Euclidean Algorithm Research Articles

Euclidean direction search algorithm with maximum correntropy criterion for active noise control system

Euclidean direction search algorithm with maximum correntropy criterion for active noise control system

Factoring perfect reconstruction filter banks into causal lifting matrices: A Diophantine approach

The elementary theory of bivariate linear Diophantine equations over polynomial rings is used to construct causal lifting factorizations (elementary matrix decompositions) for causal two-channel FIR perfect reconstruction transfer matrices and wavelet transforms. The Diophantine approach generates causal factorizations satisfying certain polynomial degree-reducing inequalities, enabling a new factorization strategy called the Causal Complementation Algorithm. This provides a causal (i.e., polynomial, hence realizable) alternative to the noncausal lifting scheme developed by Daubechies and Sweldens using the Extended Euclidean Algorithm for Laurent polynomials. The new approach replaces the Euclidean Algorithm with Gaussian elimination employing a slight generalization of polynomial division that ensures existence and uniqueness of quotients whose remainders satisfy user-specified divisibility constraints. The Causal Complementation Algorithm is shown to be more general than the causal version of the Euclidean Algorithm approach by generating additional causal lifting factorizations beyond those obtainable using the polynomial Euclidean Algorithm.

Crack Detection and Feature Extraction of Heritage Buildings via Point Clouds: A Case Study of Zhonghua Gate Castle in Nanjing

Zhonghua Gate Castle is on the tentative list for Chinese World Cultural Heritage. Due to long-term sunshine, rain erosion, and man-made damage, its surface appears to have different degrees of cracks and other diseases. This paper centers on Zhonghua Gate Castle; terrestrial laser scanning is used to obtain the exterior wall point cloud data. A crack detection method based on point cloud data curved surface reconstruction is proposed. It involves data preprocessing, crack detection, and the analysis of crack features. This method initially uses data preprocessing techniques to improve data quality. These techniques include removing ground points and super-voxel segmentation. Subsequently, local surface reconstruction was employed to address the issue of missing point cloud data within cracks and the Euclidean clustering algorithm was used for precise crack identification. The article provides a detailed analysis of the geometric characteristics of cracks. They involve the calculation of length, width, and area. The results of the experiment demonstrate that the method could successfully identify cracks and extract geometric features and has millimeter-level accuracy compared to actual crack sizes.

A tour d'horizon of de Casteljau's work

Whilst Paul de Casteljau is now famous for his fundamental algorithm of curve and surface approximation, little is known about his other findings. This article offers an insight into his results in geometry, algebra and number theory.Related to geometry, his classical algorithm is reviewed as an index reduction of a polar form. This idea is used to show de Casteljau's algebraic way of smoothing, which long went unnoticed. We will also see an analytic polar form and its use in finding the intersection of two curves. The article summarises unpublished material on metric geometry. It includes theoretical advances, e.g., the 14-point strophoid or a way to link Apollonian circles with confocal conics, and also practical applications such as a recurrence for conjugate mirrors in geometric optics. A view on regular polygons leads to an approximation of their diagonals by golden matrices, a generalisation of the golden ratio.Relevant algebraic findings include matrix quaternions (and anti-quaternions) and their link with Lorentz' equations. De Casteljau generalised the Euclidean algorithm and developed an automated method for approximating the roots of a class of polynomial equations. His contributions to number theory not only include aspects on the sum of four squares as in quaternions, but also a view on a particular sum of three cubes. After a review of a complete quadrilateral in a heptagon and its angles, the paper concludes with a summary of de Casteljau's key achievements.The article contains a comprehensive bibliography of de Casteljau's works, including previously unpublished material.

Mobile LiDAR-based Real-time Identification of Transmission Lines

Abstract. This paper proposes a method for identifying 3D point cloud of transmission line acquired by light detection and ranging (LiDAR) real-time mobile scanning. Since the single frame of point cloud obtained by LiDAR is sparse, the method employs a sliding spatial window strategy with Kalman filtering for dynamic point cloud registration. Then, a 3D point cloud deep learning neural network that utilizes uniform sampling and local feature aggregation (LFA) is designed specifically for transmission line objects. The network handles the problem of long-span objects and a large amount of point cloud. Finally, the instantiated transmission line objects are extracted from the top-down projection of the semantically segmented 3D point cloud by fast Euclidean clustering algorithm. Experiments demonstrate that the method achieves a classification accuracy of 94.7% and a mean intersection over union of 81.6% on 3D point cloud datasets of transmission line obtained from LiDAR mobile scanning, validating its ability to achieve real-time identification and distance measurement of transmission line objects.

Sums of Two Squares Visualized

In 1972, John Brillhart described an algorithm for expressing a prime p ≡ 1 ( mod 4 ) as the sum of two squares. Brillhart’s algorithm, which is based on the Euclidean algorithm, is simplicity itself. However, Brillhart’s proof of his algorithm’s correctness uses several previous results, and subsequent simplifications of his argument still retain something of an air of mystery. We provide a geometric interpretation of Brillhart’s algorithm, which not only proves that it works, but also sheds light on a surprising palindromic property proved by Perron, and used by Brillhart in his proof.

Formalizing Factorization on Euclidean Domains and Abstract Euclidean Algorithms

Formalizing Factorization on Euclidean Domains and Abstract Euclidean Algorithms

Scheduling optimization of flexible flow shop with buffer capacity limitation based on an improved discrete particle swarm optimization algorithm

To address the challenge of production scheduling in flexible flow shops with buffer capacity limitations, a mathematical model is established to minimize the maximum completion time and maximize equipment utilization. To efficiently solve the problem, an improved discrete particle swarm optimization algorithm with multi-population reduction (MPREDPSO) is developed. It integrates an elite retention strategy, which significantly contributes to its efficiency and robustness. The algorithm stands out for its ability to effectively navigate through potential local optima, a common pitfall in optimization problems, through a Euclid algorithm coupled with a sophisticated gene backtracking mechanism. The performance of MPREDPSO is rigorously tested and proven through comparative analyses. Its effectiveness is proven through comparisons with other algorithms, namely the discrete particle swarm algorithm, genetic algorithm, whale algorithm, grey wolf optimizer and whale swarm algorithm. In addition, MPREDPSO’s practical utility is demonstrated in a high-temperature ham sausage production workshop, showcasing its industrial application capabilities.

Statistical Analysis of Ionospheric Correlation for Shortwave System

AbstractStatistical analysis of the temporal and spatial variations in the ionosphere is necessary to improve the shortwave system. Based on the standardized Euclidean distance algorithm, multisource ionospheric assimilation data, International GNSS Service vertical total electron content data, and ionosonde data are used to statistically analyze the ionospheric correlation distance, and the variation of ionospheric correlation distances with local times, magnetic latitudes, and seasons are obtained. The statistical analysis results show that the zonal ionospheric correlation distance presents minima in the equatorial ionospheric anomaly crest regions. Additionally, the meridional correlation distance in middle magnetic latitudes is greater than that in other regions. The zonal ionospheric correlation distance presents obvious local‐time variability. The variation trends of the meridional and zonal correlation distance during spring and autumn are similar. The patterns of the ionospheric correlation range variation with local times, magnetic latitudes, and seasons can be used to select the optimal locations or spacing for ionosonde stations, shortwave multi‐station systems, and shortwave radio beacons.

ANALISIS CLUSTERING STUNTING DENGAN DISTANCE EUCLID

Entering the Industrial Revolution Era 4.0, human resources must be supported by healthy and intelligent human resources so that they can increase competitiveness. The world still faces the problem of hunger and malnutrition today. According to a Unicef report, many people suffer from malnutrition in the world. The World Health Organization (WHO) says that malnutrition is a dangerous threat to the health of the world's population. Stunting also has an impact in Indonesia, the prevalence of toddlers experiencing stunting in Indonesia is 24.4% in 2021. The solution created is to classify and cluster stunting so as to produce patterns that can be used as best practice to be transmitted to other affected areas. The algorithm used is Euclid, the Euclid algorithm is able to cluster stunting prevalence data into 3 clusters with a little category of 66%, a medium category of 28%, a lot of category of 6%. The results of the classification and clustering of the best stunting prevalence in cluster two with a small number, can be used as a source of accurate and updated information that can be used by the government in its efforts to optimize stunting handling in each district/city based on artificial intelligence which can provide patterns for handling and optimizing stunting. in each district/city. Malnutrition is estimated to be the main cause of 3.1 million child deaths every year. Therefore, efforts need to be made to minimize stunting by predicting stunting sufferers. The prediction results can be used as an early prevention effort.

Bridging theory and application: An in-depth study of the greatest common divisor and least common multiple in mathematics and real-world settings

The study of the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) explores the fundamental field of number theory. In addition to being essential to theoretical number theory, these mathematical ideas have extensive applications in computer science, algebra, and real-world settings like network routing and scheduling. Using techniques like the Euclidean algorithm for GCD calculation and prime factorization for LCM, the study starts by examining the fundamental concepts and notations of GCD and LCM. Then the author goes further into the detail about how these ideas are used to polynomial equations, highlighting how they may be used to solve challenging issues and simplify algebraic expressions. The studys practical applications, which demonstrate how GCD and LCM optimize scheduling tasks and boost network routing efficiency, are presented in the conclusion. The results of this study highlight the broad impact and adaptability of LCM and GCD in theoretical and applied mathematics, providing valuable insights for further research in these domains.

Obstacle detection and obstacle-surmounting planning for a wheel-legged robot based on Lidar

PurposeThis paper aims to investigate an autonomous obstacle-surmounting method based on a hybrid gait for the problem of crossing low-height obstacles autonomously by a six wheel-legged robot. The autonomy of obstacle-surmounting is reflected in obstacle recognition based on multi-frame point cloud fusion.Design/methodology/approachIn this paper, first, for the problem that the lidar on the robot cannot scan the point cloud of low-height obstacles, the lidar is driven to rotate by a 2D turntable to obtain the point cloud of low-height obstacles under the robot. Tightly-coupled Lidar Inertial Odometry via Smoothing and Mapping algorithm, fast ground segmentation algorithm and Euclidean clustering algorithm are used to recognize the point cloud of low-height obstacles and obtain low-height obstacle in-formation. Then, combined with the structural characteristics of the robot, the obstacle-surmounting action planning is carried out for two types of obstacle scenes. A segmented approach is used for action planning. Gait units are designed to describe each segment of the action. A gait matrix is used to describe the overall action. The paper also analyzes the stability and surmounting capability of the robot’s key pose and determines the robot’s surmounting capability and the value scheme of the surmounting control variables.FindingsThe experimental verification is carried out on the robot laboratory platform (BIT-6NAZA). The obstacle recognition method can accurately detect low-height obstacles. The robot can maintain a smooth posture to cross low-height obstacles, which verifies the feasibility of the adaptive obstacle-surmounting method.Originality/valueThe study can provide the theory and engineering foundation for the environmental perception of the unmanned platform. It provides environmental information to support follow-up work, for example, on the planning of obstacles and obstacles.

Distributed Constrained Optimization With Delayed Subgradient Information Over Time-Varying Network Under Adaptive Quantization.

In this article, we consider a distributed constrained optimization problem with delayed subgradient information over the time-varying communication network, where each agent can only communicate with its neighbors and the communication channel has a limited data rate. We propose an adaptive quantization method to address this problem. A mirror descent algorithm with delayed subgradient information is established based on the theory of Bregman divergence. With a non-Euclidean Bregman projection-based scheme, the proposed method essentially generalizes many previous classical Euclidean projection-based distributed algorithms. Through the proposed adaptive quantization method, the optimal value without any quantization error can be obtained. Furthermore, comprehensive analysis on the convergence of the algorithm is carried out and our results show that the optimal convergence rate can be obtained under appropriate conditions. Finally, numerical examples are presented to demonstrate the effectiveness of our results.

The Solution of the Indefinite Equation by the Method of Euclidean Algorithm

The Solution of the Indefinite Equation by the Method of Euclidean Algorithm

On Restricted Averages of Dedekind Sums

Abstract We investigate the averages of Dedekind sums over rational numbers in the set $\mathscr {F}_{\alpha }(Q) = \{\,{v}/{w}\in \mathbb {Q}: 0<w\leq Q\,\}\cap \lbrack 0, \alpha )$ for fixed $\alpha \leq 1/2$. In previous work, we obtained asymptotics for $\alpha =1/2$, confirming a conjecture of Ito in a quantitative form. In the present article we extend our former results, first to all fixed rational $\alpha $ and then to almost all irrational $\alpha $. As an intermediate step we obtain a result quantifying the bias occurring in the second term of the asymptotic for the average running time of the by-excess Euclidean algorithm, which is of independent interest.

Design recommendations of target size and tracking speed under circular and square trajectories for smooth pursuit with Euclidean algorithm in eye-control system

In this study, we investigated how the size, speed, and trajectory shape of smooth-pursuit targets for Euclid's algorithm affect the ergonomics and usability of the gaze interface, and conducted a user study with 20 participants, examining the priority of five target sizes in circular and square trajectories, and the recommended speed for each size in the five target speeds. The usability of the SUS and ISO 9241 (acceptability and comfort) of the eye control system had been evaluated. Based on the experiment results, the design recommendations of the priority of the target size and speed of the circular and square trajectories, suggested trajectory shapes. The research is an important reference for the design and development of smooth-pursuit targets based on the Euclidean algorithm.

A Real-Time Vessel Detection and Tracking System Based on LiDAR.

Vessel detection and tracking is of utmost importance to river traffic. Efficient detection and tracking technology offer an effective solution to address challenges related to river traffic safety and congestion. Traditional image-based object detection and tracking algorithms encounter issues such as target ID switching, difficulties in feature extraction, reduced robustness due to occlusion, target overlap, and changes in brightness and contrast. To detect and track vessels more accurately, a vessel detection and tracking algorithm based on the LiDAR point cloud was proposed. For vessel detection, statistical filtering algorithms were integrated into the Euclidean clustering algorithm to mitigate the effect of ripples on vessel detection. Our detection accuracy of vessels improved by 3.3% to 8.3% compared to three conventional algorithms. For vessel tracking, L-shape fitting of detected vessels can improve the efficiency of tracking, and a simple and efficient tracking algorithm is presented. By comparing three traditional tracking algorithms, an improvement in multiple object tracking accuracy (MOTA) and a reduction in ID switch times and number of missed detections were achieved. The results demonstrate that LiDAR point cloud-based vessel detection can significantly enhance the accuracy of vessel detection and tracking.

A method for obtaining maize phenotypic parameters based on improved QuickShift algorithm

Phenotype is an important bridge for studying the mechanism of “genotype-phenotype-environment”, and studying precise measurement of crop individual phenotypic information is of great significance for accelerating breeding processes and assisting in precision agriculture monitoring. The purpose of this study is to develop a clustering algorithm for corn population point clouds to accurately extract the three-dimensional morphology of individual crops. Firstly, this study used drones to obtain field corn data, reconstructed corn plant models based on the SfM algorithm, simplified and rotated point clouds, and extracted crop point clouds using methods based on color and distance thresholds. For density segmentation, this study improved the QuickShift method to segment individual corn crops. Color weights were added to the clustering density calculation process, and the original point cloud was projected onto a horizontal plane to calculate the density clustering core. Then, the remaining points were subjected to QuickShift clustering segmentation in 3D to obtain the point cloud data of individual corn plants. The use of cylindrical fitting and conditional Euclidean clustering algorithms to automatically segment point clouds of stems and leaves has achieved estimation of plant height, stem thickness, minimum bounding box volume per plant, and number of leaves. Finally, the true and measured values of plant height and stem thickness were compared. The results showed that the accuracy of the improved QuickShift method for segmentation was 93.10%, with an average measured plant height of 14.21 cm and an average stem diameter of 0.72 cm. The determination coefficient and root mean square error of the automatic and manual measurement results of plant height are 0.9186 and 0.5785 cm, respectively. The determination coefficients and root mean square errors of the automatic and manual measurement results of stem thickness are 0.8227 and 0.0574 cm, respectively. This study uses an improved QuickShift to cluster and segment corn point cloud data, achieving the segmentation of individual corn plants in the field, providing an automated, efficient, and low-cost solution for accurately measuring crop phenotypic information.

Research on the optimum path Taking the hospital delivery robot as an example

Medical delivery robot refers to the delivery robot used in the medical field. Compared with ordinary delivery robots, the medical delivery robot needs to work in an environment with many people, which means that it needs to deal with many random obstacles at any time. This article will discuss automatically avoiding obstacles, and formulate and compare algorithms to analyze the advantages and disadvantages of automatic road exploration algorithms of different algorithms. In order to accomplish this goal, this paper will use Matlab as the main development tool and use the A-star algorithm and the Euclid algorithm as the main heuristics in the main mathematical model of the program. This program needs to be able to complete the obstacle avoidance task in a map with random size, random position, random shape, and random number of obstacles, and be able to reach the end point from the starting point. In addition, this paper will also discuss the efficiency of the A-star algorithm and the Dijkstra algorithm in obstacle avoidance and route planning, and demonstrate why the A-star algorithm is more efficient by taking time and congestion indicators.

Node authentication and encrypted data transmission in mobile ad hoc network using the swarm intelligence‐based secure ad‐hoc on‐demand distance vector algorithm

AbstractIn a mobile ad hoc network (MANET), all nodes are communicated with one another across wireless networks to create a temporary network without the support of centralised management. Due to dynamic topology in MANET, secure routing is a crucial issue. The existing secure routing protocol and their security concerns are analysed in this work. The suggested Swarm Intelligence‐based Secure Ad‐hoc On‐demand Distance Vector (SIS‐AODV) algorithm offers security by applying a secret key and hash mechanism to prevent the involvement of malicious nodes in routing operations. A secure routing system of MANET guards against internal and external network attacks. The proposed SIS‐AODV algorithm consists of two sections: the secret key value generated by applying Elliptical Curve Cryptography (ECC)‐based algorithm and the PRESENT algorithm to encrypt the data packets. Besides, authentication and non‐repudiation are applied using the H‐PRESENT 128 algorithm. The PRESENT algorithm and H‐PRESENT 128 hash function require less computational power. Centralised management is optional in this scheme, so overhead decreases. The second section of SIS‐AODV consists of Ant Colony Grey Wolf Optimization over the AODV algorithm to improve network performance while implementing a security algorithm over MANET. Analysis results show maximum performance with a packet delivery ratio of 98% and throughput of 85%. In addition, end‐to‐end delay is reduced by up to 25%, and routing overhead decreases by up to 20%. Keywords: AODV, Elliptical Curve, PRESENT, H‐PRESENT, Euclidean Algorithm, ACO, GWO, Blackhole attack.