Indexing moving object trajectories with hilbert curves

Hilbert's space-filling curve is a continuous function that maps the unit interval onto the unit square. The construction of such curves in the 1890s surprised mathematicians of the time and led, in part, to the development of dimension theory. In this note, we discuss how modem notions of self-similarity illuminate the structure of this curve. In particular, we show that Hilbert's curve has a basic self-similar structure and can be generated using what is called an iterated function system, or IFS. Furthermore, its coordinate functions display a generalized type of self-similarity called digraph self-affinity and may be described using an appropriately generalized iterated function system. The notions of self-similarity used here are described in the text by Edgar [2, Ch. 4]. The definition of a digraph IFS was originally formulated in a research paper by Mauldin and Williams [4], although similar ideas have appeared elsewhere. The author has published a Mathematica package implementing the digraph IFS scheme [6]. A broad introduction to space-filling curves may be found in the book by Sagan [7].

In the novel area of membrane computing, the bio-inspired computing models with the generic name of P systems have turned out to be a convenient framework for handling different kinds of problems and for developing suitable solutions. One such problem area is generation of geometric patterns of approximations of space-filling curves encoded as words over appropriate chain code symbols. Parallel chain code P systems with rewriting in parallel of words in the regions using context-free rules have been shown to generate the languages of chain code words representing the finite approximation patterns of the well-known space-filling Peano and Hilbert curves. Here we consider the approximating polygons that converge to the space-filling curves of Hilbert and construct a parallel chain code P system generating these approximating polygons. We also construct a parallel chain code P system for generating the approximating polygons corresponding to another space-filling curve, known as Lebesgue’s curve, which is almost everywhere differentiable unlike any other space-filling curve.

A Space Filling Curve (SFC) has been proposed, which allows for exchange among departments and offers increased flexibility to the optimization process, as a facility layout technique which can treat any departments of unequal area. However little attention has been paid to the department shape in spite of the fact that SFC determines the department shape. The Hilbert Curve (HC) is used as the SFC, but it gives poor estimation of department shape and is not applicable for use in some rectangular buildings. A new SFC is thus developed, which offers better estimation of department shape than SFC based on HC, and applicability for any rectangular building. Numerical results are given to demonstrate the effectiveness of the presented SFC.

Hilbert's two-dimensional space-filling curve is appreciated for its good locality-preserving properties and easy implementation for many applications. However, Hilbert did not describe how to generalize his construction to higher dimensions. In fact, the number of ways in which this may be done ranges from zero to infinite, depending on what properties of the Hilbert curve one considers to be essential. In this work we take the point of view that a Hilbert curve should at least be self-similar and traverse cubes octant by octant. We organize and explore the space of possible three-dimensional Hilbert curves and the potentially useful properties which they may have. We discuss a notation system that allows us to distinguish the curves from one another and enumerate them. This system has been implemented in a software prototype, available from the author's website. Several examples of possible three-dimensional Hilbert curves are presented, including a curve that visits the points on most sides of the unit cube in the order of the two-dimensional Hilbert curve; curves of which not only the eight octants are similar to each other, but also the four quarters; a curve with excellent locality-preserving properties and endpoints that are not vertices of the cube; a curve in which all but two octants are each other's images with respect to reflections in axis-parallel planes; and curves that can be sketched on a grid without using vertical line segments. In addition, we discuss several four-dimensional Hilbert curves.

R-trees can be used to store and query sets of point data in two or more dimensions. An easy way to construct and maintain R-trees for two-dimensional points, due to Kamel and Faloutsos, is to keep the points in the order in which they appear along the Hilbert curve. The R-tree will then store bounding boxes of points along contiguous sections of the curve, and the efficiency of the R-tree depends on the size of the bounding boxes---smaller is better. Since there are many different ways to generalize the Hilbert curve to higher dimensions, this raises the question which generalization results in the smallest bounding boxes. Familiar methods, such as the one by Butz, can result in curve sections whose bounding boxes are a factor $\Omega(2^{d/2})$ larger than the volume traversed by that section of the curve. Most of the volume bounded by such bounding boxes would not contain any data points. In this paper we present a new way of generalizing Hilbert's curve to higher dimensions, which results in much tighter bounding boxes: they have at most 4 times the volume of the part of the curve covered, independent of the number of dimensions. Moreover, we prove that a factor 4 is asymptotically optimal.

Scaffolds with designed interior pore architecture, predefined porosity and a well interconnected predetermined network has been the most favored design approach for tissue engineering applications. Solid freeform fabrication technologies have provided the capability of fabricating tissue scaffolds with desired characteristics due to its integration with CAD enabled tools. However, currently the interior macro pore design of scaffolds have been limited to simple regular shapes of either squares or circles due to limited CAD capability. In this paper we seek to enhance the design of the scaffold architecture by using space filling curves within its interior space. The process involves: definition and characterization of space filling curves such as the Hilbert Curve and Sierpinski Curves, applying the principle of layered manufacturing to determine the scaffold individual layered process planes and layered contours; Feasibility studies applying the curve generators to sample models and the generation of fabrication planning instructions for extrusion based SFF systems is presented.

An image encryption scheme based on an improved Hilbert curve and a novel 4D hyper-chaotic system

An efficient algorithm for the generation of Hilbert's space-filling curve is given. The algorithm implements a recursive procedure that involves simple integer operations and quickly converges to the set of points that make the Hilbert curve. The algorithm is elegant, short, and considerably easier to implement than previous recursive and nonrecursive algorithms and can be efficiently implemented in all programming languages that have integer operations and allow recursion. The fundamental Hilbert shape (a line joining the four corners of a square) is represented by two variables with values of either 0 or 1. This coding technique could be successfully applied to the generation of other regular space-filling curves, such as the Peano curve.

An algorithm for generating Hilbert's space-filling curve in a byte-oriented manner is presented. In the context of one application of space-filling curves, the algorithm may be modified so that the results are correct for continua rather than for quantized spaces.

Recent breakthrough in the wireless energy transfer have attracted attention of the researchers due to its potential to prolong the lifetime of Wireless Sensor Networks (WSNs) and to eliminate energy bottlenecks. Through wireless charging vehicles sensor node's energy can be replenished periodically by using Wireless Charging Vehicles (WCVs). However, for a large scale WSN, the capacity of WCVs, positioning and coordination between them, and limited resources of WCVs have to be considered to implement a scalable charging system. Additionally, creating an efficient charging route for the WCVs considering the dynamic energy consumption of sensor nodes is also crucial. Since, WCVs have limited capacity and low system resources, the classic TSP-based Optimization algorithms, which requires high-performance computing, are not suitable for them. In this paper, we proposed GHSC (Guided Hilbert for Specified Cluster), which is based on the Hilbert space filling curve. We consider a clustered WSN with multiple WCVs, where each cluster is assigned to a single WCV. GHSC creates a primary charging route by considering a cluster as an unit square and dividing the square into small sub-squares. As the Hilbert curve goes through each sub-squares, the algorithm assigns a rank for each of them based on their position along the curve. The algorithm then map the nodes into vertices of the sub-squares and rank the nodes as well. Once ranking is done, the algorithm creates a charging route based on the node's ranking. This heuristic approach of the GHSC algorithm can find effective charging tours for the WCVs with a time complexity of (O(n log n) + Ω(nk)). Compared to other Global Optimization algorithms, GHSC was able to find tours, which was at most a logarithmic factor longer than the shortest tour.

Can a simple curve completely fill space? Giuseppe Peano first solved this question, and now space-filling curves are called Peano curves in his honor. The Hilbert curve, shown in this Demonstration, is the best-known Peano curve.

A novel class of k-space trajectories for magnetic resonance imaging (MRI) sampling using space filling curves (SFCs) is presented here. More specifically, Peano, Hilbert and Sierpinski curves are used. We propose 1-shot and 4-shot variable density SFCs by utilizing the space coverage provided by SFCs in different iterations. The proposed trajectories are compared with state-of-the-art echo planar imaging (EPI) trajectories for 128 × 128 and 256 × 256 phantom and brain images. The simulation results show that the readout time is reduced by up to 45% for the 128 × 128 image with little compromise in reconstruction quality. Also, the peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) index are improved by 2.32 dB and 0.1009, respectively, with an 18% shorter readout time using the 4-shot Hilbert SFC trajectory for reconstructing a 256 × 256 brain MRI image.

In this work we study the impact of noise on the training of object detection\nnetworks for the medical domain, and how it can be mitigated by improving the\ntraining procedure. Annotating large medical datasets for training data-hungry\ndeep learning models is expensive and time consuming. Leveraging information\nthat is already collected in clinical practice, in the form of text reports,\nbookmarks or lesion measurements would substantially reduce this cost.\nObtaining precise lesion bounding boxes through automatic mining procedures,\nhowever, is difficult. We provide here a quantitative evaluation of the effect\nof bounding box coordinate noise on the performance of Faster R-CNN object\ndetection networks for breast mass detection. Varying degrees of noise are\nsimulated by randomly modifying the bounding boxes: in our experiments,\nbounding boxes could be enlarged up to six times the original size. The noise\nis injected in the CBIS-DDSM collection, a well curated public mammography\ndataset for which accurate lesion location is available. We show how, due to an\nimperfect matching between the ground truth and the network bounding box\nproposals, the noise is propagated during training and reduces the ability of\nthe network to correctly classify lesions from background. When using the\nstandard Intersection over Union criterion, the area under the FROC curve\ndecreases by up to 9%. A novel matching criterion is proposed to improve\ntolerance to noise.\n

A space-filling curve is a linear traversal of a discrete finite multidimensional space. In order for this traversal to be useful in many applications, the curve should preserve "locality". We quantify "locality" and bound the locality of multidimensional space-filling curves. Classic Hilbert space-filling curves come close to achieving optimal locality.

This work presents novel compact high-temperature superconductor microstrip resonators and filters based on the Hilbert space-filling fractal curve. Several Hilbert resonator configurations have been assessed by using a full-wave electromagnetic simulator and their miniaturization performance has been investigated, emphasizing the parameters which allow obtaining good tradeoff between compact size and losses. These resonators have proved to be useful for filter design. A four-pole 2.45-GHz quasi-elliptic filter and a four-pole 1.9-GHz Chebyshev filter have been realized by patterning 10 mm/spl times/10 mm Y/sub 2/Ba/sub 3/Cu/sub 7-/spl delta//O thin films on MgO substrates. Insertion losses of the order of 0.2 dB have been measured, showing good agreement with the simulations.