Hexagonal Grid Research Articles

Construction of inherent geometric pattern for equal-area hexagonal discrete global grid systems

ABSTRACT Discrete Global Grid Systems (DGGSs) are hierarchical frameworks that provide seamless global coverage and enable the efficient processing and analysis of heterogeneous geospatial data. As cell resolution becomes finer, similarities in cell attributes, such as cell size and shape, increase. Understanding these shared characteristics enhances our knowledge of DGGSs structures, improving their reliability and accuracy across a range of applications. However, existing methods for quantifying these characteristics are limited, often emphasizing overall analysis and visualization rather than detailed assessment. This paper introduces a geometric pattern quantification method for equal-area hexagonal DGGSs by calculating inter-level attribute similarities to identify a reference level, segmenting the spatial distribution image at this reference level, and fitting the boundaries of these sub-images to create iso-feature lines. The results show that this method improves the accuracy and efficiency of cell attribute calculations, with enhancements of at least 5.6 times for study regions and 13.8 times for sample points at lower levels. This pattern provides a robust basis for grid selection and optimization in regional applications, promoting the broader adoption of DGGSs across various fields.

Integrating gene expression and imaging data across Visium capture areas with visiumStitched.

Visium is a widely-used spatially-resolved transcriptomics assay available from 10x Genomics. Standard Visium capture areas (6.5mm by 6.5mm) limit the survey of larger tissue structures, but combining overlapping images and associated gene expression data allow for more complex study designs. Current software can handle nested or partial image overlaps, but is designed for merging up to two capture areas, and cannot account for some technical scenarios related to capture area alignment. We generated Visium data from a postmortem human tissue sample such that two capture areas were partially overlapping and a third one was adjacent. We developed the R/Bioconductor package visiumStitched, which facilitates stitching the images together with Fiji (ImageJ), and constructing SpatialExperiment R objects with the stitched images and gene expression data. visiumStitched constructs an artificial hexagonal array grid which allows seamless downstream analyses such as spatially-aware clustering without discarding data from overlapping spots. Data stitched with visiumStitched can then be interactively visualized with spatialLIBD. visiumStitched provides a simple, but flexible framework to handle various multi-capture area study design scenarios. Specifically, it resolves a data processing step without disrupting analysis workflows and without discarding data from overlapping spots. visiumStitched relies on affine transformations by Fiji, which have limitations and are less accurate when aligning against an atlas or other situations. visiumStitched provides an easy-to-use solution which expands possibilities for designing multi-capture area study designs.

Optimizing Multi-Tier Scheduling and Secure Routing in Edge-Assisted Software-Defined Wireless Sensor Network Environment Using Moving Target Defense and AI Techniques

Software Defined Wireless Sensor Networks (SDWSN) enable flexibility in Wireless Sensor Network (WSN) environments by defining the controllable functions to WSN nodes by the Software Defined Network (SDN) controller. Due to the rapid evolution of SDWSNs, adverse effects also have occurred in terms of interference, energy consumption, and security issues. Several state-of-the-art works lend their utmost best to the SDWSN environment. However, the complete picture (i.e., relatability and security in SDWSN) poses severe challenges. The state-of-the-art issues is addressed in this research by proposing interference-aware Multi-Tier Scheduling for the SDWSN environment (MTS-SDWSN). First, we perform network construction in which the proposed network is constructed in a 2D hexagonal grid structure to resolve the connectivity issue. Upon constructing the network, the SDWSN nodes are clustered and managed to reduce the energy consumption using the Divide Well To Merge Better (DWTMB) algorithm in which the optimal Cluster Leader (CL) is selected based on adequate constraint. The data from the clustered nodes are sent to the Local Base Station (LBS) via CL in which they are scheduled in multi-tier format to diminish the complexity and interference issues. The first tier involved in scheduling among Cluster Members (CMs) and CL using adequate metrics, whereas the successive tiers (i.e., second and third) involved in scheduling among CLs to LBSs and LBSs to Sink Node (SN) are done using the Non-Cooperative Fuzzy Theory (NCFT) method. Last, the scheduled nodes are routed to appropriate destinations using Secure and Optimal Routing Protocol (SORP). The proposed SORP includes the Alibaba and Forty Thieves (AFT) and Multi Criteria Decision Making (MCDM) algorithms for selecting and ranking the optimal routes. Further, the security of the routes is enabled by adopting trust and Moving Target Defense (MTD) mechanisms. The MTD includes route switching among the SDWSN devices and active switch handling using Cycle Generative Adversarial Networks (CGAN) among the switches. The proposed work is implemented using a NS-3.26 simulation tool, and performance of the proposed model and existing works shows that the proposed work outperforms the existing works.

An adaptive hexagonal deployment model for resilient wireless sensor networks in precision agriculture

This study presents an innovative hexagonal deployment model designed specifically for wireless sensor networks (WSNs) with a primary application in precision agriculture. The proposed protocol integrates advanced features, notably an adaptive frequency-hopping spread spectrum (AFHSS) mechanism and a decentralized real-time adaptation strategy to optimize data transmission in dynamic agricultural environments. The simulation study, conducted in diverse terrains with realistic sensor node distributions, meticulously evaluates the protocol’s performance using comprehensive Quality of Service (QoS) metrics. The hexagonal deployment model operates by strategically positioning sensor nodes in a hexagonal grid pattern, ensuring uniform coverage of the agricultural field. The AFHSS mechanism dynamically adjusts frequency channels, mitigating interference and fortifying the network’s robustness against external disruptions. Complementing this, the decentralized real-time adaptation empowers individual nodes to autonomously respond to the ever-changing environmental conditions, optimizing data transmission efficiency. Quantitative results from the simulations exhibit outstanding performance metrics. The protocol achieves an average latency of 50 milliseconds, a packet loss rate below 2%, a success rate exceeding 95%, and highly efficient obstacle management, with adjusted nodes accounting for less than 5%. These compelling outcomes underscore the protocol’s exceptional ability to deliver responsive and reliable data transmission, positioning it as a promising solution for enhancing environmental monitoring in precision agriculture. This study provides quantitative evidence of the protocol’s prowess and delves into the nuanced working mechanisms, offering a deeper understanding of its potential impact. The findings contribute significant insights to the field, serving as a robust foundation for researchers and practitioners engaged in designing and implementing resilient WSNs tailored for precision agriculture applications.

Research on hexagonal grid off-road path planning algorithm considering multiple factors

Abstract This article proposes a hexagonal grid off-road path planning algorithm that takes into account multiple factors for the path planning problem of motor vehicles in complex off-road environments. This method first uses a hexagonal grid model to abstract and quantify the terrain factors that affect off-road vehicles. Then, the Analytic Hierarchy Process is used to determine the weight of each factor on the impact of off-road vehicle traffic. Finally, by comprehensively considering the elevation, slope, and surface coverage elements, as well as the search efficiency of the algorithm, the cost function and heuristic function of the A * algorithm are further designed and optimized. The experimental results show that the method proposed in this paper outperforms existing algorithms in terms of pathfinding results and search efficiency, thus verifying the effectiveness and feasibility of the algorithm proposed in this paper.

The evolution of urban ecological resilience: An evaluation framework based on vulnerability, sensitivity and self-organization

Ecological resilience assessment has become a key link in urban sustainable governance. This study introduces a new evaluation framework to inform policy-making and practical applications. Based on the structural and functional dimensions of landscape patterns, it integrates the vulnerability, sensitivity and self-organization of resilience to point to desirable directions of ecological resilience. A composite ecological resilience index is compiled based on six indices of landscape diversity, landscape disturbance, source-sink patch distance, habitat quality, minimum cumulative resistance, and landscape restoration. The framework is particularly applicable to cities located in ecologically sensitive areas. Hence, Nanchang City, China was selected as a case study. Using 1km2 hexagonal grids, the framework is applied to map spatiotemporal changes and to analyze various natural and anthropogenic driving forces of ecological resilience in Nanchang from 2000 to 2020. Research findings confirm the feasibility and value of the urban ecological resilience analysis framework. They also highlight the advantages of the framework in revealing spatially dynamic processes and ecological resilience contributing factors, making it a valuable and practical tool for sustainable urban planning and refined management decision-making.

Spatial Predictive Modeling of Liver Fluke Opisthorchis viverrine (OV) Infection under the Mathematical Models in Hexagonal Symmetrical Shapes Using Machine Learning-Based Forest Classification Regression

Infection with liver flukes (Opisthorchis viverrini) is partly due to their ability to thrive in habitats in sub-basin areas, causing the intermediate host to remain in the watershed system throughout the year. Spatial modeling is used to predict water source infections, which involves designing appropriate area units with hexagonal grids. This allows for the creation of a set of independent variables, which are then covered using machine learning techniques such as forest-based classification regression methods. The independent variable set was obtained from the local public health agency and used to establish a relationship with a mathematical model. The ordinary least (OLS) model approach was used to screen the variables, and the most consistent set was selected to create a new set of variables using the principal of component analysis (PCA) method. The results showed that the forest classification and regression (FCR) model was able to accurately predict the infection rates, with the PCA factor yielding a reliability value of 0.915. This was followed by values of 0.794, 0.741, and 0.632, respectively. This article provides detailed information on the factors related to water body infection, including the length and density of water flow lines in hexagonal form, and traces the depth of each process.

Integrating gene expression and imaging data across Visium capture areas with visiumStitched.

Visium is a widely-used spatially-resolved transcriptomics assay available from 10x Genomics. Standard Visium capture areas (6.5mm by 6.5mm) limit the survey of larger tissue structures, but combining overlapping images and associated gene expression data allow for more complex study designs. Current software can handle nested or partial image overlaps, but is designed for merging up to two capture areas, and cannot account for some technical scenarios related to capture area alignment. We generated Visium data from a postmortem human tissue sample such that two capture areas were partially overlapping and a third one was adjacent. We developed the R/Bioconductor package visiumStitched, which facilitates stitching the images together with Fiji (ImageJ), and constructing SpatialExperiment R objects with the stitched images and gene expression data. visiumStitched constructs an artificial hexagonal array grid which allows seamless downstream analyses such as spatially-aware clustering without discarding data from overlapping spots. Data stitched with visiumStitched can then be interactively visualized with spatialLIBD. visiumStitched provides a simple, but flexible framework to handle various multi-capture area study design scenarios. Specifically, it resolves a data processing step without disrupting analysis workflows and without discarding data from overlapping spots. visiumStiched relies on affine transformations by Fiji, which have limitations and are less accurate when aligning against an atlas or other situations. visiumStiched provides an easy-to-use solution which expands possibilities for designing multi-capture area study designs.

A Fuzzy Logic-Based Directional Charging Scheme for Wireless Rechargeable Sensor Networks.

Wireless Power Transfer (WPT) has become a key technology to extend network lifetime in Wireless Rechargeable Sensor Networks (WRSNs). The traditional omnidirectional recharging method has a wider range of energy radiation, but it inevitably results in more energy waste. By contrast, the directional recharging mode enables most of the energy to be focused in a predetermined direction that achieves higher recharging efficiency. However, the MC (Mobile Charger) in this mode can only supply energy to a few nodes in each direction. Thus, how to set the location of staying points of the MC, its service sequence and its charging orientation are all important issues related to the benefit of energy replenishment. To address these problems, we propose a Fuzzy Logic-based Directional Charging (FLDC) scheme for Wireless Rechargeable Sensor Networks. Firstly, the network is divided into adjacent regular hexagonal grids which are exactly the charging regions for the MC. Then, with the help of a double-layer fuzzy logic system, a priority of nodes and grids is obtained that dynamically determines the trajectory of the MC during each round of service, i.e., the charging sequence. Next, the location of the MC's staying points is optimized to minimize the sum of charging distances between MC and nodes in the same grid. Finally, the discretized charging directions of the MC at each staying point are adjusted to further improve the charging efficiency. Simulation results show that FLDC performs well in both the charging benefit of nodes and the energy efficiency of the MC.

Optimal Local Identifying and Local Locating-dominating Codes

We introduce two new classes of covering codes in graphs for every positive integer r. These new codes are called local r-identifying and local r-locating-dominating codes and they are derived from r-identifying and r-locating-dominating codes, respectively. We study the sizes of optimal local 1-identifying codes in binary hypercubes. We obtain lower and upper bounds that are asymptotically tight. Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible. For some small n optimal constructions are obtained. Moreover, the upper bound is obtained by a linear code construction. Also, we study the densities of optimal local 1-identifying codes and local 1-locating-dominating codes in the infinite square grid, the hexagonal grid, the triangular grid and the king grid. We prove that seven out of eight of our constructions have optimal densities.

Finding Codes on Infinite Grids Automatically

We apply automata theory and Karp’s minimum mean weight cycle algorithm to minimum density problems in coding theory. Using this method, we find the new upper bound 53/126 ≈ 0.4206 for the minimum density of an identifying code on the infinite hexagonal grid, down from the previous record of 3/7 ≈ 0.4286.

A methodological framework to characterize the wildlife-livestock interface: The case of wild boar in mainland Spain

BackgroundThe representation of wildlife-livestock interface (WLI) at an accurate spatial resolution poses several challenges. Furthermore, there is a lack of published material providing detailed descriptions of geospatial techniques for the purpose of producing visual results that are interpretable and contrastable for epidemiological analysis. ObjectivesOur aim is to develop a standardized, applicable, and scalable methodological framework for describing and characterizing the WLI across a large spatial extent. Subsequently, we aim to employ this framework to depict specific WLI based on different epidemiological scenarios determined by the abundance of wild boar (Sus scrofa) and domestic ungulates as an illustrative case, specifically focusing on mainland Spain. MethodsTo establish a methodological framework, we merged data from both wild and domestic sources into a hexagonal grid. We utilized data on wild boar hunting and the locations of pig, cattle, sheep, and goat farms in mainland Spain. New variables were derived from this combined dataset to illustrate the overlapping abundance between wild boar and domestic species. Finally, a cluster analysis of the generated variables was carried out, with the aim of distinguishing and characterizing various scenarios of the wild boar-domestic ungulate interface in mainland Spain. ResultsThe hexagonal grid proved appropriate to represent and evaluate the WLI at fine spatial resolution over such broad extent. Despite the inability to ascribe a dominant livestock type and production system to a specific region, we were able to identify fifteen main areas of interest in terms of overlap. As for extensive livestock, normally at the highest risk of interaction with wild boar, the primary regions in Spain were those with dehesa agroecosystem and the Atlantic areas. Certain scenarios were particularly relevant in terms of risk for interaction and subsequent transmission of disease, namely, the case of extensive pig production in south western Spain (dehesa agroecosystem), which is especially concerned about the potential introduction of African Swine fever (ASF) in the Country. Discussion and conclusionsWe provide a basis for visualizing and understanding of different WLI scenarios, which is extensible to other regions and interfaces, and automatable where precise source of data from wildlife and livestock are available. This spatial statistics framework enables the utilization of high-resolution data, ensuring consistency on uniform grids. This aligns with the needs of high-resolution disease dissemination models based on wildlife behaviour. Such aspects are crucial for developing risk assessment and improving strategies for the prevention, control, and eradication of shared priority emerging diseases at national and international levels, such as ASF.

On group-theoretic eigenvalue vibration analysis of structural systems with C6v symmetry

In considerations of the linear vibration of symmetric systems, group theory allows the space of the eigenvalue problem to be decomposed into independent subspaces that are spanned by symmetry-adapted freedoms. These problems usually feature one or more degenerate subspaces (i.e. subspaces that contain repeating solutions). For such subspaces, the associated idempotents, as calculated from the character table of the symmetry group, are not capable of full decomposition of the subspace. In this paper, and based on group theory, simple algebraic operators that fully decompose the two degenerate subspaces of structural problems belonging to the C6v symmetry group are proposed. The operators are applied to the vibration of a spring-mass system, for which the results for natural frequencies are found to agree exactly with results from the literature. Their application to the vibration of a hexagonal plane grid reveals new insights on the character of the modes of degenerate subspaces. The overall conclusion is that, for problems belonging to the C6v symmetry group, the proposed operators allow the mixed modes of degenerate subspaces to be separated into two distinct symmetry categories, and are very effective in simplifying the actual computation of the repeating eigenvalues of these subspaces.

Isoline extraction based on a global hexagonal grid

ABSTRACT Isoline extraction is influenced by numerical differences between grid cell neighbors. Due to varying neighborhood distances and directions of a traditional square grid cell across geographical space, numerical differences fail to precisely reflect the trends of terrain features. Alternatively, hexagonal grid spatial discretization has been proven to be superior to the square grid, attributed to its isotropic, consistent connectivity, uniform neighborhood. Taking these advantages into consideration, this paper explores the advantages of hexagonal grid discretization for isoline extraction in comparison with square grid. The comparisons show that a hexagonal grid performs better than a square grid, and it can extract finer isoline based on the same resolution. Result show that hexagonal grid can express the numerical differences between neighboring grid cells consistently and correctly, leading to improvements in isoline extraction. This paper demonstrates the significant impact of grid cell geometry on the quality of isoline obtained from grid-based geographical spatial data, providing new insights for spatial analysis of geographical data.

The implications of estimating rarity in Brazilian reptiles from GBIF data based on contributions from citizen science versus research institutions

AbstractUnderstanding the distribution of rare species is important for conservation prioritisation. Traditionally, museums and other research institutions have served as depositories for specimens and biodiversity information. However, estimating abundance from these sources is challenging due to spatiotemporally biased collection methods. For instance, large‐bodied reptiles that are found near research institutions or in popular, easily accessible sites tend to be overrepresented in collections compared to smaller species found in remote areas. Recently, a substantial number of observations have been amassed through citizen (or community) science initiatives, which are invaluable for monitoring purposes. Given the unstructured nature of this sampling, these datasets are often affected by biases, such as taxonomic, spatial and temporal preferences. Therefore, analysing data from these two sources can lead to different abundance estimates. This study compiled data on Brazilian reptile species from the Global Information Biodiversity Facility (GBIF). It employed a community‐ecology approach to analyse data from research institutions and citizen science initiatives, separately and collectively, to assess taxonomic and spatial species coverage and predict species rarity. Using a 1‐degree hexagonal grid, we analysed the spatial distribution of reptile communities and calculated rarity indices for 754 reptile species. Our findings reveal that 87 species were exclusively recorded in the citizen science subset, while 212 were recorded only by research institutions. The number of observations per species in the citizen science data followed a Gambin distribution, which aligns with the expected pattern of abundance in natural communities, unlike the data from research institutions. This suggests that citizen science data may be a more accurate source for estimating species abundance and rarity. The discrepancies in rarity classifications between the datasets were likely due to differences in sample size and potentially other sampling parameters. Nevertheless, combining data collected by both research institutions and citizen science initiatives can help to fill knowledge gaps in reptile species occurrence, thus enhancing the foundation for conservation efforts on a national scale.

What Goes Around Comes Around: Covering Tours and Cycle Covers with Turn Costs

We investigate several geometric problems of finding tours and cycle covers with minimum turn cost, which have been studied in the past, with complexity, approximation results, and open problems dating back to work by Arkin et al. in 2001. Many new practical applications have spawned variants: For full coverage, all points have to be covered, for subset coverage, specific points have to be covered, and for penalty coverage, points may be left uncovered by incurring a penalty. We show that finding a minimum-turn (full) cycle cover is NP-hard even in 2-dimensional grid graphs, solving the long-standing open Problem 53 in The Open Problems Project edited by Demaine, Mitchell and O’Rourke. We also prove NP-hardness of finding a subset cycle cover of minimum turn cost in thin grid graphs, for which Arkin et al. gave a polynomial-time algorithm for full coverage; this shows that their boundary techniques cannot be applied to compute exact solutions for subset and penalty variants. We also provide a number of positive results. In particular, we establish the first constant-factor approximation algorithms for all considered subset and penalty problem variants for grid-based instances, based on LP/IP techniques. These geometric versions allow many possible edge directions (and thus, turn angles, such as in hexagonal grids or higher-dimensional variants); our approximation factors improve the combinatorial ones of Arkin et al.

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.

Spatial and Temporal Heterogeneity of Eco-Environmental Quality in Yanhe Watershed (China) Using the Remote-Sensing-Based Ecological Index (RSEI)

The long-term impacts of climate change and human activities have resulted in the Yanhe watershed, a typical watershed in the Loess Plateau region, exhibiting a high degree of vulnerability and significant heterogeneity in ecological environmental quality. This has led to environmental degradation and complex socio-ecological challenges. Consequently, there is an urgent need to carry out research on the spatial and temporal differentiation patterns of ecological environment quality. By utilizing remote sensing data spanning 21 years, this study evaluated the evolutionary trends and consistency of ecological environment quality (EEQ) within the Yanhe watershed based on the remote-sensing-based ecological index (RSEI). Furthermore, it examined global and local spatial autocorrelation of the RSEI by constructing a hexagonal grid, thereby revealing the spatiotemporal characteristics of EEQ at different scales within the Yanhe watershed. The results were as follows: (1) The EEQ has exhibited an overall upward trend in the past two decades, while it has displayed significant fluctuations; (2) the Global Moran’s I values for the years 2000, 2010, and 2020 were 0.18, 0.32, and 0.21, respectively, indicating a presence of spatial autocorrelation within the RSEI; (3) the overall EEQ of the Yanhe watershed will continue to improve, although the ecological quality in certain areas remains unstable due to local natural conditions and human activities. This research not only contributes to the technical framework for analyzing the spatiotemporal heterogeneity of EEQ but also provides actionable insights for ecosystem restoration and sustainability within the Loess Plateau watershed. Our work advances the understanding of ecological dynamics in semi-arid regions and offers a model for assessing ecological quality in similar environmental contexts.

Improved A* Navigation Path-Planning Algorithm Based on Hexagonal Grid

Navigation systems are extensively used in everyday life, but the conventional A* algorithm has several limitations in path planning applications within these systems, such as low degrees of freedom in path planning, inadequate consideration of the effects of special regions, and excessive nodes and turns. Addressing these limitations, an enhanced A* algorithm was proposed using regular hexagonal grid mapping. First, the approach to map modeling using hexagonal grids was described. Subsequently, the A* algorithm was refined by optimizing the calculation of movement costs, thus allowing the algorithm to integrate environmental data more effectively and flexibly adjust node costs while ensuring path optimality. A quantitative method was also introduced to assess map complexity and adaptive heuristics that decrease the number of traversed nodes and increase the search speed. Moreover, a turning penalty measure was implemented to minimize unnecessary turns on the planned paths. Simulation results confirmed that the improved A* algorithm exhibits superior performance, which can dynamically adjust movement costs, enhance search efficiency, reduce turns, improve overall path planning quality, and solve critical path planning issues in navigation systems, greatly aiding the development and design of these systems and making them better suited to meet modern navigation requirements.

The oriented chromatic number of the hexagonal grid is 6

The oriented chromatic number of a directed graph G is the minimum order of an oriented graph to which G has a homomorphism. The oriented chromatic number χo(F) of a graph family F is the maximum oriented chromatic number over any orientation of any graph in F. For the family of hexagonal grids H2, Bielak (2006) proved that 5≤χo(H2)≤6. Here we close the gap by showing that χo(H2)≥6.