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Incremental algorithms are among the most popular approaches for Delaunay triangulation, and the point insertion sequence has a substantial impact on the amount of work needed to construct Delaunay triangulations in incremental algorithm triangulation. In this paper, 2D adaptive Hilbert curve insertion, including the method of dividing 3D multi-grids and adjusting the 3D adaptive Hilbert curve to avoid the “jump” phenomenon, is extended to 3D Delaunay triangulation. In addition, on the basis of adaptive Hilbert curve insertion, we continue to optimize the addition of control points by selecting control points in every order and every grid level. As a result, the number of conflicting elongated tetrahedra that have to be created and deleted multiple times and the number of search steps for positioning inserted points can both be reduced. Lastly, a new comparison method is used in the point location process to solve the precision problem in 3D Delaunay triangulation. As shown by detailed experiments and analysis, compared with previous adaptive Hilbert curve insertion, CGAL, regular grid insertion, multi-grid insertion and random insertion, the proposed 3D Delaunay triangulation is the most efficient for both artificial and real surface sampling point sets.

Based on the n-ary counting system, combined with the matrix semi-tensor product theory and Hilbert curve, a chaotic image encryption algorithm is designed. Different from the traditional encryption method, the algorithm proposed in this paper is an encryption algorithm with scrambling and diffusion at the same time. First, the pixel value is converted from decimal to n-ary. In the n-ary counting system, the plaintext image is randomly divided into some groups, and the Hilbert curve is used for scrambling to each group. The blocks are converted into scrambled images, so that the scrambling and diffusion can be carried out at the same time. Then, in order to improve the security of the algorithm, another round of diffusion is carried out based on matrix semi-tensor product mechanism. Chaotic sequence is generated by Chen system. This chaotic sequence performs matrix semi-tensor product operation with the first round of encrypted image, and generate second encrypted images. Finally, this encryption method is applied to color image encryption. Compared with some representative algorithms, the experimental results show that the algorithm proposed in this paper is secure and it can resist common attacks.

Fractal geometries consistently provide solutions to several electromagnetic design problems. In this paper, fractal geometries such as Hilbert and Moore curves are used to design efficient High-Impedance Surfaces. Modern communication devices have many sensors that are needed to communicate wirelessly. The critical component of wireless communications is antennas. Planar microstrip patch antennas are popular due to their low profile, compactness, and good radiation characteristics. The structural disadvantages of microstrip antennas are that they have surface waves that propagate over the ground plane. High-Impedance Surface (HIS) planes are a prominent solution to minimize and eliminate surface waves. The HIS structures behave as active LC filters that suppress surface waves at their resonance frequency. The resonance frequency of the structure is obtained by its LC equivalent or by analyzing the reflection phase characteristics. This work presents conventional HIS structures similar to mushroom HIS and fractal HIS such as Hilbert curve and Moore curve HIS. The HIS reflection phase characteristics are obtained by applying periodic boundary conditions with plane wave illumination. The results were obtained in terms of the reflection phase angle. The conventional mushroom structures show narrow band characteristics at given dimensions of 10 mm × 10 mm and 20 mm × 20 mm. These structures are helpful in the replacement of PEC ground planes for patch antennas under sub-6 GHz. The Hilbert and Moore fractals are also designed and have a multiband response that can be useful for L, S, and C band applications. Another design challenge of HIS is protrusions, which make design difficult. The work also presents the effect of having vias and the absence of vias on reflection phase characteristics. The response shows the least and no significant effect of vias under the x-band operation.

A compact size of filter realization can be achieved by decreasing the number of resonators, using via ground holes technique and using substrate that has higher dielectric constant [4] or using a rectangular resonator sandwiched between two interdigital structures, [5]. [...]it can also be improved by defected ground structure. [...]fractal geometries offer smaller resonator size. From Figure 4 it is seen that if the length of the e resonator is shortened from 13.9 mm to 9 mm then it will form a harmonic frequency resulting in a less selective filter characteristics.It can be concluded that the parameter d and e will influence the harmonic or spurius frequency while the parameter of c will control the cut-off frequency, insertion loss and return loss either. 3.Sierpinski Carpet Lowpass Filter Design Figure 5 shows various DGS of the Sierpinski Carpet designs. [...]Edition.

In this paper the iterative rigorous approach is applied flexibly to complex Hilbert filters using structured computational grids. The wave concept iterative process (WCIP) is reviewed, focusing on their fundamental conception and simple algorithm, with the presence of defected ground structure (DGS) appearing as Hilbert curve ring (HCR) cells. The selected studied filters have shown increasing levels of sophistication and complexity that have properly contributed in evaluating the WCIP performance. To attain this purpose, the simulated computational results are demonstrated by measurements, and the iterative results, accurate enough, are obtained to some extent for the more sophisticated designs.

A discrete space-filling curve provides a one-dimensional indexing or traversal of a multi-dimensional grid space. Sample applications of space-filling curves include multi-dimensional indexing methods, data structures and algorithms, parallel computing, and image compression. Common measures for the applicability of space-filling curve families are locality and clustering. Locality preservation reflects proximity between grid points, that is, close-by grid points are mapped to close-by indices or vice versa. We present analytical and empirical studies on the locality properties of the two-dimensional Hilbert curve family. The underlying locality measure, based on the p -normed metric d p , is the maximum ratio of d p ( v , u ) m to d p ( v ~ , u ~ ) over all corresponding point-pairs ( v ,  u ) and ( v ~ , u ~ ) in the m -dimensional grid space and one-dimensional index space, respectively. Our analytical results close the gaps between the current best lower and upper bounds with exact formulas for p ∈ { 1 , 2 } , and extend to all reals p ≥ 2 . We also verify the results with computer programs over various grid-orders and p -values. Our empirical results will shed some light on determining the exact formulas for the locality measure for all reals p ∈ ( 1 , 2 ) .

The Hilbert curve is an important method for mapping high-dimensional spatial information into one-dimensional spatial information while preserving the locality in the high-dimensional space. Entry points of a Hilbert curve can be used for image compression, dimensionality reduction, corrupted image detection and many other applications. As far as we know, there is no specific algorithms developed for entry points. To address this issue, in this paper we present an efficient entry point encoding algorithm (EP-HE) and a corresponding decoding algorithm (EP-HD). These two algorithms are efficient by exploiting the m consecutive 0s in the rear part of an entry point. We further found that the outputs of these two algorithms are a certain multiple of a certain bit of s , where s is the starting state of these m levels. Therefore, the results of these m levels can be directly calculated without iteratively encoding and decoding. The experimental results show that these two algorithms outperform their counterparts in terms of processing entry points.

Computer vision techniques have advanced greatly in recent years through deep learning, achieving unprecedented performance. This has motivated applying deep learning to malware detection through image-based approaches to circumvent extensive feature engineering for diverse threats. However, existing work converting Android binaries to rectangular images neglects the intrinsic byte sequence structure, introducing spurious spatial relationships that weaken detection accuracy. To address this, space-filling curves have mapped binaries to images while preserving ordering. This paper proposes a novel method using Hilbert space-filling curves to visualize and classify Android apps. Bytecode is extracted from Dalvik Executable (DEX) files and transformed to grayscale images via Hilbert coding for model training. Additionally, a novel and balanced image dataset is proposed consisting of Hilbert transformations for 4995 benign and 4995 malicious Android apps randomly sampled from the AndroZoo repository. Experiments using this dataset evaluated pre-trained InceptionV3, VGG16, ResNet50 and EfficientNetB0 via transfer learning. A custom Convolutional Neural Network (CNN) was also trained from scratch. InceptionV3 achieved the highest performance at 97.99% accuracy, 98.50% precision, 97.50% recall and 97.99% F1-score. Comparative assessment with previous image-based malware detection research indicates our approach outperforms state-of-the-art approaches. By leveraging Hilbert space-filling curves to map binaries to images while preserving sequential relationships, detection accuracy is improved over methods introducing extraneous spatial representations.

Sustainable development denotes the enhancement of living standards in the present without compromising future generations’ resources. Sustainable Development Goals (SDGs) quantify the accomplishment of sustainable development and pave the way for a world worth living in for future generations. Scholars can contribute to the achievement of the SDGs by guiding the actions of practitioners based on the analysis of SDG data, as intended by this work. We propose a framework of algorithms based on dimensionality reduction methods with the use of Hilbert Space Filling Curves (HSFCs) in order to semantically cluster new uncategorised SDG data and novel indicators, and efficiently place them in the environment of a distributed knowledge graph store. First, a framework of algorithms for insertion of new indicators and projection on the HSFC curve based on their transformer-based similarity assessment, for retrieval of indicators and load-balancing along with an approach for data classification of entrant-indicators is described. Then, a thorough case study in a distributed knowledge graph environment experimentally evaluates our framework. The results are presented and discussed in light of theory along with the actual impact that can have for practitioners analysing SDG data, including intergovernmental organizations, government agencies and social welfare organizations. Our approach empowers SDG knowledge graphs for causal analysis, inference, and manifold interpretations of the societal implications of SDG-related actions, as data are accessed in reduced retrieval times. It facilitates quicker measurement of influence of users and communities on specific goals and serves for faster distributed knowledge matching, as semantic cohesion of data is preserved.

A growing number of generative statistical models do not permit the numerical evaluation of their likelihood functions. Approximate Bayesian computation has become a popular approach to overcome this issue, in which one simulates synthetic data sets given parameters and compares summaries of these data sets with the corresponding observed values. We propose to avoid the use of summaries and the ensuing loss of information by instead using the Wasserstein distance between the empirical distributions of the observed and synthetic data. This generalizes the well-known approach of using order statistics within approximate Bayesian computation to arbitrary dimensions. We describe how recently developed approximations of the Wasserstein distance allow the method to scale to realistic data sizes, and we propose a new distance based on the Hilbert space filling curve. We provide a theoretical study of the method proposed, describing consistency as the threshold goes to 0 while the observations are kept fixed, and concentration properties as the number of observations grows. Various extensions to time series data are discussed. The approach is illustrated on various examples, including univariate and multivariate g-and-k distributions, a toggle switch model from systems biology, a queuing model and a Lévy-driven stochastic volatility model.