Curve Interpolation Research Articles

Enhancing Vehicle Trajectory Quality: A Two‐Step Data Reconstruction Method Using Wavelet Transform and Normal Acceleration Value

ABSTRACTData reconstruction is essential in enhancing the quality of vehicle trajectory data. Previous studies have identified the location of abnormal data inaccurately, resulting in poor trajectory reconstruction results. This study proposed a two‐step reconstruction method. The first step detected the locations of obviously abnormal speed data using wavelet transform. Then, the abnormal data were repaired by the cubic spline curve interpolation algorithm. The second stage identified the locations of abnormal acceleration data based on the general acceleration value. And the vehicle trajectory data were reconstructed using Lagrange interpolation and Kalman filter algorithms. The approach was utilized on NGSIM trajectory data. The results show that the acceleration values of the proposed method range from −6.69 m/s2 to 4.96 m/s2, with a standard deviation of 0.87. The reconstructed results are more closely matching drivers' physiological capabilities compared to other methods. These findings verify the reliability of the proposed approach and notably improve the quality of the trajectory data. It provides critical foundational data support for traffic planning, design, and management.

Holomorphic Legendrian Curves in Convex Domains

We prove several results on approximation and interpolation of holomorphic Legendrian curves in convex domains in C2n+1, n≥2, with the standard contact structure. Namely, we show that such a curve, defined on a compact bordered Riemann surface M, whose image lies in the interior of a convex domain D⊂C2n+1, may be approximated uniformly on compacts in the interior IntM by holomorphic Legendrian curves IntM→D such that the approximants are proper, complete, agree with the starting curve on a given finite set in IntM to a given finite order, and hit a specified diverging discrete set in the convex domain. We first show approximation of this kind on bounded strongly convex domains and then generalise it to arbitrary convex domains. As a consequence we show that any compact bordered Riemann surface properly embeds into a convex domain as a complete curve under a suitable geometric condition on the boundary of the codomain.

Point-normal convexity preserving curve interpolation

Point-normal convexity preserving curve interpolation

Intelligent Perception and Seam Tracking System for Thick Plate Weldments Based on Constant-Focus Optical Path

To achieve efficient and accurate thick plate welding, as well as to precisely extract and plan the paths of complex three-dimensional weld seams in large steel structures, this study introduces a novel vision-guided approach for robotic welding systems utilizing a constant-focus laser sensor. This methodology specifically targets and mitigates several critical shortcomings inherent in conventional vision-guided welding techniques, including limited detection ranges, diminished precision in both detection and tracking, and suboptimal real-time performance. For preprocessed weld images, an improved grayscale extreme centroid method was developed to extract the center of the light stripe. Furthermore, a sophisticated feature point extraction algorithm, which integrates a maximum distance search strategy with a least-squares fitting procedure, was developed to facilitate the precise and timely identification of weld seam characteristic points. To further optimize the outcomes, a cylindrical filtering mechanism was employed to eliminate substantial discrepancies, whereas local Non-Uniform Rational B-Spline (NURBS) curve interpolation was utilized for the generation of smooth and accurate trajectory plans. A spatial vector-based pose adjustment strategy was then implemented to provide robust guidance for the welding robot, ensuring the successful execution of the welding operations. The experimental results indicated that the proposed algorithm achieved a tracking error of 0.3197 mm for welding workpieces with a thickness of 60 mm, demonstrating the method’s substantial potential in the manufacturing sector, especially in the domain of automated welding.

The weighted GS-PIA algorithm for cubic B-spline curve interpolations and convergence analysis

The weighted GS-PIA algorithm for cubic B-spline curve interpolations and convergence analysis

Моделирование и визуализация поверхностей сложной формы с помощью интерполяционных кривых

This article presents an approach to modeling and visualizing surfaces of complex shapes using interpolation curves with predetermined geometric properties. A modified Bezier curve of n-order was used as interpolation curves. Modification of a Bezier arc into an interpolation curve is possible both with and without preserving tangents. When preserving tangents, the Bezier arc retains its properties as a contour arc and acquires the ability to pass through preset points. The considered modification is possible in several variations: universal, based on the uniform distribution of the parameter during the modification process, and adaptive, when the parameter values are adapted to the initial data. The use of interpolation curves makes it possible to implement a special case of the moving simplex method, an analogue of which in geometric modeling and computer-aided design systems is the section operation (or lofting). The difference is that a continuous curve is used as a generating surface instead of a piecewise one. To ensure the functionality of such a connection, we give examples of models of the surface of an onion dome and a vase using various guides. An analysis of the obtained results was carried out. The introduction of research results into CAD/CAM will significantly expand their tools in terms of shape formation and visualization of surfaces and bodies that have predetermined geometric requirements.

A Hybrid Parametrization Method for B‐Spline Curve Interpolation via Supervised Learning

AbstractB‐spline curve interpolation is a fundamental algorithm in computer‐aided geometric design. Determining suitable parameters based on data points distribution has always been an important issue for high‐quality interpolation curves generation. Various parameterization methods have been proposed. However, there is no universally satisfactory method that is applicable to data points with diverse distributions. In this work, a hybrid parametrization method is proposed to overcome the problem. For a given set of data points, a classifier via supervised learning identifies an optimal local parameterization method based on the local geometric distribution of four adjacent data points, and the optimal local parameters are computed using the selected optimal local parameterization method for the four adjacent data points. Then a merging method is employed to calculate global parameters which align closely with the local parameters. Experiments demonstrate that the proposed hybrid parameterization method well adapts the different distributions of data points statistically. The proposed method has a flexible and scalable framework, which can includes current and potential new parameterization methods as its components.

Jerk limited continuous tool path generation for flexible systems in non-cartesian coordinate systems

Inspired by computer numerical control (CNC) technology and drastic changes in the world of electronics and microcontrollers, the idea of using non-cartesian coordinate system for CNC machines became a topic of interest. Non-cartesian coordinate systems provide freedom of movement in systems with large working spaces where high dimensional accuracy is not a prime concern. Applications of such systems can be found in murals, where a painting is applied to a large wall of any arbitrary build material. In the present paper, a new design was introduced to automate the process of drawing murals on large surfaces. The proposed mechanism utilized chain and sprocket to overcome the size limitations of traditionally available x-y tables. In addition to that, the developed mechanism does not utilize the common robust structures used in the conventional CNC systems and the tool motion is created using flexible arms that are not heavily constrained. The effect of systems unwanted vibration has been eliminated using continuous trajectory and kinematic profiles. Parametric curve interpolation algorithms for trajectory planning were implemented to increase the flexibility of drawing complex curves. Parametric curves such as B-spline and non-uniform rational B-spline (NURBS) were employed to generate a jerk limited trajectory for motion control and extraction of kinematic profiles. Such trajectory constrains the jerk of the system and guarantees a smooth motion free of feed-rate fluctuations. Compared to other methods available for extracting the kinematic profiles, jerk limited method decreases the travel time and trajectory error. Ultimately, motion commands were produced by using kinematic profiles and the second order Taylor interpolator. STM32746ZG card was used as the main processor and two stepper motors controlled by a Bit Pattern interpolation algorithm were utilized to drive the proposed mechanism. The input pulses of stepper motors were stored as binary bits and transmitted to drivers in real-time. The feedback was also evaluated by two rotary encoders, placed at the end of the motors’ shafts. Encoder data were acquired and transmitted using a TCP/IP protocol to guarantee zero data loss during the transmission.

Generation of Picture Fuzzy B-Spline Curve Interpolation with Open Uniform Knot Vector

Generation of Picture Fuzzy B-Spline Curve Interpolation with Open Uniform Knot Vector

An efficient and automatic method based on monocular camera and GNSS for collecting and updating geographical coordinates of mileage pile in highway digital twin map

Abstract The high-precision positioning of mileage piles on a digital map ensures accurate data for high-speed event releases, toll audits, and road condition monitoring. This paper introduces an efficient and automatic system for positioning highway mileage piles. The self-developed data acquisition system collects mileage pile images and road trajectory. Considering the limited variety of mileage piles, a Simplified-YOLOv5m (S-YOLOv5m) is proposed. Then a high-precision character detection network of S-YOLOv5m is proposed to enhance feature extraction and improve the accuracy of character detection on mileage piles. Thirdly, the end-to-end monocular distance measurement combines the target detection with the distance estimation, enabling simultaneous object detection and distance measurement. Fourthly, by combining with the geographical coordinates of the acquisition point, the direct solution to geodetic problems is applied to calculate the spatial coordinates of the mileage piles. Finally, the missing mileage piles are compensated for using the road curve and equal distance interpolation. The complete mileage piles and geographical information list of the inspection trajectory are output. Through a series of verification tests, the average positioning error of the mileage pile system is 1.265 m. The contribution of the automatic positioning system of mileage piles is to construct the relationship between mileage piles and geographical coordinates, match mileage piles with digital maps to realize the full-featured, real-scene, and high-dynamic management of road attributes.

The Magnetic Anomaly Map of Paraná State: An Integration of Airborne Surveys Performed Over the Years

Over the last decades, several aerial survey campaigns have been conducted in the State of Paraná. However, we did not have a complete magnetic data integration and analysis focused on the Paraná’s geological context. Therefore, the main objective of this work is to provide the integration of all available data and realize a general analysis to make the Integrated Magnetic Map of the Paraná State, in collaboration with the Geological Survey of Brazil (SGB-CPRM). After integrate the data we have applied processing techniques on public airborne magnetic data, available from the Geological Survey of Brazil database (GeoSGB). For this integration, it was used the minimum curvature interpolation, with cell sizes according to each survey. All gridded data was extrapolated upward from the original height to the altitude of 1800 m. Afterwards, we applied the upward continuation for 2700 m and the following filters: vertical gradient (VDR), total horizontal derivative (THDR), analytic signal (AS), or tilt derivative (TDR), tilt angle of the horizontal derivative (TAHG) and vertical integral of the analytic signal (VIAS). The final results provide valuable maps (and grids) that can be used by the community for future works. In this work we tried to do a simple overview of the structural framework of Paraná State, representing the main magnetic structures of the Paraná State basement and compare them with the most important structures already mapped in surface over the years. All the data present in this work and other filters tests are available and can be access in the supplementary material (https://acervodigital.ufpr.br/ https://hdl.handle.net/1884/89075 and also at zenodo repository https://doi.org/10.5281/zenodo.12820642).

Polishing Machine Control System Based on B-Spline Curve Trajectory

In this study, a B-spline trajectory control algorithm is presented, showcasing its ability to achieve smooth trajectory control within an embedded motion control system. Traditional surface machining methods involve converting interpolated trajectories into G-code using numerical control (NC) software, which is then downloaded and executed on the numerical control system. This research introduces a specialized third-degree B-spline interpolation method tailored for curved surface trajectories, aiming to enhance machining efficiency. A significant strength of this project lies in the holistic design of an embedded system, encompassing both hardware circuitry and software logic. Utilizing the cost-effective STM32 architecture, a B-spline discrete velocity interpolation algorithm is implemented, validated through experiments conducted on a polishing machine system. The project involves MATLAB software for data simulations and experiments on a polishing machine using B-spline curve interpolation, confirming the practicality and effectiveness of the third-degree B-spline algorithm within an embedded system.

FRKVF: High-accuracy motion interpolation for polishing operations using fourth-order Runge-Kutta and velocity flexibility planning

Traditional interpolation algorithms often fail to meet the precision requirements of ultra-precision machining when applied to super-precision polishing machines. Moreover, the processing of fused silica glass optical components is frequently threatened by mechanical impacts due to their fragility. In order to address this issue, this paper proposes a high-precision motion interpolation method based on fourth-order Runge-Kutta and velocity-flexible planning. This method is designed for open-architecture small multi-axis optical polishing machines to polish quartz glass. The algorithm initially employs composite Simpson's rule to calculate the lengths of sub-paths within the polishing trajectory. Based on these length values, flexible velocity planning is executed to ensure the smooth continuity of velocity, acceleration, and jerk during motion interpolation. This reduces the risk of mechanical impacts that could damage the components during the machining process. The introduction of the adaptive fourth-order Runge-Kutta method significantly enhances the parameter point calculation accuracy of NURBS curves. The incorporation of adaptive principles also maintains a higher processing speed, thereby greatly improving processing efficiency. This method comprehensively addresses both the precision of curve interpolation and execution efficiency. Finally, experimental validation is conducted on an open-architecture small multi-axis optical polishing machine. The proposed method based on FRKVF not only mitigates mechanical impacts resulting from discontinuous acceleration, thereby ensuring the machining quality of optical components, but also satisfies the high-precision requirements for processing fused silica optical elements.

Aquifer system deformation in the San Luis Valley: A new framework for modeling subsidence in agricultural regions

The San Luis Valley (SLV), Colorado, is challenged with implementing sustainable groundwater management in the face of increasing surface water scarcity due to climate change. Groundwater extraction in unconsolidated aquifers such as the SLV can cause cm-scale subsidence and rebound. This study utilizes Interferometric Synthetic Aperture Radar (InSAR) data, validated by Global Navigation Satellite System (GNSS) measurements, to measure subsidence and analyze the groundwater dynamics that cause it. Addressing the challenges of phase decorrelation and data gaps, notably from September 2018 to April 2019, we adopted a modified DS-interpolation algorithm, alongside a Singular Spectrum Analysis (SSA)-based gap filling technique. Furthermore, we enhanced the temporal resolution of groundwater level data through the Theis curve interpolation. These methodologies enabled the integration of observational well data with satellite measurements to calibrate a one-dimensional deformation model, capturing both the elastic and inelastic responses of the aquifer system. Our investigation, spanning 2015 to 2021, reveals both seasonal and long-term subsidence, with the confined aquifer section experiencing up to 1 cm/year of subsidence alongside notable seasonal fluctuations. The methodology presented here provides a path to model subsidence in regions with sparse groundwater level and noisy InSAR data. It also provides valuable insights for developing effective water management strategies in the SLV.

Bézier Curve Interpolation Model for Complex Data by Using Neutrosophic Approach

Since certain data are ignored owing to noise, coping with the complex data with neutrosophic features is problematic. This paper suggests a neutrosophic set strategy for interpolating the Bézier curve to overcome this issue. Thus, depending on the neutrosophic set notion, this work introduces the Bézier curve interpolation method for neutrosophic data. Using the neutrosophic set and its attributes, the neutrosophic control point is specified first. After that, the Bernstein basis function is linked to the control point and yields a neutrosophic Bézier. This curve is then shown using an interpolation approach that includes curves indicating membership, indeterminacy, and non-membership. Before the conclusion of this article, there is a numeric example and an algorithm for developing the neutrosophic Bézier curve using interpolation. Based on the results obtained, the neutrosophic set can deal with the complex data by treating all data including the uncertainty data as indeterminacy membership functions. In conclusion, this model can be used for real applications involving big data analysis.

Could Road Structures Impact the Avian Community? A Study Case from the South Coast Remained Forest in Malang Region, East Java Province, Indonesia

The remaining tropical forest on the south coast of the Malang region is one of the secluded areas that served as an important habitat for its biodiversity, particularly for avians. Nevertheless, the presence of the road structure that crosses over the forest might impact avian communities, which needs to be investigated. The avian survey was conducted to investigate the diversity, community profiles, abundance, and feeding guild based on two different ecosystem patches (on the roads versus outside the road structures) during January–April 2022. The audiovisual encounter methods were performed during the surveys on a total of ten sampling points. Avian species were identified using field guides where the conservation status was based on the national regulation (P106 KLHK), CITES, and IUCN Redlist. The comparison through the avian community based on species richness was analyzed using Venn Diagrams and predicted using rarefaction and interpolation curves on INEXT packages. Additionally, the comparison of sizes of avian abundance was investigated using the α-diversity parameter index, and the feeding guild was determined by five diet guilds. In total, 2536 individuals from 67 species, 34 families, and 13 orders were found during the survey. The avian communities were richer outside the road structure, but their abundance qualitatively reveals that both different ecosystem patches showed relatively good condition. Overall, road structures generate both positive (unique habitat patches) and negative (physical barriers that could disrupt natural ecological processes) impacts on avian diversity. In light of these, sustainable management and conservation action plans were urgently needed to prevent the negative effects on avian communities posed by the road structure.

The Application of MATLAB in the Mathematics Teaching of Computer Majors

Under the background of new engineering courses, professional mathematics teaching needs to keep pace with the times and improve the progressiveness and scientific nature of teaching methods.In order to improve the mathematics teaching effect of computer majors, This article constructs a new teaching environment through simulation system construction methods, analyzes the shortcomings of traditional teaching models through comparative analysis methods, and improves the teaching platform based on the actual needs of mathematics professionals in current society. The curve interpolation technology is improved, and a forward-looking real-time Matlab simulation algorithm is designed. Through the evaluation and analysis, it can be seen that the mathematics teaching system for computer majors based on MATLAB can effectively improve the intuitive effect of advanced mathematics teaching, and help to improve the teaching quality of advanced mathematics.

Path Planning of a Mobile Robot Based on the Improved Rapidly Exploring Random Trees Star Algorithm

With the increasing utilization of sampling-based path planning methods in the field of mobile robots, the RRT* algorithm faces challenges in complex indoor scenes, including high sampling randomness and slow convergence speed. To tackle these issues, this paper presents an improved RRT* path-planning algorithm based on the generalized Voronoi diagram with an adaptive bias strategy. Firstly, the algorithm leverages the properties of the generalized Voronoi diagram (GVD) to obtain heuristic paths, and a sampling region with target bias is constructed, increasing the purposefulness of the sampling process. Secondly, the node expansion process incorporates an adaptive bias strategy, dynamically adjusting the step size and expanding direction. This strategy allows the algorithm to adapt to the local environment leading to improved convergence speed. To ensure the generation of smooth paths, the paper employs the cubic spline curve interpolation algorithm for trajectory optimization to ensure that the mobile robotic can obtain the best trajectory. Finally, the proposed algorithm is experimentally compared with existing algorithms, including the RRT* and Informed-RRT* algorithms, to verify the feasibility and stability.

Spatial interpolation techniques comparison and evaluation: The case of ground-based gravity and elevation datasets of the central Main Ethiopian rift

The ground-based gravity data reveals diverse anomaly signatures in areas of the Main Ethiopian rift where active volcanic and tectonic activities are dominant. In such a region ground-based data collection is restricted to existing roads and relies on accessible stations. These resulted in gaps in data, either missing, uneven, or insufficient spatial coverage that must be estimated with proper interpolation techniques. Comparison and evaluations of the spatial interpolation methods that are commonly used in potential field geophysical data analysis were made for the terrestrial gravity and elevation data of the central Main Ethiopian rift. In this research, two widely used interpolation techniques, minimum curvature interpolation, and Ordinary Kriging were compared and assessed. A 10 % hold-out validation was employed, where 90 % of the data points were used to generate interpolated surfaces, which were then evaluated against the remaining 10 %. Following interpolation with each technique, the generated grid was converted into discrete data points (estimated values). These are then compared with the available gravity data, which were deliberately excluded from the gridding process (10 % remaining dataset). The accuracy of each method was assessed by evaluation metrics such as mean value, variance, Mean Absolute Error (MAE), Root Mean Square Error (RMSE), correlation coefficient (r), and R-squared. The results showed that the ordinary Kriging interpolation method outperformed the minimum curvature interpolants for gravity data with all performance metrics, while both interpolants seem to perform equally well for the elevation dataset. Therefore, it is proposed to use the Kriging interpolation method for potential field gravity studies conducted in the central Main Ethiopia rift.

CUBIC SPLINE INTERPOLATION TO APPROXIMATE SEA DEPTH BETWEEN TELUK SUAK AND LEMUKUTAN ISLAND

Interpolation is a technique that determines the value of a function at a point located between several known data points. Various interpolation methods include polynomial interpolation, Lagrange interpolation, Newton interpolation, and spline interpolation. Cubic spline interpolation aims to produce accurate approximations characterized by minimal oscillations in the resulting curve. This research uses the cubic spline interpolation method to estimate the sea depth in the waters between Teluk Suak and Pulau Lemukutan. The sea depth measurements are conducted to acquire information regarding the underwater topography. We collected data points using remote sensing techniques through Google Earth Pro. These points are selected based on the sea depth profile considerations, including steep areas and points of depth variation to maintain the seabed conditions within the interpolation curve, with a total of 31 data points. Subsequently, we subjected these data points to the conditions necessary for cubic spline interpolation, resulting in a system of 120 linear equations. After solving this system of linear equations, we obtained cubic spline interpolation polynomials for each subinterval and then estimated the sea depth at other points. Based on the cubic spline interpolation results, we achieved a Mean Average Percentage Error (MAPE) of 1.58%, indicating a highly accurate interpolation outcome.