Hilbert Curve Research Articles

ScienceEarth: A Big Data Platform for Remote Sensing Data Processing

Mass remote sensing data management and processing is currently one of the most important topics. In this study, we introduce ScienceEarth, a cluster-based data processing framework. The aim of ScienceEarth is to store, manage, and process large-scale remote sensing data in a cloud-based cluster-computing environment. The platform consists of the following three main parts: ScienceGeoData, ScienceGeoIndex, and ScienceGeoSpark. ScienceGeoData stores and manages remote sensing data. ScienceGeoIndex is an index and query system, a spatial index based on quad-tree and Hilbert curve which is combined for heterogeneous tiled remote sensing data that makes efficient data retrieval in ScienceGeoData. ScienceGeoSpark is an easy-to-use computing framework in which we use Apache Spark as the analytics engine for big remote sensing data processing. The result of tests proves that ScienceEarth can efficiently store, retrieve, and process remote sensing data. The results reveal ScienceEarth has the potential and capabilities of efficient big remote sensing data processing.

A novel image encryption scheme using both pixel level and bit level permutation with chaotic map

In cryptography, chaotic cryptosystem is one of the methods to carry out encryption and decryption of images. This paper introduces a new symmetric key encryption technique based on chaotic map, scan method and cyclic shift operation. The confusion and diffusion techniques are implemented using Hilbert curve and Henon map. To ensure image scrambling, both pixel level and bit level permutations are performed. A novel method is adopted for bit level permutation using cyclic shift operation. The key streams for cyclic shift and diffusion operations are generated from the Henon map. Final encrypted image is generated from the double scrambled image. The performance of the proposed method has been analyzed using various analyses like statistical analysis, entropy analysis, differential attack analysis, key sensitivity analysis and known plain text attack analysis. Experimental results show that the proposed image encryption technique resists various attacks and ensures high security. It also provides better performance when compared with several traditional and state-of-the-art image encryption methods.

Extending the limits for big data RSA cracking: Towards cache-oblivious TU decomposition

Nowadays, Big Data security processes require mining large amounts of content that was traditionally not typically used for security analysis in the past. The RSA algorithm has become the de facto standard for encryption, especially for data sent over the internet. RSA takes its security from the hardness of the Integer Factorisation Problem. As the size of the modulus of an RSA key grows with the number of bytes to be encrypted, the corresponding linear system to be solved in the adversary integer factorisation algorithm also grows. In the age of big data this makes it compelling to redesign linear solvers over finite fields so that they exploit the memory hierarchy. To this end, we examine several matrix layouts based on space-filling curves that allow for a cache-oblivious adaptation of parallel TU decomposition for rectangular matrices over finite fields. The TU algorithm of Dumas and Roche (2002) requires index conversion routines for which the cost to encode and decode the chosen curve is significant. Using a detailed analysis of the number of bit operations required for the encoding and decoding procedures, and filtering the cost of lookup tables that represent the recursive decomposition of the Hilbert curve, we show that the Morton-hybrid order incurs the least cost for index conversion routines that are required throughout the matrix decomposition as compared to the Hilbert, Peano, or Morton orders. The motivation lies in that cache efficient parallel adaptations for which the natural sequential evaluation order demonstrates lower cache miss rate result in overall faster performance on parallel machines with private or shared caches and on GPU’s.

MI-HCS: Monotonically Increasing Hilbert Code Segments for 3D Geospatial Query Window

Window queries are basic but important query tasks in geospatial databases. The Hilbert curve has good clustering properties, which can be used to effectively improve the execution efficiency of window queries. The ideal goal of a window query using Hilbert curve code is to quickly convert the query window to the corresponding monotonic continuous Hilbert code segments. However, existing algorithms have shortcomings in conversion calculations and Hilbert code segments properties. We propose the state vectors that are used to describe the filling rules of a three-dimensional Hilbert curve. In addition, we designed a direct generation algorithm for monotonically increasing Hilbert code segments (MI-HCS) for a three-dimensional window query. The MI-HCS algorithm is characterized by the direct generation of a monotonically increasing code segment set without the need to traverse all grid elements or the requirement of separate sorting steps. For a given query window <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$W\left ({{x,y,z,l,w,h} }\right)$ </tex-math></inline-formula> and a Hilbert curve of size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T\times T\times T$ </tex-math></inline-formula> , the maximum complexity of our MI-HCS algorithm is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O\left ({{\alpha _{1} \times \alpha _{2} \times \left ({{\log _{2} T+1} }\right)} }\right)$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha _{1} = \textrm {median}\left ({{l,w,h} }\right)$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha _{2} = \textrm {max}\left ({{l,w,h} }\right)$ </tex-math></inline-formula> . The experimental results of the MI-HCS algorithm complexity are consistent with the specific theoretical analysis. The experimental results show that the Hilbert code segment generation efficiency of the proposed MI-HCS algorithm is 260.5% to 423.9% higher than that of existing algorithms.

Recovery of the Temperature Field in the Case of Combustion of Premixed Propane-Air Mixtures by Methods of the Hilbert Optics

Jet combustion of premixed propane-air mixtures is frequently encountered in various devices. In the present study, the flame is investigated by methods of the Hilbert optics. The diagnostic technique adapted to studying combustion products is based on visualization of phase perturbations induced in the probing light field by the examined medium with the use of polychromatic Hilbert and Foucault-Hilbert transforms combined with pixel-by-pixel processing of the dynamic structure of the recorded images. The diagnostic setup is based on the standard IAB-463M device with modified systems of optical filtration, source, and data processing. The dynamic phase structure of the propane-air flame is visualized. The temperature at reference points of the burning jet is measured by thermocouples. On axisymmetric segments, the phase function is recovered from the Hilbert curves, and the temperature field of the flame is retrieved by using the inverse Abel transform.

Heterogeneous recurrence analysis of spatial data.

Nonlinear dynamical systems often generate significant amounts of observational data such as time series, as well as high-dimensional spatial data. To delineate recurrence dynamics in the spatial data, prior efforts either extended the recurrence plot, which is a widely used tool for time series, to a four-dimensional hyperspace or utilized the network approach for recurrence analysis. However, very little has been done to differentiate heterogeneous types of recurrences in the spatial data (e.g., recurrence variations of state transitions in the spatial domain). Therefore, we propose a novel heterogeneous recurrence approach for spatial data analysis. First, spatial data are traversed with the Hilbert Space-Filling Curve to transform the variations of recurrence patterns from the spatial domain to the state-space domain. Second, we design an Iterated Function System to derive the fractal representation for the state-space trajectory of spatial data. Such a fractal representation effectively captures self-similar behaviors of recurrence variations and multi-state transitions in the spatial data. Third, we develop the Heterogeneous Recurrence Quantification Analysis of spatial data. Experimental results in both simulation and real-world case studies show that the proposed approach yields superior performance in the extraction of salient features to characterize and quantify heterogeneous recurrence dynamics in spatial data.

Compact Wideband Dual-Frequency Antenna Based on a Simplified Composite Right/Left-Handed Transmission Line with Hilbert Curve Loading

This paper addresses the issues of low bandwidth, gain, and efficiency of miniaturized microwave antennas by proposing a novel wideband dual-frequency coplanar waveguide antenna design based on a simplified composite right/left-handed (SCRLH) transmission line structure with Hilbert curve loading. The multifrequency characteristics of the SCRLH transmission line structure are evaluated theoretically, and the antenna parameters promoting bandwidth broadening under zeroth-order resonance (ZOR) and first-order resonance (FOR) mode operation are evaluated. The bandwidth broadening in the ZOR and FOR modes is accordingly revealed to be independent of the antenna length, and the structure therefore facilitates wideband operation under miniaturization. Finally, the dual-frequency ZOR and FOR mode antenna design with center frequencies of f0 = 1.865 GHz and f1 = 2.835 GHz is validated via simulation, and the performance of a compact prototype antenna is evaluated experimentally. The −10 dB return loss bandwidths at f0 and f1 are 187 MHz (from 1.773 GHz to 1.96 GHz) and 368 MHz (from 3.002 GHz to 3.37 GHz), and the corresponding relative bandwidths are 10.1% and 11.5%, respectively. The experimentally measured peak gains and radiation efficiencies at f0 are 1.54 dB and 81.3%, respectively, and those at f1 are 1.71 dB and 74.2%, respectively.

Real-Time Noninvasive Measurement of Glucose Concentration Using a Modified Hilbert Shaped Microwave Sensor.

We developed a microwave glucose sensor based on the modified first-order Hilbert curve design and measured glucose concentration in aqueous solutions by using a real-time microwave near-field electromagnetic interaction technique. We observed S21 transmission parameters of the sensor at resonant frequencies depend on the glucose concentration. We could determine the glucose concentration in the 0–250 mg/dL concentration range at an operating frequency of near 6 GHz. The measured minimum detectable signal was 0.0156 dB/(mg/dL) and the measured minimum detectable concentration was 1.92 mg/dL. The simulation result for the minimum detectable signal and the minimum detectable concentration was 0.0182 dB/(mg/dL) and 1.65 mg/dL, respectively. The temperature instability of the sensor for human glycemia in situ measurement range (27–34 °C for fingers and 36–40 °C for body temperature ranges) can be improved by the integration of the temperature sensor in the microwave stripline platform and the obtained data can be corrected during signal processing. The microwave signal–temperature dependence is almost linear with the same slope for a glucose concentration range of 50–150 mg/dL. The temperature correlation coefficient is 0.05 dB/°C and 0.15 dB/°C in 27–34 °C and 36–40 °C temperature range, respectively. The presented system has a cheap, easy fabrication process and has great potential for non-invasive glucose monitoring.

W-band Lumped Element Kinetic Inductance Detector Array for Large Ground-Based Telescopes

We describe the development of a W-band lumped element kinetic inductance detector array for application in large ground-based telescopes, like the Sardinia radio telescope. Based on the previous studies, we use a Ti/Al bi-layer film (10 nm thick Ti $$+$$ 25 nm thick Al) for the resonators, to cover frequencies greater than 65 GHz. Optical simulations have been performed using ANSYS HFSS software suite, to optimize the absorber geometry, the illumination configuration, and the thickness of the dielectric substrate. Simulations suggest that the best geometry of the absorber is a front-illuminated third-order Hilbert curve, with a Si substrate $${235}\,{\upmu \hbox {m}}$$ thick, coupled to a single-mode circular waveguide. Electrical simulations have been performed using SONNET, to complete the design of the detectors by choosing the size of the capacitor, the bias coupling, and the feedline. In addition, the electrical simulations allow us to verify the lumped condition, to tune the feedline impedance and resonant frequencies, to constrain the coupling quality factor, and to minimize the electrical cross-talk between different pixels of the same array.

TI-Stan: Model Comparison Using Thermodynamic Integration and HMC

We present a novel implementation of the adaptively annealed thermodynamic integration technique using Hamiltonian Monte Carlo (HMC). Thermodynamic integration with importance sampling and adaptive annealing is an especially useful method for estimating model evidence for problems that use physics-based mathematical models. Because it is based on importance sampling, this method requires an efficient way to refresh the ensemble of samples. Existing successful implementations use binary slice sampling on the Hilbert curve to accomplish this task. This implementation works well if the model has few parameters or if it can be broken into separate parts with identical parameter priors that can be refreshed separately. However, for models that are not separable and have many parameters, a different method for refreshing the samples is needed. HMC, in the form of the MC-Stan package, is effective for jointly refreshing the ensemble under a high-dimensional model. MC-Stan uses automatic differentiation to compute the gradients of the likelihood that HMC requires in about the same amount of time as it computes the likelihood function itself, easing the programming burden compared to implementations of HMC that require explicitly specified gradient functions. We present a description of the overall TI-Stan procedure and results for representative example problems.

An Efficient Parallel Algorithm for Polygons Overlay Analysis

Map overlay analysis is essential for geospatial analytics. Large scale spatial data pressing poses challenges for geospatial map overlay analytics. In this study, we propose an efficient parallel algorithm for polygons overlay analysis, including active-slave spatial index decomposition for intersection, multi-strategy Hilbert ordering decomposition, and parallel spatial union algorithm. Multi-strategy based spatial data decomposition mechanism is implemented, including parallel spatial data index, the Hilbert space-filling curve sort, and decomposition. The results of the experiments showed that the parallel algorithm for polygons overlay analysis achieves high efficiency.

Novel coated differentially fed dual‐band fractal antenna for implantable medical devices

A novel differentially fed fractal antenna coated with a layer of a dielectric film and operating at two bands of Medical Implant Communication Services 402–405 MHz and Industrial, Scientific, and Medical 2.4–2.48 GHz, is investigated to enable sufficient data transmission rates, while featuring a ‘mode-control’ signal to prolong the battery life. The symmetrical fractal technology of Hilbert curve is employed to miniaturise the dimension. To facilitate interfacing with peripheral integrated circuits, differentially feeding is adopted. To demonstrate how common-mode noises are suppressed, for the first time, a detailed analysis about the noise suppression of the differential configuration with two feedings is presented. A layer of thin film is coated around the proposed antenna to isolate the effect of human tissues and alleviate the electromagnetic coupling with human body. Moreover, to draw a design methodological guideline, the key geometrical parameters are analysed numerically and theoretically. Health safety considerations, radiation performance, and link budgets are also discussed to validate the proposed antenna's applicability in biomedical telemetry.

Leveraging deep learning with symbolic sequences for robust head poses estimation

Head pose estimation is a challenging topic in computer vision with a large area of applications. There are a lot of methods which have been presented in the literature to undertake pose estimation so far. Even though the efficiency of these methods is acceptable, the sensitivity to external conditions is still being a big challenge. In this paper, we come up with a new model to overcome the problem of head poses estimation. First, the face images are converted into one-dimensional vectors as a time series using the Peano–Hilbert space-filling curve. Then, we convert these numerical series into symbolic sequences with adequate dimensionality reduction approaches. These sequences are then used as input of an encode–decoder neural network to learn and generate labels of the faces orientations. We have evaluated our model on several databases, and the experimental results have shown that the proposed method is very competitive compared to other well-known approaches.

Multi‐dimensional Sum‐Up Rounding using Hilbert curve iterates

AbstractMixed‐integer optimal control problems can be reformulated by means of partial outer convexification, which introduces binary‐valued switching functions for the different realizations of a discrete‐valued control variable. They can be relaxed naturally by allowing them to take values in [0, 1]. Sum‐Up Rounding (SUR) algorithms approximate feasible switching functions of the relaxation with binary ones. If the controls are distributed in one dimension, the approximants are known to converge in the weak∗ topology of L∞. We show that this still holds true for controls that are distributed in more than one dimension if an appropriate grid refinement strategy that is coupled with a deliberate ordering of the grid cells is chosen. This condition is satisfied by the iterates of space‐filling curves, e.g. the Hilbert curve.

A novel hybrid series stacked differential fractal inductor for MMIC applications

Purpose The purpose of this paper is to design an on-chip inductor with high inductance, high-quality factor and high self-resonance frequency for the equivalent on-chip area using fractal curves. Design/methodology/approach A novel hybrid series stacked differential fractal inductor using Hilbert and Sierpinski fractal curves is proposed with two different layers connected in series using vias. The inductor is implemented in Sonnet EM simulator using 180 nm CMOS standard process technology. Findings The proposed inductor reduces the parasitic capacitance and negative mutual inductance between the adjacent layers with significant improvement in overall inductance, quality factor and self-resonance frequency when compared with conventional series stacked fractal inductors. Research limitations/implications The fractal inductor is used to create high inductance in the single-layer process, but access to multilayers is restricted owing to unusual and expensive fabrication processes. Practical implications The proposed inductor can be used in implementation of low noise amplifier, voltage controlled oscillators and power amplifiers. Originality/value This paper introduces a combination of two fractal curves to implement a hybrid fractal inductor that enhances the performance of the inductor.

Using space-filling curves to improve the quad tree for spatial indexing

Spatial indexes, like the ones that are based on Quad Tree, are important in spatial data-bases for the effective implementation of queries with spatial constraints, particularly in the case where queries include spatial links. The quad trees are a very interesting subject, given the fact that they give the ability to solve problems in a way that focuses only on the important areas with the highest density of information. But it is not without the disadvantages because the search process in the quartile suffers from the problem of repetition when reaching the terminal node and return to the behaviour of another way in the search and lead to the absorption of large amounts of time and storage. A database management system can handle data very easily if the object is one-dimension (sequential).&#x0D; In this paper, improve the quad tree by combining one of the space filling curve types, including the Hilbert curve and the Z-ordering curve with a quad tree. It will convert from two-dimensional to one-dimensional and sequentially search and end the problem of repetition whenever it reaches a terminal node Ordinary quad tree. Resulting in reduced storage space requirements and improved implementation time.

Amendable automaton for the language of finite strings of rectangular hilbert curve

Level based Context Free Grammar (CFG) is introduced and is defined to generate the language of finite strings of Rectangular Hilbert Curve. An Amendable automaton equivalent to this Level based CFG is produced to accept the language of these finite strings.

A fast acoustic emission beamforming localization method based on Hilbert curve

Abstract The low efficiency of the acoustic emission (AE) beamforming localization method in large-scale structural monitoring seriously restricts its application in practice. The Hilbert curve is introduced to improve the computational efficiency of AE beamforming method in this paper. Firstly, the Hilbert curve is introduced to traverse the monitored area, and quickly capture the approximate position of the AE source based on Beamforming. Then, the ‘Nine-Four bisection’ method is proposed to overcome the boundary problem in region division. Finally, the Hilbert curve-beamforming method (HCBF) is verified by simulation and experiment data. The results show that, the presented method not only greatly improves the computational efficiency, but also maintains the original localization accuracy and reliability. It creates a prerequisite for the further promotion and development of the beamforming method in the field of AE source localization in large-scale structural health monitoring.

Distributed Balanced Partitioning via Linear Embedding †

Balanced partitioning is often a crucial first step in solving large-scale graph optimization problems, for example, in some cases, a big graph can be chopped into pieces that fit on one machine to be processed independently before stitching the results together, leading to certain suboptimality from the interaction among different pieces. In other cases, links between different parts may show up in the running time and/or network communications cost, hence the desire to have small cut size. We study a distributed balanced-partitioning problem where the goal is to partition the vertices of a given graph into k pieces so as to minimize the total cut size. Our algorithm is composed of a few steps that are easily implementable in distributed computation frameworks such as MapReduce. The algorithm first embeds nodes of the graph onto a line, and then processes nodes in a distributed manner guided by the linear embedding order. We examine various ways to find the first embedding, for example, via a hierarchical clustering or Hilbert curves. Then we apply four different techniques including local swaps, and minimum cuts on the boundaries of partitions, as well as contraction and dynamic programming. As our empirical study, we compare the above techniques with each other, and also to previous work in distributed graph algorithms, for example, a label-propagation method, FENNEL and Spinner. We report our results both on a private map graph and several public social networks, and show that our results beat previous distributed algorithms: For instance, compared to the label-propagation algorithm, we report an improvement of 15–25% in the cut value. We also observe that our algorithms admit scalable distributed implementation for any number of partitions. Finally, we explain three applications of this work at Google: (1) Balanced partitioning is used to route multi-term queries to different replicas in Google Search backend in a way that reduces the cache miss rates by ≈ 0.5 % , which leads to a double-digit gain in throughput of production clusters. (2) Applied to the Google Maps Driving Directions, balanced partitioning minimizes the number of cross-shard queries with the goal of saving in CPU usage. This system achieves load balancing by dividing the world graph into several “shards”. Live experiments demonstrate an ≈ 40 % drop in the number of cross-shard queries when compared to a standard geography-based method. (3) In a job scheduling problem for our data centers, we use balanced partitioning to evenly distribute the work while minimizing the amount of communication across geographically distant servers. In fact, the hierarchical nature of our solution goes well with the layering of data center servers, where certain machines are closer to each other and have faster links to one another.

Pantograph–Catenary Arcing Detection Based on Electromagnetic Radiation

Currently, the detection equipment for offline arcing on high-speed trains is of great interest because it can evacuate the contact state between the contact wire and the strip. In order to ensure the safety of the train under the effect of the arcing, the self-designed fourth-order Hilbert curve fractal antenna is chosen to detect the arcing, in this paper, based on its electromagnetic radiation signal. Aspects of time-, frequency-, and joint time–frequency-domain analyses of the signals were considered. Meanwhile, the experiments were conducted several times with different circuit currents and different electrode gaps, and the electromagnetic pulses under different moments were studied. Results indicated that the electromagnetic radiation pulses were corresponding to the abrupt variations of voltage during arcing, and all the characteristic frequencies were located at similar values (18 MHz). Mechanisms of using the short-time Fourier transform method to obtain an accurate arcing moment, extinguished time, and arcing duration have been explained in detail. Thus, the characteristics of arcing electromagnetic radiation could become the significant and efficient parameters to detect the pantograph–catenary operation state of high-speed trains.