Apercu sur les quadarbres et les structures de donnees hierarchiques. Elles sont basees sur le principe de decomposition recursive. L'accentuation est mise sur la representation de donnees dans les applications de traitement d'images, d'infographie, les systemes d'informations geographiques et la robotique. On examine en detail un certain nombre d'operations dans lesquelles de telles structures de donnees trouvent leur utilisation

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The Quadtree and Related Hierarchical Data Structures

This unique book provides an introduction to a subject whose use has steadily increased over the past 40 years. An update of Ramon Moore s previous books on the topic, it provides broad coverage of the subject as well as the historical perspective of one of the originators of modern interval analysis. The authors provide a hands-on introduction to INTLAB, a high-quality, comprehensive MATLAB toolbox for interval computations, making this the first interval analysis book that does with INTLAB what general numerical analysis texts do with MATLAB. Readers will find the following features of interest: elementary motivating examples and notes that help maximize the reader s chance of success in applying the techniques; exercises and hands-on MATLAB-based examples woven into the text; INTLAB-based examples and explanations integrated into the text, along with a comprehensive set of exercises and solutions, and an appendix with INTLAB commands; an extensive bibliography and appendices that will continue to be valuable resources once the reader is familiar with the subject; and a Web page with links to computational tools and other resources of interest. Audience: Introduction to Interval Analysis will be valuable to engineers and scientists interested in scientific computation, especially in reliability, effects of roundoff error, and automatic verification of results. The introductory material is particularly important for experts in global optimization and constraint solution algorithms. This book is suitable for introducing the subject to students in these areas. Contents: Preface; Chapter 1: Introduction; Chapter 2: The Interval Number System; Chapter 3: First Applications of Interval Arithmetic; Chapter 4: Further Properties of Interval Arithmetic; Chapter 5: Introduction to Interval Functions; Chapter 6: Interval Sequences; Chapter 7: Interval Matrices; Chapter 8: Interval Newton Methods; Chapter 9: Integration of Interval Functions; Chapter 10: Integral and Differential Equations; Chapter 11: Applications; Appendix A: Sets and Functions; Appendix B: Formulary; Appendix C: Hints for Selected Exercises; Appendix D: Internet Resources; Appendix E: INTLAB Commands and Functions; References; Index.

Introduction to Interval Analysis

A very promising idea for fast searching in traditional and multimedia databases is to map objects into points in k-d space, using k feature-extraction functions, provided by a domain expert [25]. Thus, we can subsequently use highly fine-tuned spatial access methods (SAMs), to answer several types of queries, including the 'Query By Example' type (which translates to a range query); the 'all pairs' query (which translates to a spatial join [8]); the nearest-neighbor or best-match query, etc.However, designing feature extraction functions can be hard. It is relatively easier for a domain expert to assess the similarity/distance of two objects. Given only the distance information though, it is not obvious how to map objects into points.This is exactly the topic of this paper. We describe a fast algorithm to map objects into points in some k-dimensional space (k is user-defined), such that the dis-similarities are preserved. There are two benefits from this mapping: (a) efficient retrieval, in conjunction with a SAM, as discussed before and (b) visualization and data-mining: the objects can now be plotted as points in 2-d or 3-d space, revealing potential clusters, correlations among attributes and other regularities that data-mining is looking for.We introduce an older method from pattern recognition, namely, Multi-Dimensional Scaling (MDS) [51]; although unsuitable for indexing, we use it as yardstick for our method. Then, we propose a much faster algorithm to solve the problem in hand, while in addition it allows for indexing. Experiments on real and synthetic data indeed show that the proposed algorithm is significantly faster than MDS, (being linear, as opposed to quadratic, on the database size N), while it manages to preserve distances and the overall structure of the data-set.

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FastMap: a fast algorithm for indexing, data-mining and visualization of traditional and multimedia datasets

Volume rendering is a technique for visualizing sampled scalar or vector fields of three spatial dimensions without fitting geometric primitives to the data. A subset of these techniques generates images by computing 2-D projections of a colored semitransparent volume, where the color and opacity at each point are derived from the data using local operators. Since all voxels participate in the generation of each image, rendering time grows linearly with the size of the dataset. This paper presents a front-to-back image-order volume-rendering algorithm and discusses two techniques for improving its performance. The first technique employs a pyramid of binary volumes to encode spatial coherence present in the data, and the second technique uses an opacity threshold to adaptively terminate ray tracing. Although the actual time saved depends on the data, speedups of an order of magnitude have been observed for datasets of useful size and complexity. Examples from two applications are given: medical imaging and molecular graphics.

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Efficient ray tracing of volume data

Digital topology deals with the topological properties of digital images: or, more generally, of discrete arrays in two or more dimensions. It provides the theoretical foundations for important image processing operations such as connected component labeling and counting, border following, contour filling, and thinning—and their generalizations to three- (or higher-) dimensional “images.” This paper reviews the fundamental concepts of digital topology and surveys the major theoretical results in the field. A bibliography of almost 140 references is included.

Digital topology: introduction and survey

A quadtree may be represented without pointers by encoding each black node with a quaternary integer whose digits reflect successive quadrant subdivisions. We refer to the sorted array of black nodes as the “linear quadtree” and show that it introduces a saving of at least 66 percent of the computer storage required by regular quadtrees. Some algorithms using linear quadtrees are presented, namely, (i) encoding a pixel from a 2n × 2>n array (or screen) into its quaternary code; (ii) finding adjacent nodes; (iii) determining the color of a node; (iv) superposing two images. It is shown that algorithms (i)-(iii) can be executed in logarithmic time, while superposition can be carried out in linear time with respect to the total number of black nodes. The paper also shows that the dynamic capability of a quadtree can be effectively simulated.

An effective way to represent quadtrees

Suppose all zeros of a polynomialp but one are known to lie in specified circular regions, and the value of the logarithmic derivativep?p ?1 is known at a point. What can be said about the location of the remaining zero? This question is answered in the present paper, as well as its generalization where several zeros are missing and the values of some derivatives of the logarithmic derivative are known. A connection with a classical result due to Laguerre is established, and an application to the problem of locating zeros of certain transcendental functions is given. The results are used to construct (i) a version of Newton's method with error bounds, (ii) a cubically convergent algorithm for the simultaneous approximation of all zeros of a polynomial. The algorithms and their theoretical foundation make use of circular arithmetic, an extension, based on the theory of Moebius transformations, of interval arithmetic from the real line to the extended complex plane.

Circular arithmetic and the determination of polynomial zeros

Linear octtrees for fast processing of three-dimensional objects

Consider a polynomialP(z) of degreeN whose zeros are known to lie insideN closed disks, each disk containing one and only one root. In this paper we show that if the given disks are "sufficiently well separated", then the first derivative ofP(z) never vanishes inside the initial inclusion regions. The formulation of the square-root iteration in terms of circular regions is then possible and leads to an iterative scheme with degree four convergence. The corresponding algorithm makes use of circular arithmetic and in particular of the definition of square root of a disk. A criterion for the selection of the appropriate square-root set is also given. The procedure can be used to simultaneously refine all (complex or real) roots ofP(z) together with their error bounds.

Parallel laguerre iterations: The complex case

In Gargantini (1982a) it has been shown that storing black nodes of a quadtree is sufficient to retrieve any basic property associated with quadtrees. To achieve this, each black node must be represented as a quaternary integer whose digits (from left to right) describe the path from the root to that node. The sorted sequence of quaternary integers representing a given region is called the linear quadtree associated with that region. Such a structure has been shown to save more than two-thirds of the memory locations used by regular quadtrees. In this paper we present procedures for translating and rotating a region and consider the superposition of binary images with different characteristics (such as different resolution parameter, different pixel size and/or different center). Translation, rotation, and superposition are shown to be O(N log N) operations; for translation N is the number of black pixels; for rotation N is the number of black nodes; for superposition N is the sum of black nodes or black pixels of the two images, depending on whether or not the two regions are centered on the same raster.

Irene Gargantini

Papers

An effective way to represent quadtrees

Circular arithmetic and the determination of polynomial zeros

Linear octtrees for fast processing of three-dimensional objects

Parallel laguerre iterations: The complex case

Translation, rotation and superposition of linear quadtrees