finally, the proposed technique is employed to process a real RADARSAT-2 dataset to show its capabilities and potentialities.

Presents a fast shadow generation algorithm for a large number of convex polyhedra (objects) in a 3D scene based on the concept of characteristic views (CVs). The concept of CVs is that the infinite possible views of a 3D polyhedron can be grouped into a finite number of classes which have the same visible polygons from some reference points (i.e. viewpoints or point light sources). The space can be partitioned into subspaces in which all reference points have the same object ordering. Basically, object-precision hidden surface removal algorithms and object-precision shadow generation algorithms are the same except for the reference point. The shadow volume binary space partitioning (SVBSP) tree algorithm is known to be efficient in a static environment in which the point light source can be changed. However, if a scene consists of many objects and each object consists of many polygons, the time complexity for generating and traversing the SVBSP tree increases rapidly because the SVBSP tree algorithm deals with polygons which are components of objects. Furthermore, the SYBSP tree algorithm suffers from polygon-splitting problems resulting in a high cost as the number of polygons increases. Our approach is object-oriented in the sense that an object is used as the basic unit and polygons can be considered as components of the associated object. For all light source positions, the object ordering for shadow generation is determined at the preprocessing stage. At run-time, the shadow detection algorithm is executed and, if necessary, shadow fragments are generated.

This paper discusses the interest of binary partition trees as a region-oriented image representation. Binary partition trees concentrate in a compact and structured representation a set of meaningful regions that can be extracted from an image. They offer a multiscale representation of the image and define a translation invariant 2-connectivity rule among regions. As shown in this paper, this representation can be used for a large number of processing goals such as filtering, segmentation, information retrieval and visual browsing. Furthermore, the processing of the tree representation leads to very efficient algorithms. Finally, for some applications, it may be interesting to compute the binary partition tree once and to store it for subsequent use for various applications. In this context, the paper shows that the amount of bits necessary to encode a binary partition tree remains moderate.