With the proliferation of wireless communications and rapid advances in technologies, algorithms for efficiently answering queries about large number of spatial data are needed. Spatial data consists of spatial objects including data of higher dimension. Neighbor finding is one of the most important spatial operations in the field of spatial data structures. In recent years, many researchers have focused on finding efficient solutions to the nearest neighbor problem (NN) which involves determining the point in a data set that is the nearest to a given query point. It is frequently used in Geographical Information Systems (GIS). A block B is said to be the neighbor of another block A, if block B has the same property as block A has and covers an equal-sized neighbor of block A. Jozef Voros has proposed a neighbor finding strategy on images represented by quadtrees, in which the four equal-sized neighbors (the east, west, north, and south directions) of block A can be found. However, based on Voros's strategy, the case that the nearest neighbor occurs in the diagonal directions (the northeast, northwest, southeast, and southwest directions) will be ignored. Moreover, there is no total ordering that preserve proximity when mapping a spatial data from a higher dimensional space to a 1D-space. One way of effecting such a mapping is to utilize space-filling curves. Space-filling curves pass through every point in a space and give a one-one correspondence between the coordinate and the 1D-sequence number of the point. The Peano curve, proposed by Orenstein, which maps the 1D-coordinate of a point by simply interleaving the bits of the X and Y coordinates in the 2D-space, can be easily used in neighbor finding. But with the data ordered by the RBG curve or the Hilbert curve, the neighbor finding would be complex. The RBG curve achieves savings in random accesses on the disk for range queries and the Hilbert curve achieves the best clustering for range queries. Therefore, in this thesis, we first show the missing case in the Voros's strategy and show the ways to find it. Next, we show that the Peano curve is the best mapping function used in the nearest neighbor finding. We also show the transformation rules between the Peano curve and the other curves such that we can efficiently find the nearest neighbor, when the data is linearly ordered by the other curves. From our simulation, we show that our proposed two strategies can work correctly and faster than the conventional strategies in nearest neighbor finding. Finally, we present a revised version of NA-Trees, which can work for exact match queries and range queries from a large, dynamic index, where an exact match query means finding the specific data object in a spatial database and a range query means reporting all data objects which are located in a specific range. By large, we mean that most of the index must be stored in secondary memory. By dynamic, we mean that insertions and deletions are intermixed with queries, so that the index cannot be built beforehand.

The design of multi-dimensional access methods is difficult as compared to those of one-dimensional case because of no total ordering that preserves spatial locality. One way is to look for the total order that preserves spatial proximity at least to some extent. A space-filling curve is a continuous path which passes through every point in a space once so giving a one-to-one correspondence between the coordinates of the points and the 1D-sequence numbers of points on the curve. The Hilbert curve is a famous space filling curve, since it has been shown to have strong locality preserving properties; that is, it is the best space-filling curve in minimizing the number of clusters. Hence, it has been extensively used to maintain spatial locality of multidimensional data in a wide variety of applications. A window query is an important query operation in spatial (image) databases. Given a Hilbert curve, a window query reports its corresponding orders without the need to decode all the points inside this window into the corresponding Hilbert orders. Chung et al. have proposed an algorithm for decomposing a window into the corresponding Hilbert orders. However, the Hilbert curve requires that the region is of size 2^k x 2^k, where k∈N. The intuitive method such as Chung et al.¡¦s algorithm is to directly use Hilbert curves in the decomposed areas and then connect them. They must generate a sequence of the scanned quadrants additionally before encoding and decoding the Hilbert order of one pixel and scan this sequence one time while encoding and decoding one pixel. In this dissertation, on the design of methods for window queries on a Hilbert curve, we propose an efficient algorithm, named as Quad-Splitting, for decomposing a window into the corresponding Hilbert orders on a Hilbert curve without individual sorting and merging steps. The proposed algorithm does not perform individual sorting and merging steps which are needed in Chung et al.¡¦s algorithm. From our experimental results, we show that the Quad-Splitting algorithm outperforms Chung et al.¡¦s algorithm. On the design of the methods for generating the Hilbert curve of an arbitrary-sized image, we propose approximately even partition approach to generate a pseudo Hilbert curve of an arbitrary-sized image. From our experimental results, we show that our proposed pseudo Hilbert curve preserves the similar strong locality property to the Hilbert curve. On the design of the methods for coding Hilbert curve of an arbitrary-sized image, we propose encoding and decoding algorithms. From our experimental results, we show that our encoding and decoding algorithms outperform the Chung et al.¡¦s algorithms.

Recently, millions of databases have been used and we need a new technique that can automatically transform the processed data into useful information and knowledge. Data mining is the technique of analyzing data to discover previously unknown information and spatial data mining is the branch of data mining that deals with spatial data. In spatial data mining, clustering is one of useful techniques for discovering interesting data in the underlying data objects. The problem of clustering is that give n data points in a d-dimensional metric space, partition the data points into k clusters such that the data points within a cluster are more similar to each other than data points in different clusters. Cluster analysis has been widely applied to many areas such as medicine, social studies, bioinformatics, map regions and GIS, etc. In recent years, many researchers have focused on finding efficient methods to the clustering problem. In general, we can classify these clustering algorithms into four approaches: partition, hierarchical, density-based, and grid-based approaches. The k-means algorithm which is based on the partitioning approach is probably the most widely applied clustering method. But a major drawback of k-means algorithm is that it is difficult to determine the parameter k to represent ``natural' cluster, and it is only suitable for concave spherical clusters. The k-means algorithm has high computational complexity and is unable to handle large databases. Therefore, in this thesis, we present an efficient clustering algorithm for large spatial databases. It combines the hierarchical approach with the grid-based approach structure. We apply the grid-based approach, because it is efficient for large spatial databases. Moreover, we apply the hierarchical approach to find the genuine clusters by repeatedly combining together these blocks. Basically, we make use of the Hilbert curve to provide a way to linearly order the points of a grid. Note that the Hilbert curve is a kind of space-filling curves, where a space-filling curve is a continuous path which passes through every point in a space once to form a one-one correspondence between the coordinates of the points and the one-dimensional sequence numbers of the points on the curve. The goal of using space-filling curve is to preserve the distance that points which are close in 2-D space and represent similar data should be stored close together in the linear order. This kind of mapping also can minimize the disk access effort and provide high speed for clustering. This new algorithm requires only one input parameter and supports the user in determining an appropriate value for it. In our simulation, we have shown that our proposed clustering algorithm can have shorter execution time than other algorithms for the large databases. Since the number of data points is increased, the execution time of our algorithm is increased slowly. Moreover, our algorithm can deal with clusters with arbitrary shapes in which the k-means algorithm can not discover.

Queries on spatial data commonly concern a certain range or area, for example, queries related to intersections, containment and nearest neighbors. The Continuous Nearest Neighbor (CNN) query is one kind of the nearest neighbor queries. For example, people may want to know where those gas stations are along the super highway from the starting position to the ending position. Due to that there is no total ordering of spatial proximity among spatial objects, the space filling curve (SFC) approach has proposed to preserve the spatial locality. Chen and Chang have proposed efficient algorithms based on SFC to answer nearest neighbor queries, so we may perform a sequence of individually nearest neighbor queries to answer such a CNN query in the centralized system by one of Chen and Chang's algorithms. However, each searched range of these nearest neighbor queries could be overlapped, and these queries may access several same pages on the disk, resulting in many redundant disk accesses. On the other hand, Zheng et al. have proposed an algorithm based on the Hilbert curve for the CNN query for the wireless broadcast environment, and it contains two phases. In the first phase, Zheng et al.'s algorithm designs a searched range to find candidate objects. In the second phase, it uses some heuristics to filter the candidate objects for the final answer. However, Zheng et al.'s algorithm may check some data blocks twice or some useless data blocks, resulting in some redundant disk accesses. Therefore, in this thesis, to avoid these disadvantages in the first phase of Zheng et al.'s algorithm, we propose a local expansion approach based on the Peano curve for the CNN query in the centralized system. In the first phase, we determine the searched range to obtain all candidate objects. Basically, we first calculate the route between the starting point and the ending point. Then, we move forward one block from the starting point to the ending point, and locally spread the searched range to find the candidate objects. In the second phase, we use heuristics mentioned in Zheng et al.'s algorithm to filter the candidate objects for the final answer. Based on such an approach, we proposed two algorithms: the forward moving (FM) algorithm and the forward moving* (FM*) algorithm. The FM algorithm assumes that each object is in the center of a block, and the FM* algorithm assumes that each object could be in any place of a block. Our local expansion approach can avoid the duplicated check in Zheng et al.'s algorithm, and determine a searched range with higher accuracy than that of Zhenget al.'s algorithm. From our simulation results, we show that the performance of the FM or FM* algorithm is better than that of Zheng et al.'s algorithm, in terms of the accuracy and the processing time.

Les courbes remplissant l'espace sont connues pour la capacité d'ordonner les points multidimensionnels sur une ligne en tout conservant la localité, i.e. les points proches sont toujours proches sur la ligne. La conservation de la localité est beaucoup recherchée dans plusieurs applications. La courbe de Hilbert est la courbe remplissant l'espace qui conserve le mieux la localité. Cette courbe est originalement proposée en 2D, i.e. n'est qu'applicable aux points dans un espace 2D. Pour une perspective d'application dans le cas multidimensionnel, nous proposons dans cette thèse une généralisation de la courbe de Hilbert. La courbe généralisée est définie en s'appuyant sur la propriété essentielle de la courbe de Hilbert qui crée son niveau de conservation de la localité : l'adjacence. Ainsi, elle évite la dépendance du motif primitif RBG qui est le seul motif primitif de la courbe étendu par les recherches précédentes. Le résultat est donc une famille de courbe conservant bien la localité. L'optimisation de la conservation de la localité est aussi abordée pour permettre de retrouver la courbe qui conserve le mieux la localité. Pour cet objectif, nous proposons une mesure de la conservation de la localité. En s'appuyant sur les paramètres, cette mesure peut adapter aux différentes situations applicatives comme le changement de métrique ou de taille de localité. La construction est une partie importante de la thèse, elle est la base du calcul de l'index utilisé dans l'application. Pour un calcul de l'index rapide, la courbe de Hilbert autosimilaire est utilisée. La courbe de Hilbert satisfaisant les conditions de la courbe fait l'objet du chapitre 4. La courbe généralisée est enfin appliquée dans la recherche d'image. Il s'agit d'une recherche par le contenu où chaque image est caractérisée par un vecteur multidimensionnel. Les images sont ordonnées par la courbe sur une ligne ; ainsi, la recherche est simplifiée en une recherche sur une liste ordonnée. En donnant une image d'entrée, les images similaires sont celles correspondantes aux index voisins de l'index de l'image d'entrée. La conservation de la localité garantit que ces index correspondent aux images similaires. / The space-filling curves are known for the ability to order the multidimensional points on a line while preserving the locality, i.e. the close points are closely ordered on the line. The locality preserving is wished in many applications. Hilbert curve is the best locality preserving space-filling curve. This curve is originally proposed in 2D, i.e. it is only applied to points in a 2D space. For application in the multidimensional case, we propose in this thesis a generalization of Hilbert curve. Generalized curve is based on the essential property of Hilbert curve that creates its level of locality preserving: the adjacency. Thus, it avoids the dependence on the pattern RBG, which is the only pattern of the curve extended by previous researches. The result is a family of curves preserving well the locality. The optimization of the locality preserving is also addressed to find out the best locality preserving curve. For this purpose, we propose a measure of the locality preserving. Based on the parameters, this measure can adapt to different application situations such as the change of metric or locality size. The curve construction is an important part of the thesis. It is the basis of the index calculation used in application. For a rapid index calculation, the self-similar Hilbert curves is used. They are Hilbert curves satisfying the self-similar conditions specified in chapitre 4. The generalized curve is finally applied in image search. It is the question of the content-based image search (CBIR) where each image is characterized by a multidimensionalvector. Images are ordered by the curve of a line, and the search is simplified to the search on an ordered list. By giving an input image, similar images are those corresponding to neighbors of the index of the input. The locality preserving ensures that these indexes correspond to similar images.

<p>Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics.</p><p>In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared.</p><p>A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.</p>

Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics. In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared. A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.