AbstractÐSeveral schemes for the linear mapping of a multidimensional space have been proposed for various applications, such as access methods for spatio-temporal databases and image compression. In these applications, one of the most desired properties from such linear mappings is clustering, which means the locality between objects in the multidimensional space being preserved in the linear space. It is widely believed that the Hilbert space-filling curve achieves the best clustering [1], [14]. In this paper, we analyze the clustering property of the Hilbert space-filling curve by deriving closed-form formulas for the number of clusters in a given query region of an arbitrary shape (e.g., polygons and polyhedra). Both the asymptotic solution for the general case and the exact solution for a special case generalize previous work [14]. They agree with the empirical results that the number of clusters depends on the hypersurface area of the query region and not on its hypervolume. We also show that the Hilbert curve achieves better clustering than the z curve. From a practical point of view, the formulas given in this paper provide a simple measure that can be used to predict the required disk access behaviors and, hence, the total access time.

Indexing schemes for grids based on space-filling curves (e.g., Hilbert indexings) find applications in numerous fields, ranging from parallel processing over data structures to image processing. Because of an increasing interest in discrete multi-dimensional spaces, indexing schemes for them have won considerable interest. Hilbert curves are the most simple and popular space-filling indexing scheme. We extend the concept of curves with Hilbert property to arbitrary dimensions and present first results concerning their structural analysis that also simplify their applicability. We define and analyze in a precise mathematical way r-dimensional Hilbert indexings for arbitrary r 2. Moreover, we generalize and simplify previous work and clarify the concept of Hilbert curves for multi-dimensional grids. As we show, Hilbert indexings can be completely described and analyzed by "generating elements of order 1", thus, in comparison with previous work, reducing their structural comp...

Image search has been actively studied in recent years. On the other hands, image browsing has received little attention. Image browsing refers to the process of presenting some forms of overview or summary of the image relationships, thus facilitating a user to navigate across the data set and find images of interests. In this paper, we present a new data structure built on the multi-linearization of image attributes for efficient organization of the data set and fast visual browsing of the images. We describe new techniques for multi-linearization based on multiple space-filling curves and hierarchical clustering techniques. In addition to providing fast navigation, our proposed data structure allows computationally efficient insertion and deletion of images from the data set. We then present a novel image navigator and browser built on dual-linearization data structure and intuitive presentation of image relevance and relationships, demonstrate the image navigation process, and repo...

Indexing schemes for grids based on space-filling curves (e.g., Hilbert curves) find applications in numerous fields, ranging from parallel processing over data structures to image processing. Because of an increasing interest in discrete multidimensional spaces, indexing schemes for them have won considerable interest. Hilbert curves are the most simple and popular space-filling indexing schemes. We extend the concept of curves with Hilbert property to arbitrary dimensions and present first results concerning their structural analysis that also simplify their applicability.

We present a tensor product formulation for Hilbert space-filling curves. Both recursive and iterative formulas are expressed in the paper. We view a Hilbert space-filling curve as a permutation which maps two-dimensional ¥§¦©¨�¥� ¦ data elements stored in the row major or column major order to the order of traversing a Hilbert space-filling curve. The tensor product formula of Hilbert space-filling curves uses several permutation operations: stride permutation, radix-2 Gray permutation, transposition, and anti-diagonal transposition. The iterative tensor product formula can be manipulated to obtain the inverse Hilbert permutation. Also, the formulas are directly translated into computer programs which can be used in various applications including R-tree indexing, image processing, and process llocation, etc. Key words: tensor product, block recursive algorithm, Hilbert space-filling curve, stride

. The array aliasing mechanism provided in the Connection Machine Fortran (CMF) language and run--time system provides a unique way of identifying the memory address spaces local to processors within the global address space of distributed memory architectures, while staying in the data parallel programming paradigm. We show how the array aliasing feature can be used effectively in optimizing communication and computation performance. The constructs we present occur frequently in many scientific and engineering applications, and include various forms of aggregation and array reshaping through array aliasing. The effectiveness of the optimization techniques is demonstrated on an implementation of Anderson's hierarchical O(N) N --body method. Key words. Data parallel programming, array aliasing, hierarchical N --body methods. AMS(MOS) subject classifications. 68N15, 68N20, 70--08, 70F10 1. Introduction. Data parallel programming provides an effective way to write maintainable, portabl...

Thresholding is the most commonly used technique in image segmentation. In this paper, we first propose an efficient sequential algorithm to improve the relative entropy-based thresholding technique. This algorithm combines the concepts of the relative entropy with that of the local entropy and also includes the quadtree hierarchical structure in it. Second, we derive a constant time parallel algorithm to solve this problem on the reconfigurable array of processors with wider bus networks (RAPWBN).

Abstract: We use a tensor product based multi-linear algebra theory to formulate three-dimensional Hilbert space-filling curves. A 3-D Hilbert space-filling curve is specified as a permutation which rearranges three-dimensional 2 n 2 n 2 n data elements stored in the row major order as in C language or the column major order as in FORTRAN language to the order of traversing a 3-D Hilbert space-filling curve. The tensor product formulation of 3-D Hilbert space-filling curves uses stride permutation, reverse permutation, and Gray permutation. We present both recursive and iterative tensor product formulas of 3-D Hilbert space-filling curves. In addition, we derive a tensor product formula of inverse 3-D Hilbert space-filling curve permutation. The tensor product formulas are directly translated into computer programs which can be used in various applications. The process of program generation is explained in the paper.

A Hilbert space-filling curve is a curve of 2^n x 2^n two-dimensional space that it visits neighboring points consecutively without crossing itself. The application of Hilbert space-filling curves in image processing is to rearrange image pixels in order to enhance pixel locality. An iterative program of the Hilbert space-filling curve ordering generated from a tensor product formulation is used to rearrange pixels of medical images. We implement four lossless encoding schemes, run-length encoding, LZ77 coding, LZW coding, and Huffman coding, along with the Hilbert space-filling curve ordering. Combination of these encoding schemes are also implemented to study the effectiveness of various compression methods. In addition, differential encoding is employed to medical images to study different format of image representation to the above encoding schemes. In the paper, we report the testing results of compression ratio and performance evaluation. The experiments show that the pre-processing operation of differential encoding followed by the Hilbert space-filling curve ordering and the compression method of LZW coding followed by Huffman coding will give the best compression result.