在多維資料擷取方法的設計相對於一維資料擷取的設計較為困難，是因為多維資料沒有整體的順序。Space-filling curve 是一種線段，可以走過空間的每一個點，而且每個點只可走過一次。它可以將多維座標轉換為1 維座標，也就是可以幫多維的資料決定一個全域索引(Global Index)。Hilbert 曲線是最具代表性的Space-filling curve，可以有效的保存空間的區域特性(Locality)，已經廣泛的應用在多種的領域上，尤其是影像處理及壓縮的應用。所以，如何減少執行所需時間，來有效擷取Hilbert 曲線方法的設計是很重要的工作。因此，在此博士論文中，為了在Hilbert 曲線中有效地支援視窗查詢(Window Query)，我們首先利用Hilbert 曲線的特性及Chang 等學者提出的NA-Trees 空間索引方法，提出四等份切割法(Quad-Splitting)來計算視窗中的所有Hilbert 索引值。我們的方法不須額外的排序與合併的步驟，可以有效減少執行時間。經效能分析與實驗結果顯示，我們的方法比之前學者提出的方法更能有效率的執行視窗查詢。然而，Hilbert 曲線有一個主要限制，就是邊長相等，而且大小是2 的次方數，這樣會限制應用的範圍。接著，我們提出近似均分法(approximately even partition)，擴充原有的2×2 基礎曲線，將邊長相等的正方形影像切割成數個基礎曲線並相連成一正方形的Hilbert 曲線。我們將所提出曲線命名為Hilbert*曲線，同時亦提出快速計算出座標相對應Hilbert*索引值的方法。最後，應用Chung 等學者提出的任意邊長Hilbert 曲線的方法中，將任意邊長影像先切割成數個正方形子影像。再將每個正方形子影像用大小相同合適的Hilbert*曲線取代，每個子曲線相連成任意邊長Hilbert 曲線。我們也提出快速計算出座標相對應的任意邊長Hilbert 曲線索引值的方法。經效能分析與實驗結果顯示，我們的Hilbert*曲線的區域特性極為接近原有Hilbert 曲線。在擷取任意邊長Hilbert 曲線上，我們的方法在效率上比Chung 等學者提出的方法更好。

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.

ABSTRACT 1
LIST OF FIGURES iii
LIST OF TABLES vii
1. Introduction 1
1.1 Space Filling Curves 2
1.2 The Hilbert Curve Based Compressed Image 5
1.3 The Window Query on the Compressed Image 8
1.4 Related Works of the Hilbert Curve 8
1.5 Motivations and Contributions 12
1.5.1 The Quad-Splitting Algorithm for the Window Query on the Hilbert Curve 12
1.5.2 The Approximately Even Partition Approach for Coding the Hilbert Curve of an Arbitrary-Sized Image 13
1.6 Organization of Dissertation 16
2. A Survey of Access Methods on the Hilbert Curve 17
2.1 Methods for Generating the Hilbert Curve 17
2.1.1 Algorithm 781 18
2.1.2 SFCGen Algorithm 19
2.1.3 Kamata and Bandoh’s Algorithm 21
2.1.4 Chung et al.’s Algorithm 24
2.2 The Window Query on the Hilbert Curve 28
3. The Quad-Splitting Algorithm for a Window Query on the Hilbert Curve 31
3.1 Observations 32
3.2 The Quad-Splitting Algorithm 38
3.3 Performance 49
3.3.1 Analysis of Time Complexity 51
3.3.2 Experiment Results 54
3.4 Summary 60
4. The Approximately Even Partition Approach for Coding the
Hilbert Curve of an Arbitrary-Sized Image 61
4.1 The Approximately Even Partition Approach 62
4.2 Constructing the Hilbert¤ Curve 67
4.3 The Encoding Algorithm for the Hilbert¤ Curve 70
4.4 The Decoding Algorithm for the Hilbert¤ Curve 75
4.5 Extension of the Hilbert¤ Curve for an Arbitrary-Sized Image 78
4.6 Performance 80
4.7 Summary 86
5. Conclusion 88
5.1 Summary 88
5.2 Future Research Direction 90
BIBLIOGRAPHY 91

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