Space-filling Curves Research Articles

On Recursive Tiling of some Mathematical Objects.

This essay, in a novel way, uses various tiles, such as dominoes, square-, interlocking-, and flow-tiles, to visualize approximations to recursively defined, potentially infinite mathematical objects, such as binary code tables, and variants of the space-filling Hilbert curve. The power of L-systems is shown by defining these tiled objects as L-systems.

DPFANet: Deep Point Feature Aggregation Network for Classification of Irregular Objects in LIDAR Point Clouds

Point cloud data acquired by scanning with Light Detection and Ranging (LiDAR) devices typically contain irregular objects, such as trees, which lead to low classification accuracy in existing point cloud classification methods. Consequently, this paper proposes a deep point feature aggregation network (DPFANet) that integrates adaptive graph convolution and space-filling curve sampling modules to effectively address the feature extraction problem for irregular object point clouds. To refine the feature representation, we utilize the affinity matrix to quantify inter-channel relationships and adjust the input feature matrix accordingly, thereby improving the classification accuracy of the object point cloud. To validate the effectiveness of the proposed approach, a TreeNet dataset was created, comprising four categories of tree point clouds derived from publicly available UAV point cloud data. The experimental findings illustrate that the model attains a mean accuracy of 91.4% on the ModelNet40 dataset, comparable to prevailing state-of-the-art techniques. When applied to the more challenging TreeNet dataset, the model achieves a mean accuracy of 88.0%, surpassing existing state-of-the-art methods in all classification metrics. These results underscore the high potential of the model for point cloud classification of irregular objects.

Vector geographic data commutative encryption and watermarking algorithm based on prediction differences

Commutative encryption and watermarking (CEW) is a seamless integration of encryption and watermarking technologies, overcoming the limitations of single security protection methods that may fail to provide comprehensive data protection. It is a research hotspot in the field of geographic information security. However, constructing or selecting stable features for vector point data to apply CEW algorithms remains a challenging problem. To address this, this paper utilizes space-filling curves to organize discrete points into line or polygon types and proposes a CEW algorithm for vector data based on prediction differences. Specifically, feature points are used to fit curves, providing independent operational spaces for watermarking and encryption. The segmented fitted curves are arbitrarily flipped to achieve perceptual encryption, while watermarks are embedded in the differences between the fitted curves and the actual data. Experimental results demonstrate that the proposed algorithm effectively addresses the issue of existing algorithms being unsuitable for point data. In simulated real-world scenarios, the extracted watermark information achieved NC values above 0.75, indicating strong robustness. Under conditions using existing lossless algorithms, the watermark capacity of the proposed method is thousands of times higher than that of existing algorithms, and it exhibits excellent encryption security, providing a new perspective for further research on lossless CEW.

A gamma-ray events discrimination method based on convolutional neural network in a HPGe spectrometer

Discriminating between full- and partial-energy deposition events in a high-purity germanium detection system can achieve the subtraction of background events and reduce the system’s minimum detectable activity (MDA). Traditional anticoincidence techniques and pulse shape discrimination techniques can both achieve event type discrimination, but the former has a complex equipment structure and is difficult to maintain, and the latter still has room for improvement in recognition accuracy. This paper proposes a method using convolutional neural networks (CNNs) to learn and recognize pulse waveform features for event type discrimination. We studied the impact of space-filling curve encoding methods and convolution kernel size on the training effect of the model. The results show that the zigzag curve encoding method results in a higher discrimination accuracy of the model; when the convolution kernel size is 9 × 9, the model has the best convergence speed and training effect. Finally, we compared the performance of fully connected neural network (FC network), 1D-CNN, and 2D-CNN in event type discrimination using MDA as the indicator. The results show that compared to FC network, CNNs perform better in reducing the MDA across the entire energy range, especially 2D-CNN, which performs better in the medium and low energy range compared to 1D-CNN. After using 2D-CNN for event discrimination and background suppression, the MDA is reduced by 21.23% (122 keV), 16.32% (244 keV), 8.43% (344 keV), 13.76% (662 keV), 5.54 % (1173 keV), and 26.35% (1332 keV).

HINTs: Sensemaking on large collections of documents with Hypergraph visualization and INTelligent agents.

Sensemaking on a large collection of documents (corpus) is a challenging task often found in fields such as market research, legal studies, intelligence analysis, political science, or computational linguistics. Previous works approach this problem from topic- and entity-based perspectives, but the capability of the underlying NLP model limits their effectiveness. Recent advances in prompting with LLMs present opportunities to enhance such approaches with higher accuracy and customizability. However, poorly designed prompts and visualizations could mislead users into falsely interpreting the visualizations and hinder the system's trustworthiness. In this paper, we address this issue by taking into account the user analysis tasks and visualization goals in the prompt-based data extraction stage, thereby extending the concept of Model Alignment. We present HINTs, a VA system for supporting sensemaking on large collections of documents, combining previous entity-based and topic-based approaches. The visualization pipeline of HINTs consists of three stages. First, entities and topics are extracted from the corpus with prompts. Then, the result is modeled as a hypergraph and hierarchically clustered. Finally, an enhanced space-filling curve layout is applied to visualize the hypergraph for interactive exploration. The system further integrates an LLM-based intelligent chatbot agent in the interface to facilitate the sensemaking of interested documents. To demonstrate the generalizability and effectiveness of the HINTs system, we present two case studies on different domains and a comparative user study. We report our insights on the behavior patterns and challenges when intelligent agents are used to facilitate sensemaking. We find that while intelligent agents can address many challenges in sensemaking, the visual hints that visualizations provide are still necessary. We discuss limitations and future work for combining interactive visualization and LLMs more profoundly to better support corpus analysis.

AMP: Total Variation Reduction for Lossless Compression via Approximate Median-based Preconditioning

With the increasing scale of cloud computing applications of next-generation embedded systems, a major challenge that domain scientists are facing is how to efficiently store and analyze the vast volume of output data. Compression can reduce the amount of data that needs to be transferred and stored. However, most of the large datasets are in floating-point format, which exhibits high entropy. As a result, existing lossless compressors cannot provide enough performance for such applications. To address this problem, we propose a total variation reduction method for improving the compression ratio of lossless compressors (namely, FPC + and FPZIP + ), which employs a median-based hyperplane to precondition the data. In particular, we first try to exploit the space-filling curve (SFC), a well-known technique to preserve data locality for a multi-dimensional dataset. We show and explain why a raw SFC, such as Hilbert and Z-order curves, cannot improve the compression ratio. Then, we explore the opportunity and theoretical feasibility of the proposed total variation reduction-based algorithm. The experiment results show the effectiveness of the proposed method. The compression ratios are improved up to 48.2% (20.6% on average) for FPZIP and 42.4% (18.4% on average) for FPC. Moreover, through observing the time composition of the proposed method, it is found that the median finding holds a high percentage of the execution time. Hence, we further introduce an approximate median finding algorithm, providing a linear-time overhead reduction scheme. The experiment results clearly demonstrate that this algorithm reduces execution time by an average of 56.7% and 40.7% compared to FPC + and FPZIP + , respectively.

Moiré patterns of space-filling curves

It is shown on the examples of Moore and Gosper curves that two spatially shifted or twisted, preasymptotic space-filling curves can produce large-scale superstructures akin to moiré patterns. To study physical phenomena emerging from these patterns, a geometrical coupling coefficient based on the Neumann integral is introduced. It is found that moiré patterns appear most defined at the peaks of those coefficients. A physical interpretation of these coefficients as a measure for inductive coupling between radiofrequency resonators leads to a design principle for strongly overlapping resonators with vanishing mutual inductance, which might be interesting for applications in MRI. These findings are demonstrated in graphical, numerical, and physical experiments. Published by the American Physical Society 2024

A GPU-Based Integration Method from Raster Data to a Hexagonal Discrete Global Grid

This paper proposes an algorithm for the conversion of raster data to hexagonal DGGSs in the GPU by redevising the encoding and decoding mechanisms. The researchers first designed a data structure based on rhombic tiles to convert the hexagonal DGGS to a texture format acceptable for GPUs, thus avoiding the irregularity of the hexagonal DGGS. Then, the encoding and decoding methods of the tile data based on space-filling curves were designed, respectively, so as to reduce the amount of data transmission from the CPU to the GPU. Finally, the researchers improved the algorithmic efficiency through thread design. To validate the above design, raster integration experiments were conducted based on the global Aster 30 m digital elevation dataDEM, and the experimental results showed that the raster integration accuracy of this algorithms was around 1 m, while its efficiency could be improved to more than 600 times that of the algorithm for integrating the raster data to the hexagonal DGGS data, executed in the CPU. Therefore, the researchers believe that this study will provide a feasible method for the efficient and stable integration of massive raster data based on a hexagonal grid, which may well support the organization of massive raster data in the field of GIS.

Duality of a coiled phononic crystal enables reflectionless interfaces

Recently, it has been demonstrated that one-dimensional bar-based phononic crystals can exhibit subwavelength Bragg bandgaps by coiling the bars and locking the nodal rotational degrees of freedom to create what is termed a coiled phononic crystal (CPnC) [C. L. Willey et al., Phys. Rev. Appl. 18, 014035 (2022)]. Here, it is shown that the CPnC exhibits duality of its dispersion curves relative to its coiling/twist angle, meaning that the dispersion curves are symmetric about a particular coiling/twist angle defined configuration. An exciting implication of this finding is that under a certain set of constraints, segments of dual unit cells with perpendicular wave propagation directions can be connected such that their wave transmission is equivalent to a finite CPnC entirely composed of identical unit cells with parallel wave propagation directions. The ability to link unit cells with different wave propagation orientations, but the same dispersion/dynamic stiffness, is used to create an elastic hierarchically coiled phononic crystal based on a fractal space-filling curve design. The novelty of this work is that it numerically demonstrates reflectionless wave propagation in large fractal architectures (created from specific combinations of dual unit cells) such that regular phononic properties (i.e., passbands and bandgaps) are preserved, allowing for propagation of broadband signals and filtering.

Crashworthiness analysis and optimization of brain-coral-inspired multilayer sandwich structures under axial crushing

Inspired by brain corals and cuttlebone, this study employed 3D printing technology to fabricate a novel bio-inspired multilayer sandwich structure based on the Hilbert space-filling curve (named BHSS). The mechanical behavior and deformation process of the BHSS were compared through quasi-static axial crushing experiments and finite element (FE) simulations. The energy absorbing characteristics of the BHSS with different layers were compared through FE simulations, and the results indicated that the 4-layer BHSS displayed superior crashworthiness. Then, parametric studies were conducted to investigate the influence of layer-height gradient and wall-thickness gradient on the energy absorption performance and deformation modes of the BHSS. It was confirmed that the double gradient designs significantly reduced the initial peak force and improved the specific energy absorption of the BHSS. Finally, the multi-objectives optimization based on response surface method and the non-dominated sorting genetic algorithm (NSGA-II) was employed to optimize the geometric parameters of the BHSS, aiming at the optimal configuration for better crashworthiness. Compared to the original design structure, the SEA of the optimized knee point structure was increased by 21.8% and the IPF was reduced by 72.6%. These findings provided valuable guidelines for the brain-coral-inspired design of multilayer sandwich structures with superior energy-absorbing performance.

Geometry-induced process and part characteristics in support-free powder bed fusion of polypropylene at room temperature

Laser-based powder bed fusion (LPBF) of semi-crystalline polymers enables the support-free layer-wise manufacturing of geometrically diverse, complex components. In contrast to the established quasi-isothermal powder bed fusion of polymers at elevated temperatures, non-isothermal, cold processing strategies allow to significantly extend the range of applicable material systems. Relying on the superposition of discretized, fractal exposure strategies and the implicit mesoscopic compensation of crystallization shrinkage, the support-free LPBF of polypropylene at room temperature is demonstrated. The present paper displays the temporally and spatially discrete exposure of superposed fractal, space-filling curves that enable the support-free LPBF of polypropylene through combining the mesoscopic compensation of crystallization shrinkage and the laser-induced minimization of thermal shrinkage through the implementation of pre-exposure scans. The non-isothermal processing regime was observed to exhibit an intrinsic robustness towards the influence of processing parameters on emerging peak temperatures while showing a significant extent of accumulated heat within manufactured parts. Complementary mechanical characterizations showed an orientation-dependent influence of the applied energy density on emerging mechanical properties, correlated with geometry-dependent temporal process characteristics that implicitly influence the available coalescence time and the timespan available for the thermal homogenization.

Engineered substrates for metasurface antennas

This letter explores the advantages of additively manufactured substrates with spatially varying electromagnetic properties. These engineered substrates will be constructed using space filling curves (SFC) of various orders. New advanced manufacturing systems such as the nScrypt 3Dn-300, have enabled the rapid fabrication of these SFC substrates. This letter will apply the engineered SFC substrate to the design and fabrication of metasurface antennas. By utilising a SFC to vary the local substrate permittivity, along with the printed conductive patch dimensions, the range of achievable surface impedances can be greatly expanded. This enlarged design space will be leveraged to yield increased gain for a given metasurface antenna size. Methods to characterise the substrate permittivity and conductive ink are discussed along with a complete description of the metasurface antenna design, fabrication and validation process.

Dynamic estimation method for pulsar periods based on photon energy distribution folding and image template matching

The accuracy of the pulsar period estimation directly affects the restoration effect of the signal profile. A more accurate pulsar profile will help improve the accuracy of pulsar delay estimation and thereby improve the performance of X-ray pulsar navigation. This paper proposes a pulsar period estimation method based on photon energy distribution folding and image template matching (PETM). This method uses the probability distribution information of photon energy for weighted epoch folding. The one-dimensional (1D) profile information was converted into two-dimensional (2D) image information through reverse space-filling curve (SFC) encoding. Then, a feature matching was performed between the target structure and the template structure. At the same time, the criterion of Pearson correlation coefficient (PCC) was used to quantitatively evaluate the matching effect to estimate the optimal period. The simulation results show that the period estimation accuracy of the PETM method is significantly improved, as compared with the traditional \( method. This work also analyzes the folding effect based on the photon energy distribution model and conducts simulation experiments and comparisons on influencing factors, such as noise interference and data quality. At the same time, we also specifically demonstrated the effectiveness of the PETM method for the glitch phenomenon (i.e., a sudden change in period) of pulsar periods. Finally, we also used China's XPNAV-1 satellite to conduct experiments and analysis of the actual observation data of PSR B0531+21 pulsar within a fixed period of time. The results show that the period estimation accuracy of this method is 4.8190$ns$, which is 50.23<!PCT!> higher than the traditional \( method. The method proposed in this article has the advantages of high estimation accuracy, stable estimation performance, strong anti-interference ability, and excellent dynamic period estimation performance. Therefore, it can further improve the navigation performance of X-ray pulsars.

Authenticating and securing healthcare records: A deep learning-based zero watermarking approach

Security in medical records is critical to patient privacy and confidentiality. Digital Patient Records (DPR) hold sensitive information that can reveal a patient's health status and history. Their unauthorized access or exposure can lead to severe consequences, including identity theft, discrimination, and medical malpractice. Therefore, ensuring proper security measures is critical in protecting DPR and other medical records from breaches or unauthorized access. In this regard, a robust deep learning-based zero-watermarking approach is presented for authenticating and securing healthcare records. The carrier image is initially visibly marked with the hospital logo to identify ownership and prevent illegal duplication and forgery. The image mark is scrambled by applying the step space-filling curve method for improved security. In the final phase, Alexnet is used to extract the features of visibly marked carrier image. Further, NSST and SVD-based zero watermarking is implemented to conceal the scrambled mark within the features of visibly marked carrier images. It is essential for copyright protection since it establishes ownership while preventing the unauthorized use or dissemination of valuable medical research, images, and reports. The proposed framework has exhibited superior versatility, robustness, and imperceptibility compared to existing techniques with a maximum improvement of 47%.

Applying Convolutional Neural Networks to data on unstructured meshes with space-filling curves

This paper presents the first classical Convolutional Neural Network (CNN) that can be applied directly to data from unstructured finite element meshes or control volume grids. CNNs have been hugely influential in the areas of image classification and image compression, both of which typically deal with data on structured grids. Unstructured meshes are frequently used to solve partial differential equations and are particularly suitable for problems that require the mesh to conform to complex geometries or for problems that require variable mesh resolution. Central to our approach are space-filling curves, which traverse the nodes or cells of a mesh tracing out a path that is as short as possible (in terms of numbers of edges) and that visits each node or cell exactly once. The space-filling curves (SFCs) are used to find an ordering of the nodes or cells that can transform multi-dimensional solutions on unstructured meshes into a one-dimensional (1D) representation, to which 1D convolutional layers can then be applied. Although developed in two dimensions, the approach is applicable to higher dimensional problems. To demonstrate the approach, the network we choose is a convolutional autoencoder (CAE), although other types of CNN could be used. The approach is tested by applying CAEs to data sets that have been reordered with a space-filling curve. Sparse layers are used at the input and output of the autoencoder, and the use of multiple SFCs is explored. We compare the accuracy of the SFC-based CAE with that of a classical CAE applied to two idealised problems on structured meshes, and then apply the approach to solutions of flow past a cylinder obtained using the finite-element method and an unstructured mesh.

Magnetic Performance of Eddy Current Suppressing Structures in Additive Manufacturing

Additively manufactured soft-magnetic components are inherently bulky leading to significant eddy current losses when applied to electrical machines. Prior works have addressed this issue by implementing structures based on the Hilbert space-filling curve which include eddy current suppressing gaps, thereby reducing the fill factor of the soft-magnetic component. The present research aims at investigating a number of space-filling curves in addition to sheets in order to find the optimal eddy current suppressing structure from an electromagnetic point of view. By means of both analysis and finite-element simulation, it was shown that sheets are superior at minimizing eddy current losses while space-filling curves excel at maximizing the fill factor.

Wide-angle passive beam steering using 3D modified partial Maxwell fisheye lens

This study presents a broadband, 3D gradient index beam-steering lens, derived from an optimized modification of the partial Maxwell fisheye (PMFE) design, achieving a boresight gain of 23 dBi, -80° to 80° beam steering, and <10 dB gain roll-off. Utilizing fused filament fabrication (FFF) to realize its intricate geometry, the design employs a novel polar space-filling curve (PSFC) to establish a 3D varying, effective permittivity distribution. Rigorous simulations and experimental validation attest to its effectiveness, marking the first 3D implementation of a PMFE-type lens to our knowledge. This research underscores the feasibility and diverse applications of a low-cost, wide-angle passive beam-steering dielectric lens.

Adaptive Sampling of 3D Spatial Correlations for Focus+Context Visualization.

Visualizing spatial correlations in 3D ensembles is challenging due to the vast amounts of information that need to be conveyed. Memory and time constraints make it unfeasible to pre-compute and store the correlations between all pairs of domain points. We propose the embedding of adaptive correlation sampling into chord diagrams with hierarchical edge bundling to alleviate these constraints. Entities representing spatial regions are arranged along the circular chord layout via a space-filling curve, and Bayesian optimal sampling is used to efficiently estimate the maximum occurring correlation between any two points from different regions. Hierarchical edge bundling reduces visual clutter and emphasizes the major correlation structures. By selecting an edge, the user triggers a focus diagram in which only the two regions connected via this edge are refined and arranged in a specific way in a second chord layout. For visualizing correlations between two different variables, which are not symmetric anymore, we switch to showing a full correlation matrix. This avoids drawing the same edges twice with different correlation values. We introduce GPU implementations of both linear and non-linear correlation measures to further reduce the time that is required to generate the context and focus views, and to even enable the analysis of correlations in a 1000-member ensemble.

Determining solution set of nonlinear inequalities using space-filling curves for finding working spaces of planar robots

Determining solution set of nonlinear inequalities using space-filling curves for finding working spaces of planar robots

Interactive Visual Analysis of Spatial Sensitivities.

Sensitivity analyses of simulation ensembles determine how simulation parameters influence the simulation's outcome. Commonly, one global numerical sensitivity value is computed per simulation parameter. However, when considering 3D spatial simulations, the analysis of localized sensitivities in different spatial regions is of importance in many applications. For analyzing the spatial variation of parameter sensitivity, one needs to compute a spatial sensitivity scalar field per simulation parameter. Given n simulation parameters, we obtain multi-field data consisting of n scalar fields when considering all simulation parameters. We propose an interactive visual analytics solution to analyze the multi-field sensitivity data. It supports the investigation of how strongly and in what way individual parameters influence the simulation outcome, in which spatial regions this is happening, and what the interplay of the simulation parameters is. Its central component is an overview visualization of all sensitivity fields that avoids 3D occlusions by linearizing the data using an adapted scheme of data-driven space-filling curves. The spatial sensitivity values are visualized in a combination of a Horizon Graph and a line chart. We validate our approach by applying it to synthetic and real-world ensemble data.