Interpolation Precision Research Articles

Optimization of the Seventh-order Polynomial Interpolation 1P Kernel in the Time Domain

This paper presents the optimization of the convolutional, seventh-order polynomial, one-parameter, interpolation kernel. In the first part of the paper, the seventh-order kernel is defined, and, after that, the process of the kernel optimization is described. The optimization criterion was the minimization of the interpolation error e. The optimization involved the selection of the optimal value of the kernel parameter 𝛼, and it was carried out in the time domain. In the second part of this paper, the experiment, which was realized with the aim of determining the precision of interpolation of the third-order, fifth-order, and the seventh-order interpolation kernels, is described. After that a comparative analysis of the interpolation precision is described. As a measure of the interpolation precision, the mean square error (MSE) was used. The results of the experiment are presented graphically and tabularly. Finally, using a comparative analysis, the precision of interpolation with the kernel, whose parameters were optimized in the time domain, in relation to the kernel, whose parameters were optimized in the spectral do-main, was analyzed. Based on the comparative analysis, a recommendation for the optimal parameter for the seventh-order kernel is given.

Spatial interpolation of global DEM using federated deep learning

The digital elevation model (DEM) provides important data support for 3D terrain modeling. However, due to the complex and changeable terrain in the real world and the high cost of field measurement, it is extremely difficult to obtain continuous and high-density elevation data directly. Therefore, it is necessary to rely on spatial interpolation technology to restore the DEM overall picture in the original sampling area. The traditional spatial interpolation method usually has the characteristics of low model complexity and high computational cost, which leads to low real-time performance and low precision of the interpolation process. The interpolation operation based on DEM data can be considered as a special image generation process where the input is a DEM image with missing values and the output is a complete DEM image. At present, a large number of studies have proved that deep learning methods are very effective in image generation tasks. However, the training of deep learning models requires the support of a large number of high-quality data sets. DEM data in various countries, especially in key regions, are usually restricted by privacy protection regulations and cannot be disclosed. The emergence of Federated Learning (FL) provides a new solution, which supports local training on multiple end nodes, without sending local data to a remote center server for centralized training, effectively protecting data privacy. In this study, we propose a DEM interpolation model based on FL and multiScale U-Net. The experimental results show that compared with the traditional method, this model has faster processing speed and lower interpolation precision. At the same time, this research result provides a new way for efficient and secure use of terrain information, especially in those application scenarios that have strict requirements for DEM data privacy and security.

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.

Meshless numerical manifold method with novel subspace tracking and CSS locating techniques for slope stability analysis

The meshless numerical manifold method (MNMM) inherits the advantages of handling continuous and discontinuous problems uniformly, improving interpolation precision, and avoiding linear dependence issues. This paper refines the MNMM method for slope stability analysis to overcome the dependence and sensitivity of the strength reduction method of the finite element method on the number of finite element meshes and the algorithm for solving the nonlinear equations when calculating slopes with soft and weak interlayers. First, the strength reduction method (SRM) was incorporated into MNMM to create the numerical model called MNMM-SRM. Based on the MNMM-SRM, a complementary Mohr-Coulomb (M−C) control equation compliant with Koiter's rule of multi-yielding surfaces was established. In the principal stress space, the subspace tracking method was used to track the eigenvalues and eigenvectors of the stress tensor, thereby addressing the corner problem of M−C multi-yield surfaces. Combining the k-means clustering algorithm with wavelet transformation and employing the slope's total nodal displacement field as an input variable, an intelligent and efficient critical slip surface (CSS) automatic extraction method was developed. Three typical cases were used to validate the accuracy of the numerical model. This numerical model has a wide range of potential applications in slope stability research.

Exposure-response relation for vibration-induced white finger: effect of different methods for predicting prevalence.

A pooled analysis of vibration-induced white finger (VWF) in population groups of workers has been performed using the results of a published meta-analysis as source material (Nilsson T, Wahlström J, Burström L. Hand-arm vibration and the risk of vascular and neurological diseases a systematic review and meta-analysis. PLoS One. 2017:12(7):e0180795. https://doi.org/10.1371/journal.pone.0180795). The methods of data selection follow those described previously by Scholz et al. (in Scholz MF, Brammer AJ, Marburg S. Exposure-response relation for vibration-induced white finger: inferences from a published meta-analysis of population groups. Int Arch Occup Environ Health. 2023a:96(5):757-770. https://doi.org/10.1007/s00420-023-01965-w) to enable comparison with the results of the present work. The analyzed epidemiologic studies contain different prevalences of VWF observed after different durations of employment involving exposure to the vibration of power tools and machines. These prevalences are transformed to 10% prevalence by either linear or polynomial (i.e. "S"-shaped curvilinear) interpolation in order to compare with the exposure-response relation contained in the relevant international standard (ISO 5349-1:2001). An exposure-response relation is constructed using regression analysis for the time (in years) to reach 10% prevalence in a population group, when subjected to a daily vibration exposure calculated according to the procedures specified in the standard, A(8). Good fits to the data are obtained when polynomial and linear prevalence interpolation is used. The 95-percentile confidence intervals (CIs) of the exposure-response relation predicted by polynomial prevalence interpolation lie at somewhat larger lifetime exposures than those obtained by linear prevalence interpolation. Uncertainty in the precision of polynomial prevalence interpolation is mitigated by giving equal weight to linear interpolation when interpreting the results. When the 95-percentile CIs of the exposure-response models obtained by linear and polynomial prevalence interpolation are used to define the most probable exposure-response relation, the resulting common range of values includes the ISO exposure-response relation. It is proposed that an exposure-response relation for the onset of VWF derived from a regression analysis is specified in terms of the lower limit of its CI. Hence, when exposure measures are constructed according to the ISO standard and equal weight is given to the results of the 2 methods for interpolating prevalence described here, the ISO exposure-response relation would be considered to provide a conservative estimate for a 10% prevalence of VWF to develop in a population group, at least for A(8) > 4 m/s2. It thus remains the relation to use for assessing exposure to hand-transmitted vibration in the workplace. Additional research is needed to resolve inconsistencies in the ISO method for calculating daily exposures.

Interpolation of GNSS Position Time Series Using GBDT, XGBoost, and RF Machine Learning Algorithms and Models Error Analysis

The global navigation satellite system (GNSS) position time series provides essential data for geodynamic and geophysical studies. Interpolation of the GNSS position time series is necessary because missing data will produce inaccurate conclusions made from the studies. The spatio-temporal correlations between GNSS reference stations cannot be considered when using traditional interpolation methods. This paper examines the use of machine learning models to reflect the spatio-temporal correlation among GNSS reference stations. To form the machine learning problem, the time series to be interpolated are treated as output values, and the time series from the remaining GNSS reference stations are used as input data. Specifically, three machine learning algorithms (i.e., the gradient boosting decision tree (GBDT), eXtreme gradient boosting (XGBoost), and random forest (RF)) are utilized to perform interpolation with the time series data from five GNSS reference stations in North China. The results of the interpolation of discrete points indicate that the three machine learning models achieve similar interpolation precision in the Up component, which is 45% better than the traditional cubic spline interpolation precision. The results of the interpolation of continuous missing data indicate that seasonal oscillations caused by thermal expansion effects in summer significantly affect the interpolation precision. Meanwhile, we improved the interpolation precision of the three models by adding data from five stations which have high correlation with the initial five GNSS reference stations. The interpolated time series for the North, East, and Up (NEU) are examined by principal component analysis (PCA), and the results show that the GBDT and RF models perform interpolation better than the XGBoost model.

Regional freezing index and its frequency calculation considering the certainty effects of elevation and latitude

The freezing index (FI) is an important index used in investigations of climate change, frozen ground degradation and frost heave resistance engineering design. In view of the fact that the deterministic effects of latitude and elevation are not considered in the frequency calculation of FI, we proposed an index-freezing method that considers the certainty effects of both elevation and latitude by referring to the index-flood method in this paper. The correlations between the FI and certainty factors (elevation and latitude) were obtained by multiple regression analysis. The effects of latitude and elevation were then removed by nondimensionalisation, and dimensionless FI sequences were subsequently obtained. Finally, the index-freezing method was verified by regional probability analysis. Using the daily average temperature data recorded at 10 major meteorological stations over the 1960–2020 period in Ningxia, the calculation process of the FI and its frequency distribution were provided. The results showed that the proposed FI method can not only remove the certainty effects of elevation and latitude but can also consider the uncertainty associated with interannual FI variations, thus providing more scientific, reasonable and accurate results. The generalised extreme value (GEV) distribution is the optimal frequency distribution of the nondimensional regional FI. The estimation errors of the missing data tests were mostly within 10%, and the residual sum of squares (RSS) and root-mean-square error (RMSE) values were also lower than those obtained through spatial interpolation, thus indicating that the interpolation precision of the proposed FI method was optimal.

Interpolation Technique for the Underwater DEM Generated by an Unmanned Surface Vessel

High-resolution underwater digital elevation models (DEMs) are important for water and soil conservation, hydrological analysis, and river channel dredging. In this work, the underwater topography of the Panjing River in Shanghai, China, was measured by an unmanned surface vessel. Five different interpolation methods were used to generate the underwater DEM and their precision and applicability for different underwater landforms were analyzed through cross-validation. The results showed that there was a positive correlation between the interpolation error and the terrain surface roughness. The five interpolation methods were all appropriate for the survey area, but their accuracy varied with different surface roughness. Based on the analysis results, an integrated approach was proposed to automatically select the appropriate interpolation method according to the different surface roughness in the surveying area. This approach improved the overall interpolation precision. The suggested technique provides a reference for the selection of interpolation methods for underwater DEM data.

Optimization of the 2P fifth degree convolution kernel in the spectral domain

The first part of the paper describes a two-parameter (2P) fifth-order interpolation kernel, r. After that, from the 2P kernel, the kernel components were created. By applying the Fourier transformation to each kernel component, the spectral components of the 2P kernel were obtained. The spectral characteristic of the 2P kernel, H, was created from the spectral components. After that, the algorithm, that optimizes the parameters of the 2P kernel so as to eliminate the ripple of the spectral characteristics, is described. The optimization was performed in such a way that the spectral characteristic developed in the Taylor series, HT. With the condition for the elimination of the members of the Tylor series, which greatly affect the ripple of the spectral characteristic, the optimal kernel parameters (aopt, bopt) were determined. The second part of the paper describes an Experiment, in which the interpolation accuracy of the 2P kernel was tested. Convolution interpolation, with the 2P kernel, was performed over the signals from the Test base. The Test base is created with musical signals. By analyzing the interpolation error, which is represented by the Mean Square Error, MSE, the precision of the interpolation was determined. The results (aopt, bopt, MSEmin) are presented on tables and graphs. Detailed comparative analysis showed higher interpolation precision with the proposed 2P interpolation kernel, compared to the interpolation precision with, 1P interpolation kernel. Finally, the numerical values of the optimal kernel parameters, which are determined by the optimization algorithm proposed in this paper, were experimentally verified.

A Concise Method for Calibrating the Offset of GPS Precise Satellite Orbit

A set of Global Navigation Satellite Systems (GNSS) satellite orbit and clock offset are an essential prerequisite for precise application. However, abrupt changes in accuracy at the boundaries are prevalent in products provided by international GNSS services, resulting in decreased orbit interpolation precision near the daily boundary. In addition, the effect of this phenomenon is reflected in the deterioration of accuracy and the fluctuations in subsequent applications. In this study, time-weighted and equal-weighted calibrated methods were utilized for adjacent Global Positioning System (GPS) satellite orbits and the orbit variations were then corrected for the clock offset to ensure their consistency. The calibration method is evaluated based on the accuracy and smoothness of post-processing kinematic precise point positioning (PPP) and low earth orbit (LEO) precise orbit determination (POD) near the day boundary. In a variety of scientific applications, the results indicate that the proposed calibration method can effectively reduce the excessive differences near the day boundary between adjacent days. Near the boundary, maximum improvements for post-processing kinematic PPP, dynamic LEO precision orbit, kinematic LEO precision orbit are 41.5%, 9.4%, and 20.5%, respectively.

IMAGE INTEROLATION USING OF THE FIFTH AND SEVENTH ORDER POLYNOMIAL PARAMETER KERNELS

In the first part of the paper, one-parameter (1P), fifth and seventh order polynomial interpolationconvolution kernels, are described. In the second part of the paper an Experiment is described. The precision of theimage interpolation was tested. Interpolation of the Test images from the base, using interpolation kernels, wereperformed. The precision of the interpolation kernels was measured using the mean squared error (MSE). Next,optimum kernel parameter, α, were determined by minimizing MSE. After that, a comparative analysis of theminimum MSE values, for the fifth and seventh order polynomial kernels, was performed. The results are presentedtabularly and graphically.

A High-Precision Interpolation Method for Data Transfer in Fluid-Structure Interaction Analysis

In order to improve the precision of data transfer in fluid-structure interaction (FSI) analysis, this paper puts forward an improved inverse isoparametric mapping (IIM) method, which can improve the load transfer accuracy by means of increasing the quantity of Gauss integral points. As displacement transfer happens, the method can perform the function of revising the displacement between the point to be interpolated and projection point, by which the example emulation is also carried out. Some practical solutions are presented, aiming to grapple with pertinent problems such as local interpolation burr and global interpolation distortion which are originated from oversimplified structure dynamic models and inconsistent structural deformation. The calculation results show that the improved IIM method has higher interpolation precision, which can consequently provide a valuable source of reference for FSI analysis by improving the precision of data transfer and satisfying the conservation of total force, total torque, and total virtual work in the meantime.

Multi-objective optimization sampling based on Pareto optimality for soil mapping

Precise mapping based on sampling data is meaningful for efficient soil quality management. The distributions of sampling sites in feature space and geographical space, as well as the distribution of point pairs at different distances, all significantly impact mapping precision, and these three aspects should all be considered in a sampling design. Although optimizing the spread of sampling sites in the above three aspects has been realized and addressed in previous studies, their tradeoff relationship and influence on mapping accuracy have not been comprehensively investigated, partly due to the limitations of weighting-based optimization way. In this article, we proposed a sampling strategy based on Pareto optimality to examine the tradeoff relationship among the three aspects and their influence on interpolation precision. Based on soil organic matter data from Yi’an district, we applied this approach to generate sampling schemes and analyze their distributions and prediction errors. Single-objective optimization of these three aspects was also conducted for comparison. The results revealed that: (1) the sampling strategy that simultaneously optimizes the distributions of sampling sites in the three aspects was able to obtain sampling designs with higher mapping accuracy than the method that only optimizes one aspect. (2) There was an apparent synergistic relationship between the distributions of sampling sites in feature space and geographical space. However, these objectives had antagonistic relationships with the distribution of point pairs at different distances. (3) When considering the tradeoff among the three aspects, the more even the distribution in geographical space was, the higher the mapping accuracy they produced. Furthermore, more even distribution in feature space improved the mapping accuracy, but the benefit faded after a certain degree, whereas the mapping accuracy initially increased as the distance distribution of point pairs in different intervals became more even but subsequently declined beyond a certain point. Attention should be given to the tradeoff relationship and impact on interpolation when sampling for mapping. In similar scenarios like the study case, it is recommended that sampling sites should be distributed in feature space with more than 45% evenness of the optimal state and then distributed in geographical space as evenly as possible, while the distribution of point pairs should not exceed 65% evenness of its optimal state when the sample size range is approximately 50 to 150. This study provides helpful references for scientific sampling with the goal of precise mapping.

Integrity Monitoring of PPP-RTK Positioning; Part II: LEO Augmentation

Low Earth orbit (LEO) satellites benefit future ground-based positioning with their high number, strong signal strength and high speed. The rapid geometry change with the LEO augmentation offers acceleration of the convergence of the precision point positioning (PPP) solution. This contribution discusses the influences of the LEO augmentation on the precise point positioning—real-time kinematic (PPP-RTK) positioning and its integrity monitoring. Using 1 Hz simulated data around Beijing for global positioning system (GPS)/Galileo/Beidou navigation satellite system (BDS)-3 and the tested LEO constellation with 150 satellites on L1/L5, it was found that the convergence of the formal horizontal precision can be significantly shortened in the ambiguity-float case, especially for the single-constellation scenarios with low precision of the interpolated ionospheric delays. The LEO augmentation also improves the efficiency of the user ambiguity resolution and the formal horizontal precision with the ambiguities fixed. Using the integrity monitoring (IM) procedure introduced in the first part of this series of papers, the ambiguity-float horizontal protection levels (HPLs) are sharply reduced in various tested scenarios, with an improvement of more than 60% from 5 to 30 min after the processing start. The ambiguity-fixed HPLs can generally be improved by 10% to 60% with the LEO augmentation, depending on the global navigation satellite system (GNSS) constellations used and the precision of the ionospheric interpolation.

Fabrication and Performance Analysis of 3D Inkjet Flexible Printed Touch Sensor Based on AgNP Electrode for Infotainment Display

It is possible to employ printed capacitive sensors in car bezel applications because of its lower cost and higher detecting capabilities. In this paper, a flexible sensor for automotive entertainment applications has been developed using an electrode flexible sensor with an interdigitated pattern printed on it using screen printing and 3D printing fabrication processes. Design concerns such as electrode overlap, electrode gap and width on capacitance changes, and production costs were studied. In addition, a new generation of flexible printed sensors has been developed that can outperform conventional human–machine interface (HMI) sensors. The capacitance of the design pattern may be optimized by using a 15mm overlap and 0.5mm electrode line width. Due to the precision of interpolation, overlap has a larger effect on sensor performance than it would have without it.

Simulation of yield and water balance using WHCNS and APSIM combined with geostatistics across a heterogeneous field

Estimating water balance is the foundation of improving water productivity (WP) and managing crop production efficiently in fields with significant spatial variability of soil properties. Agricultural system models are useful tools to simulate crop yield, WP, and water balance; however, they are rarely focused on geostatistical characteristics on the spatial scale. If agricultural system models are used spatially, it is important to consider whether the geostatistical characteristics of the simulations are similar to those based on measured data. In this study, two process-based models, the widely-used Agricultural Production Systems sIMulator (APSIM) and the newly-developed soil Water Heat Carbon Nitrogen Simulator (WHCNS), were used to simulate crop yield, water balance, and WP in a 54-ha field with spatially variable soil properties. Performance of the simulations was good for both models; however, the simulation accuracy of WHCNS was higher than for APSIM. Geostatistical characteristics of measured maize yield and final soil water storage (SWSf) were different from the simulated data. The fitted semivariogram models of simulated yield and SWSf had a higher semivariogram range and lower random variation than that of measured data. The fitted semivariogram models and geostatistical characteristics of simulated water balance also varied between the two models. Although the agricultural system models simulated the spatial distribution of variables efficiently, their spatial structure was changed in comparison with the spatial structure of measured data. This would affect the interpolation precision of spatial distribution maps. More work is required on the robustness and prediction accuracy of both models for their implementation in a spatial way.

Computing knots by quadratic and cubic polynomial curves

A new method is presented to determine parameter values (knot) for data points for curve and surface generation. With four adjacent data points, a quadratic polynomial curve can be determined uniquely if the four points form a convex polygon. When the four data points do not form a convex polygon, a cubic polynomial curve with one degree of freedom is used to interpolate the four points, so that the interpolant has better shape, approximating the polygon formed by the four data points. The degree of freedom is determined by minimizing the cubic coefficient of the cubic polynomial curve. The advantages of the new method are, firstly, the knots computed have quadratic polynomial precision, i.e., if the data points are sampled from a quadratic polynomial curve, and the knots are used to construct a quadratic polynomial, it reproduces the original quadratic curve. Secondly, the new method is affine invariant, which is significant, as most parameterization methods do not have this property. Thirdly, it computes knots using a local method. Experiments show that curves constructed using knots computed by the new method have better interpolation precision than for existing methods.

Improved Kriging model based on weighted self-adaptive differential evolution and application for structural monitoring data

The integrity of structural health monitoring data plays an important role in extracting data features. An improved Kriging model based on weighted self-adaptive differential evolution algorithm (KWSDE) is proposed to repair the spatial missing data. In the KWSDE model, the control parameters of variogram in Kriging are optimized by a novel differential evolution (DE) with both scale factor and crossover rate adaptation. Besides, a weighted least-square method is presented for the optimization process to restrain the variable scale of the fitness function, which makes it more suitable for the lack of monitoring data. Finally, a distributed optical fiber monitoring experiment is designed to verify the reliability and functionality of the proposed model. Compared with the Kriging based on DE model (KDE), the interpolation precision of the KWSDE model can be improved by more than 10%, which has great significance for its engineering application.

Virtual reconstruction of random moving image capturing points based on chaos embedded particle swarm optimization algorithm

During the exercise process, such as jumping, and running, the angles of different joints of the athlete's body will produce different changes, resulting in obvious mismatches in different joint feature points. The chaotic matching error makes the calculation results of the basic shape features of the athletes in different parts have large deviations. In this paper, an improved algorithm of CEPS (Chaos embedded particle swarm) optimization algorithm interpolation is introduced, which can improve the interpolation precision and reduce the calculation time of interpolation. Through interpolation, the fast Fourier transform can be used to realize the fast reconstruction of the target 3d image based on biomechanical characteristics. The experimental and simulation results show that the three-dimensional image reconstruction algorithm based on the interpolation method can reconstruct the three-dimensional image of the target, achieve the detection of the target and have good resolution, and improve the authenticity of the human motion image sequence three-dimensional dynamic simulation.

Comparison of interpolation methods for mapping layered soil particle-size fractions and texture in an arid oasis

Soil particle-size (clay, silt, and sand) fractions and texture maps are key inputs for soil physical, chemical, hydrological, agricultural, and ecological models. In the current study, ordinary kriging (OK), simple kriging (SK), and universal kriging (UK) combined with additive, centered, isometric and symmetry log-ratio (ALR, CLR, ILR and SLR) transformations and compositional kriging (CK) are used to map the layered soil texture and to interpolate the soil particle-size fractions based on soil surveys in the middle reaches of Heihe river basin in Northwest China. Results depicted that silt loam is the dominant soil texture in soil profiles of 0–140 cm, accounting for almost 50% in the study area. Layers 3–5 (soil depth of 40–100 cm) show the strongest spatial dependency of soil particle-size fractions as compared to other layers. The results revealed that soil particle-size fractions and soil texture interpolation are dependent on the proper selection of interpolation methods. SLR transformation are better than other log-ratio transformations when interpolating each soil particle-size fraction using different kriging methods. However, for interpolating soil texture types, ALR transformation is the best for OK and SK, and CLR for UK. UK-SLR is recommended to interpolate soil particle-size fractions for all soil layers, and UK-CLR to interpolate the layered soil texture of the study area. More than eighty soil samples should be taken to get a higher soil particle-size fractions/soil texture interpolation precision.