Spline Interpolation Function Research Articles

A calibration method of line-structured light system for measuring outer circle dimension during machining

The line-structured light system is widely utilized for various 3D profile measurements due to its convenience and high efficiency. However, the existing calibration methods primarily focus on establishing the relationship between object points and image points, which imposes stringent requirements on the calibration target. To address this issue, a practical calibration method based on interpolation function is proposed in order to measure a specific 3D profile - namely, the diameter of a rotating workpiece during machining. Firstly, the light stripe projected onto an ordinary machined workpiece by a light plane is captured using a Charge-Coupled Device (CCD) camera and subsequently fitted to a quadratic elliptic curve after undergoing image processing. Subsequently, the cubic spline interpolation function is selected to directly establish a mapping relationship between geometric characteristics of the elliptic curve and the diameter measurement. Several experiments are conducted to evaluate the proposed methods. The experimental results demonstrate that compared with two comparison methods, our approach reduces the calibration error of linear structured light systems by 69 % and 51 %, respectively; furthermore, it enables restoration of smooth and detailed 3D models through parallel movement of the system. Moreover, our entire calibration process proves practicable in simplifying experimental procedures while remaining suitable for industrial inspection applications.

Evaluating capabilities of the spline and cubic spline interpolation functions in reference evapotranspiration estimation implementing satellite image data

Evaluating capabilities of the spline and cubic spline interpolation functions in reference evapotranspiration estimation implementing satellite image data

Human calcaneal variation relative to subsistence strategy, activity level, and footwear

Lower limb cortical and trabecular bone varies with human behavior, leading to suggestions that activity level decreases have contributed to a more gracile skeleton. Similar trends are likely present in calcaneal morphology due to its locomotor role during heel strike. Such relationships exist in calcaneal trabecular structure; however, they have yet to be investigated in external morphology. Here entire external calcaneal shape is analyzed among three human populations that vary in subsistence strategy, activity level, and footwear use (n = 93) to investigate how calcaneal morphology varies relative to these factors. Calcanei were either surface scanned or micro-CT scanned. Calcaneal external shape was analyzed using a sliding semilandmark analysis with 1,007 semilandmarks. Semilandmarks were allowed to slide along tangent vectors or planes to minimize the bending energy of the thin plate spline interpolation function relative to an updated Procrustes average. Final landmark configurations underwent a Generalized Procrustes Analysis. Shape variation of Procrustes coordinates was summarized using principal components analysis (PCA). Procrustes distances between the average calcaneus of each population were calculated, and resampling statistics run to test for significant differences. The three populations exhibit significantly different calcaneal morphologies (p<0.001 for all pairwise comparisons) and separate along the first three PCs (42.11% of variance). Hunter-gatherers have superoinferiorly taller and mediolaterally wider posterior calcanei than sedentary populations. This likely serves as an adaptation for increased load transfer through the posterior calcaneus due to more active lifestyles. This is supported further by variation among the two industrialized populations. The 19th–20thcentury industrialized population exhibits a relatively mediolaterally wider posterior calcaneus than the mid-20thcentury-born population, suggesting there has been further gracilization of the calcaneus with increases in sedentary behavior over the last century.

Spacecraft attitude maneuver planning with multi–sensor pointing constraints using improved RRT–star algorithm

During the spacecraft attitude maneuvers for the observation of mission targets, multiple instruments may face different pointing constraints, and they are rarely considered by the current attitude maneuver planning methods. In this case, this paper proposes an improved rapidly–exploring random trees star (RRT*) algorithm to obtain discrete quaternion path nodes while satisfying the pointing constraints of multiple sensors. Next, the temporal properties of these path nodes are defined by a trapezoidal path planning method, and bounded constraints on the angular velocity and control moment during the attitude maneuver are introduced. Then, the continuous quaternion curves through the discrete quaternion path nodes are calculated by a segmented cubic spline interpolation function. Finally, the angular velocity and control moment are calculated through the inverse dynamics. The simulation results verify the feasibility of the method. It can satisfy complex constraints and obtain the attitude optimization path, and alleviate the large control moment variations in the start and stop phases of the rest–to–rest maneuver.

Improved inverse analysis methods and modified apparent earth pressure for braced excavations in soft clay

Inverse analysis procedures are used to update project parameters and applied to predict future performance in the later stages. However, the inverse analysis procedure of earth pressure of excavation has received less attention. This paper proposes an improved inverse analysis method and the modified apparent earth pressure model for braced excavations in soft clay. A new procedure for inverse analysis of excavation is suggested based on the quintic spline interpolation function, and a trial algorithm is used to eliminate the errors generated in the inverse process. The proposed inverse solutions are validated by numerical simulations and reported engineering examples and show good agreements. Based on the inverse data and empirical apparent pressure diagram (APD), a modified apparent earth pressure method for braced excavations in soft clay is further suggested. And the relationship between the degree of anisotropy of soft clay and apparent earth pressure coefficients is discussed. The study results provide a reference for dynamic feedback design and adaptive management of the excavation.

Spatiotemporal variability characteristics of extreme climate events in Xinjiang during 1960–2019

Under the global warming, it is particularly important to explore the response of extreme climate to global climate change over the arid regions. Based on daily temperature (maximum, minimum, and average) and precipitation data from meteorological stations in Xinjiang, China, we analyzed the spatiotemporal characteristics of extreme temperature and extreme precipitation events via combining thin plate smoothing spline function interpolation, Sen's slope, and Mann-Kendall test. Our results showed that during 1960-2019, the extreme low temperature index of frost days (FD), icing days (ID), cold days (TX10p), cold nights (TN10p), and cold speel duration index (CSDI) all showed the downward trend to varying degrees, and the extreme high temperature index of summer days (SD25), warm days (TX90p), warm night (TN90p), and warm speel duration index (WSDI) all showed an upward trend to varying degrees, and the extreme low temperature index of high altitude mountains decreases more than that of the basin and plains. In addition, all the extreme temperature indices are closely related to the annual average temperature in Xinjiang (R > 0.6). Among the extreme precipitation indices, except for the consecutive dry days (CDD), the other extreme precipitation indices showed increasing trends to different degrees, but the changes in extreme precipitation in Xinjiang were mainly manifested by the increase of heavy precipitation in a short period (the increase of heavy precipitation and extreme heavy precipitation was the largest, 44.8mm/10a and 17.6mm/10a, respectively) and spatially concentrated in the Ili River and Altai Mountains in northern Xinjiang. Meanwhile, annual precipitation was positively correlated with the extreme precipitation index (R > 0.4), except for the CDD. This study provides theoretical support for the prevention and control of natural disasters in the dry zone.

Multi-view and multi-scale super-resolution method of logging curves based on fractal theory.

With the increasing development of unconventional reservoirs around the world, there is an increasing need to enhance the level of geological characterization. Obviously, since the well loggings are one of the most important data to obtain a fine-scale reservoir model, obtaining well loggings with sufficiently high vertical resolution has always been an important issue and challenge. Therefore, due to its low cost and less time-consuming, employing an advanced signal processing technique to enhance the vertical resolution of loggings has always been a research hotspot in the relevant literature. However, non-homogeneity of the target reservoir is not taken into account in the traditional methods such as vertical resolution matching and spline function interpolation. Furthermore, for state-of-the-art methods that employ different versions of shallow or deep machine learning models, how to adequately exploit the multi-scale shape and temporal information in the employed loggings has been a very challenging problem. To address the above problems, by combining the fractal theory with the long short-term memory network technique, a novel multi-view and multi-scale logging resolution enhancing method was proposed in this paper to make full use of the self-shape-similarity and temporal correlation information in the logging data. Experimental results show that, compared with bi-cubic linear interpolation, sparse representation, super-resolution convolutional neural network and random forest, more promising supper-resolution results can be obtained using the proposed method.

Characteristic Analysis of the Spatio-Temporal Distribution of Key Variables of the Soil Freeze–Thaw Processes over Heilongjiang Province, China

The soil freeze–thaw phenomenon is one of the most outstanding characteristics of the soil in Heilongjiang Province. Quantitative analysis of the characteristics of changes in key variables of the soil freeze–thaw processes is of great scientific importance for understanding climate change, as well as ecological and hydrological processes. Based on the daily surface temperature and air temperature data in Heilongjiang Province for the past 50 years, the spatial–temporal distribution characteristics of key variables and their correlations with air temperature and latitude in the freeze–thaw process of soil were analyzed using linear regression, the Mann–Kendall test, the local thin disk smooth spline function interpolation method, and correlation analysis; additionally, the spatial–temporal distribution of key variables and the changes in the surface temperature during the freeze–thaw process are discussed under different vegetation types. The results show that there is a trend of delayed freezing and early melting of key variables of the soil freeze–thaw process from north to south. From 1971 to 2019 a, the freezing start date (FSD) was delayed at a rate of 1.66 d/10 a, the freezing end date (FED) advanced at a rate of 3.17 d/10 a, and the freezing days (FD) were shortened at a rate of 4.79 d/10 a; with each 1 °C increase in temperature, the FSD was delayed by about 1.6 d, the FED was advanced by about 3 d, and the FD was shortened by about 4.6 d; with each 1° increase in latitude, the FSD was delayed by about 2.6 d, the FED was advanced by about 2.8 d, and the FD was shortened by about 5.6 d. The spatial variation in key variables of the soil freeze–thaw process under the same vegetation cover was closely related to latitude and altitude, where the lower the latitude and altitude, the more obvious the variation trend; among them, the interannual variation trend of key variables of soil freeze–thaw under meadow cover was the most obvious, which varied by 9.65, 16.86, and 26.51 d, respectively. In addition, the trends of ground temperature under different vegetation types were generally consistent, with the longest period of unstable freeze–thaw and the shortest period of stable freeze in coniferous forests, compared to the shortest period of unstable freeze–thaw and the longest period of stable freeze in meadows. The results of the study are important for our understanding of soil freeze–thaw processes and changes in Heilongjiang Province, as well as the evolution of high-latitude permafrost; they also promote further exploration of the impact of soil freeze–thaw on agricultural production and climate change.

Feature curve-based registration for airborne LiDAR bathymetry point clouds

The airborne LiDAR bathymetry (ALB) system is widely used in the fields of sea–land measurement, including seafloor topographic feature description, 3D seafloor model construction, coral reef monitoring, and underwater archaeology. A manned ALB data registration method based on feature curves was proposed considering problems such as sparse ALB data features, low point cloud density, and difficult corresponding feature extraction. First, triangulation network interpolation was used to extract the isoline points of the seafloor point cloud. The curves describing the seafloor topographic trend were generated through cubic parabolic spline function interpolation. Second, the curve deformation energy function was constructed based on the curve features, and a similarity measurement was carried out on the registration curve by integrating the energy function with the longest common subsequence (LCSS) algorithm. The RANSAC and ICP algorithms were used for rough and fine registration, respectively. The effectiveness and robustness of the proposed method were evaluated using two sets of experimental data. The experimental results showed that the average distance between corresponding points in the coarse registration stage reached 0.128 m and 0.136 m. After fine registration, the average point distance between the point cloud coordinates and ground truth reached 0.073 m and 0.267 m, providing an effective and reliable solution for ALB data registration.

Approximate Solution of Fractional Differential Equation by Quadratic Splines

In this article, we consider approximate solutions by quadratic splines for a fractional differential equation with two Caputo fractional derivatives, the orders of which satisfy 1<α<2 and 0<β<1. Numerical computing schemes of the two fractional derivatives based on quadratic spline interpolation function are derived. Then, the recursion scheme for numerical solutions and the quadratic spline approximate solution are generated. Two numerical examples are used to check the proposed method. Additionally, comparisons with the L1–L2 numerical solutions are conducted. For the considered fractional differential equation with the leading order α, the involved undetermined parameters in the quadratic spline interpolation function can be exactly resolved.

Spatial interpolation of PM2.5 concentrations during holidays in south-central China considering multiple factors

Investigations on PM2.5 regional interpolation models during specific periods with abnormal PM2.5 variation patterns (e.g., during holidays) remain relatively rare. In this paper, we first explored PM2.5 concentration patterns in major provincial capitals in south-central China during the Spring Festival and National Day periods. Then, we analysed the correlation between atmospheric pollutants, meteorological conditions, Zenith Tropospheric Delay (ZTD), elevation and Normalized Difference Vegetation Index (NDVI) with PM2.5 in south-central China during the Spring Festival and National Day periods from 2016 to 2018, followed by GeoDetector application to explore the spatial stratified heterogeneity. In addition, we developed Geographically Weighted Regression (GWR) model, Multiscale Geographically Weighted Regression (MGWR) model, and Geographically and Temporally Weighted Regression (GTWR) model. Then, we applied three interpolation methods (Kriging, Empirical Bayesian Kriging, and Tensor Spline Function) to further optimize the residuals of each regression class model. The results indicated that the model created by the Tensor Spline Function (TSF) interpolation method correcting for the residuals of the GWR model (GWR-TSF) attained the best fit, the GTWR model achieved the worst fit during the Spring Festival period, and the MGWR model attained the worst fit during the National Day period. In terms of the interpolation accuracy, the GWR-TSF model also achieved the highest interpolation accuracy, while the GTWR model attained the lowest interpolation accuracy.

A Knuckle‐Walking Signal in the Trabecular Architecture of the African Ape Scaphoid

African ape (Panand Gorilla) carpals have few diagnostic features in their external morphology reflecting knuckle‐walking hand postures, and whether the internal trabecular architecture of their carpals yields a functional signal for knuckle‐walking is unclear. Here we perform a high‐density, multiple‐VOI analysis of trabecular architecture in the scaphoid of knuckle‐walking African apes and non‐knuckle‐walkers (Pongo, Macaca,and Papio). We predict that (1) Gorilla, the taxon with largest body mass, will have the highest trabecular thickness (Tb.Th) and trabecular spacing (Tb.Sp) because Tb.Th and Tb.Sp have been shown to be correlated with body size, (2) bone volume fraction (BV/TV) will be higher in habitually terrestrial taxa than in arboreal taxa, as the forelimb is thought to experience relatively high compressive loads during terrestrial locomotion, (3) Pongo will have the lowest degree of anisotropy (DA) as they engage in variable locomotor behaviors, and (4) pattern of DA distribution will differ between knuckle‐walking and non‐knuckle‐walking taxa, as the consistency and direction of wrist joint loading likely differs among primates with different quadrupedal hand postures.Fifty‐four scaphoids were scanned using high‐resolution µCT (voxel range: 48‐27 μm). Surface models were created from image data and then subjected to a weighted spherical harmonic analysis. Volumes of interest (VOIs) were manually placed into a surface model of the average scaphoid, and then warped to individual specimens using the thin plate spline interpolation function. Trabecular properties were calculated using the BoneJ plugin for ImageJ and MATLAB, and variation in distribution of trabecular parameters among taxa was summarized using a principal component analysis (PCA). Parameter values among species were compared with non‐parametric MANOVAs using a randomized residual permutation procedure, and PC scores were compared using Kruskal‐Wallis ANOVAs. The parameter distribution within each bone was visualized by comparing species‐average color maps of average parameter values.As predicted, Tb.Th and Tb.Sp are greatest in Gorilla, indicating that these parameters are related in part to body size. In contrast, results indicate that BV/TV and DA magnitudes do not follow predictions. Specifically, habitually terrestrial taxa do not have significantly higher BV/TV than arboreal taxa, and taxa that engage in more variable locomotor behaviors (i.e., Pongo) do not have the lowest values of DA. Finally, results indicate that knuckle‐walking African apes can be distinguished from other taxa studied here by pattern of DA distribution, and these differences appear to reflect differences in habitual hand postures and locomotor modes. If this pattern is unique to knuckle‐walking apes among hominoids, and is also identified in the scaphoid of fossil hominoids, then it may be used to indicate whether fossil taxa were habitual knuckle‐walkers.

Analysis of approaches to spline interpolation of functions with large gradients in the boundary layer

The problem of cubic spline interpolation of functions with large gradients in the boundary layer is considered. It is assumed that the decomposition in the form of the sum of the regular and singular components is valid for the interpolated function. This decomposition is valid for the solution of a boundary value problem for a second order ordinary differential equation with a small parameter e at the highest derivative. An overview of approaches to the construction of splines, the error of which is uniform with respect to the small parameter e, is given. The approaches are based on the use of Shishkin and Bakhvalov meshes, which are thickened in the boundary layer region. An alternative approach based on a modification of a cubic spline with fitting to the singular component of the interpolated function is also considered. The comparison of the accuracy of the applied approaches with the performance of computational experiments is carried out.

The Solution of Non-Linear Equations System Containing Interpolation Functions by Relaxing the Newton Method

Many world phenomena lead to nonlinear equations systems. For some applications, the non-linear equations which construct the non-linear equations system are interpolation functions. However, the interpolation functions are usually not represented as mathematical expressions but as computer programs in specific programming languages. The research proposed using the relaxed Newton method to solve the non-linear equations system that contained interpolation functions. The interpolation functions were represented in the R programming language. Then, the experiment used the Spline interpolation function to construct a two-dimensional non-linear equations system. Eleven initial guesses, maximum of ten-time iterations, and 10-7 precision were applied. The solution of the non-linear equations system and the iteration needed on each initial guess were observed. The experiment shows that the proposed method works well for solving the non-linear equations system constructed by Spline interpolation functions. By observing the initial guesses used in the experiment, there are four possible results: true solution, spurious solution, false solution, and no solution. Applying 11 initial guesses have five initial guesses resulting in true solutions, one initial guess in spurious solution, three initial guesses in false solutions, and one initial guess in no solution. The discussions imply that this method can be generalized to the three-dimensional non-linear equations system or higher dimensions.

Online monitoring for floating raft structure displacement based on optimal placement of measuring points

The structure displacement of the raft of vibration isolation for ships is an important factor that affects shafting alignment accuracy and the safety of the equipment on the raft. In order to realize real-time monitoring of the raft structure displacement, which is the uncertain loadings and complex in shape, a measuring point placement strategy and an online monitoring method based on the displacement parameters to identify the displacement of the raft are proposed. FEM-based simulations for a prototyped raft have been conducted to obtain the displacement contour map. According to the displacement contour map based on displacement gradient and index evaluation, the measuring points have been combined and optimized. Displacement reconstruction principle based on a surface spline interpolation function method has been elaborated and derived. Structure displacement at any point can be obtained by a few measuring points. Finally, a case study of a certain type of floating raft is analyzed by the simulation analysis, and experimental research. The simulation and experimental results verify the validity of the measuring point placement strategy and the average of relative error with a value of less than 4% under different uncertain loadings. This method can effectively solve the problem of online monitoring of floating raft structure displacement under different changing loadings.

Diagnosis system of sleep apnea based on wearable sensor

Sleep Apnea Syndrome (SAS) is a serious threat to the life and health of patients. In this paper, a breathing coil is used as a flexible sensor, and a breathing signal conditioning circuit is designed to collect breathing signals. At the same time, an algorithm for determining the incidence of apnea is designed. First, all the maximum points in the respiratory signal and the corresponding extreme positions are found; then the cubic spline interpolation function is used to fit the upper and lower values of the maximum and the minimum envelope, and find the average value of the upper and lower envelopes; finally, the peak or trough point by searching for the local minimum or local maximum in the average envelope was found, and then further segmented calculations was performed. The test results show that the misjudgment range between the number of sleep apneas recorded by the simulation experiment and the number of sleep apneas obtained by the algorithm is between -2 and 3 times. The designed system terminal equipment is accurate, comfortable to wear, can be monitored for a long time. And is suitable for family daily sleep monitoring.

Information representation of blockchain technology: Risk evaluation of investment by personalized quantifier with cubic spline interpolation

With the applications of blockchain technology in various fields, the research on blockchain has attracted much attention. Different from the researches focusing on specific applications of blockchain technology in a certain field, this study devotes to capturing the attitudes of investors regarding different risk criteria in blockchain technology investment decision making. We use personalized quantifiers to extract investors’ preferences on each risk evaluation criterion. At present, the personalized quantifier that can reflect individual attitudes and behavior intentions have been fitted by various functions, but there are still limitations. In this regard, this paper introduces a cubic spline interpolation function to fit the personalized quantifier, and addresses the consistency of the personalized quantifier in the ordered weighted averaging aggregation. Moreover, we employ a qualitative information representation model called probabilistic linguistic term sets to express decision-makers' evaluations on each criterion. We give a case study to illustrate the usability of the proposed personalized quantifier in blockchain risk evaluation. The comparative analysis with other four types of personalized quantifiers shows that our proposed personalized quantifier with cubic spline interpolation has ideal geometric characteristics in terms of smooth curve and high fitting accuracy, thus having strong applicability. Further, we show that this method is relatively easy to operate.

Smooth approximation of a varying refractive-index profile and its application in the computation of light waves

In this paper, the smooth approximation of light waves is studied for an open optical waveguide with a distinct refractive-index profile, which involves high-precision computation of the eigenmodes and corresponding eigenfunctions. During analysis, the refractive-index function is first approximated by a quadratic spline interpolation function. Since the quadratic spline function is a polynomial of degree two in every sub-interval (sub-layer), it is equivalent to a piecewise polynomial of degree two, based on which, the corresponding Sturm-Liouville eigenvalue problem of the Helmholtz operator in sub-layer can be solved analytically by the Kummer functions. Finally, the approximate dispersion equation is established to the TE case. Obviously, the approximate dispersion equations converge fast to the exact ones, as the maximum value of the sub-interval sizes tends to zero. Furthermore, eigenmodes may be obtained by Müller’s method with suitable initial values. To refine the accuracy, the equidistant partition and the non-equidistant partition are applied to divide the interval. Numerical simulations show that the eigenfunctions of the spline interpolation are much smoother than the ones with piecewise interpolation. In addition, the non-equidistant partition can help improve the accuracy and the order of convergence of general solutions reaches the third.

A Collaborative Spline Adaptive Filter for Nonlinear Echo Cancellation

Acoustic echo causes the quality of communication to degenerate and results in loss of clarity. Though conventional linear adaptive filters have been applied successfully to eliminate linear acoustic echo, they can barely deal with the problem of nonlinear acoustic echo. In this paper, a collaborative spline adaptive filter is presented to eliminate nonlinear acoustic echo. The filter is composed of a linear adaptive filter in the upper branch and a nonlinear spline adaptive filter in the lower branch. The nonlinear one is a Hammerstein system consisting of a spline interpolation function and a subsequent linear adaptive filter. The two branches are collaboratively combined and a mixing parameter is adopted, which can be updated to adjust the proportion of the lower branch output signal. Experimental results show that the presented method can achieve a good performance in nonlinear echo cancelation regardless of whether the nonlinear degree of the echo path varies with time or not.

A new implementation of DRM with dual interpolation boundary face method for Poisson equation

We present a new numerical algorithm combining the dual reciprocity method with the dual interpolation boundary face method for solving the two-dimensional Poisson equations. In this new combined approach, the boundary physical variables and particular solutions are approximated by dual interpolation elements which include source and virtual points. Additionally, this algorithm is implemented by employing the augmented thin plate spline interpolation function to approximate non-homogeneous term and the Hermite-type moving-least-square approximation to approximate physical variables for virtual nodes. Compared with the traditional dual reciprocity boundary element method, this combined method possesses higher accuracy and efficiency and is better suited for structures with small feature sizes. Several numerical examples are given to illustrate the superiority of the proposed method for solving Poisson equations.