Hexagonal discrete global grid systems are multi-resolution frameworks for earth data and have attracted much attention in research towards the next generation of geographic information system. Each resolution of the framework divides the earth into a hexagonal grid of cells and cell codes are used instead of Cartesian coordinates to organize and query data. Aperture (the ratio of cells between resolutions) 4 systems have structural advantages compared with aperture 3 and 7, but haven't been fully developed. In this manuscript, we examine the aperture 4 hexagon hierarchy on uniform tiles (HHUT) and modify the coding method. We propose two HHUT-based cell navigation methods, one for regular and the other for irregular areas. We demonstrate the utility and high efficiency of the proposed methods in two applications. Results show that the proposed method for an irregular area is on average 400 times faster than DGGRID, the conventional library. The proposed method for a regular area is on average seven times faster than the hexagon lattice quad tree method and approaches that of H3. The modified coding and novel cell navigation methods proposed here indicate high efficiency, superior data organization and broad application prospects.

Objectives: Discrete global grid systems are the preferred data models supporting multisource geospatial information fusion. Hexagonal grids have become more popular in many applications due to their geometric characteristics within uniform adjacent. Methods: We design a uniform tiles hierarchy on the surface of the icosahedron according to the characteristics of the aperture-4 hexagonal discrete global grids, using complex numbers to build a unified coding and operation model. We also design algorithms including interoperating between geographic coordinates and codes, querying neighborhood codes. Results: The experimental results show that interoperation between geographic coordinates and codes efficiency of the proposed algorithm is approximately 2.74 and 1.73 times that of the traditional algorithm respectively,and that neighborhood codes query efficiency of the proposed algorithm is approximately 7.46 times that of the traditional algorithm. As the grid level rises, the advantages of the proposed algorithm become more obvious.Conclusions: The results of this paper are expected to provide theoretical and technical supports for the unified organization, management, processing and analysis of multi-source earth observation data.

Discrete Global Grid Systems (DGGS) provide a multi-resolution discrete representation of the Earth and are preferable for the organization, integration, and analysis of large and multi-source geospatial datasets. Generating grids for the area of interest is usually the premise and basis for DGGS applications. Owing to incongruent hierarchies that restrict the multi-resolution applications of hexagonal DGGS, current grid generation of hexagonal DGGS for local areas mainly depends on inefficient single-resolution traversal methods by judging the spatial relationship between each cell and the area. This study designs a fast generation algorithm for local parts of hexagonal DGGS based on the hierarchical properties of DGGS. A partition structure at intervals of multiple levels is first designed to ensure the coverage relevance between parent and children cells of different levels. Based on this structure, the algorithm begins with coarser resolution grids and recursively decomposes them into the target resolution, with multiple decomposition patterns used and a unique condition proposed to make the generated grids without gaps or overlaps. Efficient integer coordinate operations are used to generate the vast majority of cells. Experimental results show that the proposed algorithm achieves a significant improvement in efficiency. In the aperture 4 hexagonal DGGS, the efficiency ratio of the proposed and traversal algorithms increases from six times in level 14 to approximately 339 times in level 18. This study provides a solid foundation for subsequent data quantization and multi-resolution applications in hexagonal DGGS and has broad prospects.

Discrete Global Grid System (DGGS) is a new multi-resolution geospatial data modeling and processing scheme for the digital earth. The icosahedron is commonly regarded as an ideal polyhedron for constructing DGGSs with small distortions; however, the shape of its face is triangular, making it difficult to incorporate the matrix structure used for geospatial data storage and parallel computing. To overcome this limitation, this study utilizes the rhombic triacontahedron (RT) as the basic polyhedron to construct DGGSs. An equal-area projection between the surface of RT and the sphere is developed and used to design a grid-generation algorithm for the aperture 4 hexagonal DGGS based on RT. Compared with the equal-area DGGS based on the icosahedron, the proposed scheme results in smaller angular projection distortions, with the mean and standard deviation decreasing by 41.6% and 30.9%, respectively. The grid cells of the RT DGGS also achieve more optimized geometric characteristics in shape compactness, length deviation, and angle deviation than those in the icosahedron DGGS. Additionally, the cross-surface computation efficiency provides advantages in code conversion to latitude and longitude and proximity queries. Furthermore, the use of RT offers a new and better framework within the context of DGGS research and application.

Hexagonal discrete global grids can provide an excellent solution for the massive multi-source, multi-temporal, and multi-resolution raster data integration and management. Traditional images are rectangular pixels, and cannot be expressed on a hexagonal grid. Therefore, how to obtain images based on hexagonal pixels has attracted widespread academic attention. Combining current hexagonal sampling methods, this paper studies the evaluation criteria of hexagonal sampling accuracy, summarizes the previous research, and proposes a more general hexagonal sampling method for remote sensing images. This method mainly involves signal preprocessing, spectrum analysis, calculation of sampling interval, and establishment of accuracy evaluation standards. Finally, we verify the feasibility of the proposed hexagon algorithm to provide a reference for hexagon sampling.

Discrete global grid systems (DGGSs) are an emerging multiresolution 3D model used to integrate and analyze big earth data. The characteristic of multiresolution is usually realized by hierarchically subdividing cells on the sphere using certain refinement. This paper introduces mixed aperture three- and four- icosahedral hexagonal DGGSs using two types of refinement, the various combinations of which can provide more resolutions compared with pure aperture hexagonal DGGSs and can flexibly design the aperture sequence according to the target resolutions. A general hierarchy-based indexing method is first designed, and related indexing arithmetics and algorithm are developed based on the indexing method. Then, the grid structure on the surface of the icosahedron is described and by projection spherical grids are obtained. Experiments show that the proposed scheme is superior to pure aperture schemes in choosing grid resolutions and can reduce the data volume by 38.5% in representing 1-km resolution raster dataset; using the proposed indexing arithmetics to replace spherical geometry operations in generating discrete spherical vector lines based on hexagonal cells can improve the generation efficiency.

Due to the superior anti-interception performance and inherent security features, the wide applications of frequency-hopping (FH) signals bring a great challenge to the reconnaissance and monitoring of FH emitters. This paper addresses the problem of positioning measurement estimation for unknown FH signals in passive localization, considering the range migration (RM) and Doppler frequency migration (DFM) of the maneuvering target within the observation time. A coherent range difference (RD), range rate difference (RRD) and acceleration difference (AD) estimation algorithm based on scaled Fourier transform and scaled non-uniform fast Fourier transform is proposed. This method can effectively remove RM and random DFM effects regardless of varied carrier frequency and achieve the coherent estimation of RD, RRD and AD. The whole estimation process can be easily implemented by complex multiplications combined with fast Fourier transform (FFT) and inverse FFT operations without any brute-force searching procedure. Numerical experiments demonstrate that the anti-noise performance of the proposed method is superior to several representative methods and comparable to the optimal maximum likelihood estimator with a much lower computational cost.

The polyhedral discrete global grid system (DGGS) is a multi-resolution discrete earth reference model supporting the fusion and processing of multi-source geospatial information. The orientation of the polyhedron relative to the earth is one of its key design choices, used when constructing the grid system, as the efficiency of indexing will decrease if local areas of interest extend over multiple faces of the spherical polyhedron. To date, most research has focused on global-scale applications while almost no rigorous mathematical models have been established for determining orientation parameters. In this paper, we propose a method for determining the optimal polyhedral orientation of DGGSs for areas of interest on a regional scale. The proposed method avoids splitting local or regional target areas across multiple polyhedral faces. At the same time, it effectively handles geospatial data at a global scale because of the inherent characteristics of DGGSs. Results show that the orientation determined by this method successfully guarantees that target areas are located at the center of a single polyhedral face. The orientation process determined by this novel method reduces distortions and is more adaptable to different geographical areas, scales, and base polyhedrons than those employed by existing procedures.

This paper addresses the joint time difference of arrival (TDOA), frequency difference of arrival (FDOA) and differential Doppler rate estimation problem for high-speed maneuvering targets in passive location systems, involving linear range migration (LRM), quadratic range migration (QRM) and linear Doppler frequency migration (LDFM) within observation time. A noise-resistant estimation algorithm based on second-order keystone transform (SKT) and non-uniform fast Fourier transform (NUFFT) is proposed. After QRM correction via SKT, a phase compensation function is constructed to eliminate LRM and estimate the FDOA. Then, NUFFT is used to remove LDFM and realize the joint estimation of TDOA and differential Doppler rate. Comparisons with several relatively new algorithms indicate that the proposed algorithm can obtain a good trade-off between computational cost and estimation performance. Extensive numerical examples, analysis of computational complexity and estimation performance can validate the effectiveness of the proposed method.

Hexagonal discrete global grid systems are the preferred data models supporting multisource geospatial information fusion. Related research has aroused widespread concern in the academic community, and hierarchical indexing algorithms are one of the main research focuses. In this paper, we propose an algorithm for indexing the cell of a ringed spatial area based on a hexagonal lattice quad-tree (HLQT) structure and the indexing characteristics. First, we design a single-resolution indexing algorithm in which indexing starts from the initial quad tree and expands ring by ring using coding operations, and a quad-tree structure is applied to accelerate this process. Second, the hierarchical indexing algorithm is implemented based on single-resolution indexing, and a pyramid hierarchical model is established. Finally, we perform comparison experiments with existing algorithms. The results of the experiments indicate that the single-level indexing efficiency of the proposed algorithm is approximately twice that of the traditional method and that the hierarchical indexing efficiency is approximately 67 times that of the traditional method. These findings verify the feasibility and superiority of the algorithm proposed in this paper.

This paper addresses the estimation problem of range difference and range rate difference for unknown frequency hopping signals in passive emitter localization, considering the range migration (RM) of the high-speed moving target within the observation time. A coherent estimation algorithm based on the scaled Fourier transform is proposed. This method can effectively remove the RM and random phase effects. The whole estimation process can be fast implemented without any searching operation. Numerical experiments demonstrate that the estimation performance of proposed algorithm is superior to existing algorithms, and comparable to the maximum likelihood estimator.

This study addresses the joint time difference of arrival (TDOA) and frequency difference of arrival (FDOA) estimation problem in passive emitter localisation, involving range migration during the observation time. A computational efficient estimation algorithm based on scaled Fourier transformation is proposed. This method can effectively remove the RM and accomplish the parameter estimation. Cross-terms suppression ability of the proposed method is also analysed and its characteristic indicates the applicability in the scenario of multi-targets. The whole estimation step can be fast implemented by complex multiplications, fast Fourier transformation (FFT) and inverse FFT without any searching process. Compared with the ideal maximum likelihood estimator via extensive numerical experiments, the proposed method can achieve comparable estimation performance and have gained more than ten thousand times reduction in the computational cost, which helps practical application.

Grid system is a multi-resolution raster data structure, which is widely applied in organization, processing and analysis of multi-scale geospatial data. Research on hexagon grid system with important geometric attributes has attracted extensive attention in academia. Description and calculation of hierarchical relation is one of the research difficulties. According to the complex radix number theory and the affiliation of grid cells in interval hierarchy, the mathematical model of the planar aperture 4 hexagon grid system is established. Based on these, the equivalent encoding scheme is proposed, the encoding operations are defined and the rules of them are generalized. Meanwhile, the coding index and transformation between code and Cartesian coordinates are designed. The results of contrast experiments show that the proposed encoding scheme has structural symmetry compared with similar schemes, which can significantly improve the efficiency of encoding operation and has practical application potential.

This study proposes an algebraic distributed source localisation algorithm that combines time difference of arrival (TDOA) and angle of arrival (AOA) measurements. The proposed algorithm uses AOAs to remove the unknown parameters in TDOA equations caused by the specially distributed structure. Then, the observation equations are transformed into a set of pseudo-linear equations and apply linear weighted least square to obtain the source position. The application of weighting matrix can lead to an approximate maximum likelihood estimator and produce a substantial improvement in source localisation accuracy. Both theoretical analysis and simulation results indicate the efficiency of the proposed algorithm and its performance can achieve the Cramer–Rao lower bound at a moderate noise level.

Discrete global grid system is a new data model which supports the fusion processing of multi-source geospatial information.Research into hexagon grid systems that have excellent geometric attributes has raised academic concern.Description of hierarchical relation and design of encoding scheme are research difficulties.According to the characteristics of the planar aperture 4 hexagon grid system, this paper designs an encoding scheme named Hexagon Lattice Quad Tree (HLQT).Code operations are defined, rules of them are generalized and based on these, transformation between 2-dimensional coordinates and addressing codes is implemented.Compared with similar schemes, HLQT overcomes the disadvantages caught by encoding schemes which divide the odd and even levels or mix the vertices and centers for encoding.In addition, operation rules of HLQT are simpler and easier for complementation.Contrast experiments show that the add operation efficiency of HLQT is about 6 times that of PYXIS and about 5 times that of HQBS, the efficiency of the transform algorithm from 2-dimensional coordinates to codes is about 5 times that of HQBS, and the efficiency of the transform algorithm from codes to 2-dimensional coordinates is about 3 times that of HQBS.

Discrete Global Grid Systems (DGGSs) are spatial references that use a hierarchical tessellation of cells to partition and address the entire globe. They provide an organizational structure that permits fast integration between multiple sources of large and variable geospatial data sufficient for visualization and analysis. Despite a significant body of research supporting hexagonal DGGSs as the superior choice, the application thereof has been hindered owing in part to the lack of a rational hierarchy with an efficient addressing system. This paper presents an algebraic model of encoding scheme for the Aperture 3 Hexagonal (A3H) DGGS. Firstly, the definition of a grid cell, which is composed of vertices, edges, and a center, is introduced to describe fundamental elements of grids. Secondly, by identifying the grid cell with its center, this paper proves that cell centers at different levels can be represented exactly using a mixed positional number system in the complex plane through the recursive geometric relationship between two successive levels, which reveals that grid cells are essentially special complex radix numbers. Thirdly, it is shown that through the recursive geometric relationship of successive odd or even levels, the mixed positional number system can also be applied to uniquely represent cell centers at different levels under specific constraint conditions, according to which the encoding scheme is designed. Finally, it is shown that by extending the scheme to 20 triangular faces of the regular icosahedron, multi-resolution grids on closed surfaces of the icosahedron are addressed perfectly. Contrast experiments show that the proposed encoding scheme has the advantages of theoretical rigor and high programming efficiency and that the efficiency of cross-face adjacent cell searching is 242.9 times that of a similar scheme. Moreover, the proposed complex radix number representation is an ideal formalized description tool for grid systems. The research ideas introduced herein can be used to create a universal theoretical framework for DGGSs.

Vectors are a key type of geospatial data, and their discretization, which involves solving the problem of generating a discrete line, is particularly important. In this study, we propose a method for constructing a discrete line mathematical model for a triangular grid based on a "weak duality" hexagonal grid, to overcome the drawbacks of existing discrete line generation algorithms for a triangular grid. First, a weak duality relationship between triangular and hexagonal grids is explored. Second, an equivalent triangular grid model is established based on the hexagonal grid, using this weak duality relationship. Third, the two-dimensional discrete line model is solved by transforming it into a one-dimensional optimal wandering path model. Finally, we design and implement the dimensionality reduction generation algorithm for a discrete line in a triangular grid. The results of our comparative experiment indicate that the proposed algorithm has a computation speed that is approximately 10 times that of similar existing algorithms; in addition, it has better fitting effectiveness. Our proposed algorithm has broad applications, and it can be used for real-time grid transformation of vector data, discrete global grid system (DGGS), and other similar applications.

Global discrete grid system will subdivide the earth recursively to form a multi-resolution grid hierarchy with no Overlap and seamless which help build global uniform spatial reference datum and multi-source data processing mode which takes the position as the object and in the aspect of data structure supports the organization, process and analysis of the remote sensing big data. This paper adopts the base transform to realize the mutual transformation of square pixel and hexagonal pixel. This paper designs the corresponding discrete Fourier transform algorithm for any lattice. Finally, the paper show the result of the DFT of the remote sensing image of the hexagonal pixel.