Curve Interpolation Research Articles

Image Interpolation via Scanning Line Algorithm and Discontinuous B-Spline

Image interpolation is a basic operation in image processing. Lots of methods have been proposed, including convolution-based methods, edge modeling methods, point spread function (PSF)-based methods or learning-based methods. Most of them, however, present a high computational complexity and are not suitable for real time applications. However, fast methods are not able to provide artifacts-free images. In this paper we describe a new image interpolation method by using scanning line algorithm which can generate C - 1 curves or surfaces. The C - 1 interpolation can truncate the interpolation curve at big skipping; hence, the image edge can be kept. Numerical experiments illustrate the efficiency of the novel method.

Study of a Novel Bidirectional Compression Electron Gun for $W$ -Band Sheet Beam TWT

A novel bidirectional compression electron gun is designed for a W-band sheet beam traveling-wave tube. A parabolic curved cathode and a beam focusing electrode based on spline interpolation curves are proposed to obtain a beam with good quality. Two directional compressions are introduced to reduce the beam loading of cathode. A 30-kV, 0.97-A beam is emitted by a curved 0.112-cm2 cathode with a beam loading of 8.66 A/cm2. The beam at waist position is 1.4 mm $\times0.4$ mm with a compression ratio of 2 and 10 in two directions separately. Studies of beam emittance and sensitivity of beam dynamics to mechanical errors are also carried out to understand the influence of structure parameters on electron gun performance.

A PIA optimization algorithm for non-uniform B-spline curves and surfaces

This paper aims to address an important issue in B-spline interpolation. In order to reduce shape distortion due to parameterization in the iterative interpolation of B-spline curves and surfaces, we consider both the initial iteration direction of the control vertices and the parameter values corresponding to the nearest points on the initial iterative curves or surfaces. Genetic algorithm with a carefully designed objective function is employed in finding the optimal parameter solutions for high order B-spline curves and surfaces. Experiments show that the shape distortion due to parameterization has been significantly reduced.

An improved state space method for force identification based on function interpolation in the presence of large noise

The conventional state space method for force identification has the disadvantage of large discretization error with a low sampling frequency. This paper presents an improved method based on the function interpolation of the external force in time domain. Two types of the interpolation functions are investigated, one is the linear interpolation, and the other type is the sigmoid curve interpolation. Gauss integration method is used for integration computation. Numerical studies show that both of the improved methods based on the two types of interpolation function are more accurate especially when the sampling is long and/or with a low sampling frequency. In addition, the proposed method is also extended for the case of high noise level. The key idea is to divide the time step of measured responses into several smaller time steps to form an overdetermined equation of the inverse force identification. Then, the least square algorithm is adopted, which helps to reduce the effect of the high random noise to improve the accuracy of identified solution.

Low rank interpolation of boundary spline curves

The coefficients of a tensor-product spline surface in Rd with m×n control points form a tensor of order 3 and dimension (m,n,d). Motivated by applications in isogeometric analysis we analyze the rank of this tensor. In particular, we propose a new construction for low rank tensor-product spline surfaces from given boundary curves. While the results of this construction are generally not affinely invariant, we propose a simple standardization procedure that guarantees affine invariance for d=2. In addition we provide a detailed comparison with existing constructions of spline surfaces from boundary data.

Generating spiral tool paths based on spiral enter assistant line

In order to increase cutting width, decrease tool path length, and improve machining efficiency, the spiral enter assistant line (SEAL) is embedded into a convex pocket to plan a spiral tool path. First, the inner offset curve of the pocket boundary undergoes equidistant division to acquire the external equidistant points. Subsequently, principal component analysis (PCA) and length factors optimization are employed to obtain SEAL. Next, vector operation is performed on the external equidistant points and SEAL to calculate the corresponding internal points. After connecting the external equidistant points and the corresponding internal points, a B-spline spiral tool path is planned based on linear interpolation and B-spline curve fitting. In addition, the length factors can be adjusted to modify the distribution and length of the tool path. Theoretical analysis and machining experiments demonstrate that compared to other conventional algorithms, the spiral tool path presented in this study has obvious advantages on cutting width, tool path length, and machining efficiency. These advantages are especially pronounced when the major principal axis is much longer than the minor principal axis.

Contact analysis of Type17 coupler based on finite element method

Abstract Fatigue cracks of the hook tongue components in Type17 coupler usually occur on its S-surface under improper working conditions. Stress of S-surface mainly consists of tensile stress, bending stress and contact mechanics on the contact surface in stretch conditions. Finite element method with pure penalty algorithm, which discovers the details of stress distribution on S-surface contact regions in contact analysis, is proposed. Stress interpolation curves illustrated the effect of contact to stress concentration on S-surface by extracting the surface stress data. Several peaks value of stress caused by surface contact in contact areas coincide with Hertz stress. According to the comparative analysis, an improved design, which avoids line contact under stress condition, made the four main peak value of stress in the original design decreased by 18.1%, 18.2%, 10.0% and 14.0% respectively. The results conduct that the proposed approach is feasible and the improved design is one way for improving stress condition of hook tongue components.

Surface reconstruction for thin aero engine blade from disorganized contact measured points

Blade surface reconstruction is one of the necessary steps in blade adaptive machining system. The majority of the current reconstruction methods are based on point-cloud, which is obtained using noncontact measurements. Correspondingly, these methods are invalid when the points are obtained using contact measurements. In this paper, we studied the method of surface reconstruction for thin features based on measured contact points. Owing to the random order of the measured points, and the fact that when the distance between two measured points is larger than the wall thickness of the blade will lead to interpolation errors, a quad tree-based region division method, and a points sorting algorithm, are proposed for correctly ordering the points. On this basis, inverse interpolation of cross-sectional curves using a cubic NURBS curve is used to fit them and reconstruct surfaces. Surface reconstruction examples are presented herein, and a comparison between the reconstructed and designed surfaces is used to verify the validity of our method.

Age Estimation of Polymeric Insulation in Absorption of Chemical Components

The investigation of the lifetime execution of polymeric insulator is frequently in light of deficient information. Specialists confront difficulties of not having long perception time to accumulate finish lifetime information so as to make forecast with conviction. In this paper, we are proposing to utilize the best fitting polynomial interpolation method to deal with decide the lifetime show and extrapolate data towards the period of polymeric insulators. To assemble the data, for example, the change of convergence of substance component of polymeric insulation because of various anxiety conditions, high temperature vulcanized (HTV) silicone rubberexamples were subjected to quickened debasement utilizing an Accelerated Weathering Tester (QUV). Every example was subjected to quickened maturing cycle in agreement to ASTM G154 cycle 4 for an aggregate of 3000 hours. The maturing cycle experiences a cycle of 70°C and bright introduction at light power of 1.55 W/m2 for eight hours taken after by four hour buildup at 50°C. Chemical component focuses, for example, Carbon, Oxygen, Silicon and Aluminum were acquired and assessed utilizing a Scanning Electron Microscope (SEM) with energy dispersive xray (EDX). The convergences of chemical components are broke down utilizing polynomial interpolation strategy for different requests. In the first place request to fifth request polynomial interpolation techniques are examined to decide the best fitting bend without essentialness difference from the genuine esteem or genuine esteem. The interpolation curve is utilized to gauge the age of an obscure matured specimen. This examination has uncovered a novel technique for deciding a model that could portray the lifetime of polymeric insulators and extrapolation strategy for evaluating the age for polymer insulator.

To the question of the interpolation of the phase equilibrium curves for the hydrates of methane and carbon dioxide

The experimental data base was developed. This database includes 200 points of the phase equilibrium curve of methane hydrate, as well as more than 300 experimental points for carbon dioxide hydrate. This database is based on the work of a large number of researchers. Taking into account the experimental data, empirical formulas are constructed for determining the equilibrium parameters of the methane and carbon dioxide hydrates, as well as the equilibrium curve of “liquid carbon dioxide – gaseous carbon dioxide”.

A Novel Approach for NURBS Interpolation with Minimal Feed Rate Fluctuation Based on Improved Adams-Moulton Method

In order to reduce the feed rate fluctuation of interpolation, a novel approach for NURBS interpolation with minimal feed rate fluctuation based on improved Adams-Moulton (IAM) method is proposed. At first, the representation and calculation of NURBS curve interpolation are described. Then, the constraints of feeding step length are firstly given out to calculate the minimal hoping feeding step length and the detailed IAM method of NURBS curve interpolation is presented. Finally, simulations and experiments are carried out to verify the feasibility and applicability of proposed IAM method.

基于椭球极投影的三维船舶航线高精度绘制

为解决电子海图中三维船舶航线高精度绘制问题，给出了一种基于椭球极投影的绘制方法。首先，通过建立椭球极投影，将椭球面上的点与平面上的点作一一对应。其次，将港口和船舶位置信息投影到平面上，利用Hermite插值技术，构造一条平面插值曲线。然后将该平面曲线通过椭球极逆投影，得到一条地球面曲线。最后，基于船舶航向等信息，提出了三维航线长度优化及多组航线交点求解的算法，实现了三维航线自动生成及相关计算。数值实验表明本文方法较已有方法更具真实感，精度更高。 In order to solve the problem of 3D high accuracy rendering of ship route in electronic navigational chart, a method based on the ellipsoidal projection is presented. Firstly, the ellipsoidal projection is established, and the point on the ellipsoid is one-to-one corresponding to the point on the plane. Secondly, the port and ship position information is projected onto the plane, and a plane interpolation curve is constructed by using Hermite interpolation technique. Then using the inverse ellipsoidal projection, a curve on the earth surface is obtained. Based on the information of the ship heading, the algorithm of 3D route length optimization and multi-route intersection is presented. The automatic generation of 3D route and related calculation are realized. Finally, numerical experiments show that the method is more realistic and more accurate than the existing methods.

Fast parametric curve interpolation with minimal feedrate fluctuation by cubic B-spline

Due to the reliable feedrate fluctuation and computation load of the existing parametric curve interpolation, a fast interpolation method by cubic B-spline for parametric curve is presented which results in a minimum feedrate fluctuation and light computation load. As there are many geometry implementation tools and many good properties in the B-spline compared with the polynomial, the relation between the arc length s and curve parameter u can be fitted by the cubic B-spline accurately. Because the feedrate fluctuation of the generally used Taylor approximation method is sensitive to the curvature of the toolpath, its accuracy cannot be controlled. For a given feedrate fluctuation of 0.05%, the proposed interpolation method can guarantee the error requirements by increasing the number of the control points. After the de Boor method is applied in real time, the computation load of the cubic B-spline interpolation is decreased compared with the Taylor approximation method and higher order polynomial-fitting method. In order to save the memory consumption for storing the parameters of the fitted cubic B-spline, an iterative optimization process is applied to obtain the knot vector elements and optimize the control points. Simulations and experiments show that the interpolation method can attain high accuracy and computation efficiency. According to the simulations, for most of the complex curves, the feedrate fluctuation of the proposed interpolation method is decreased by about 50% when the feedrate is scheduled and the computation load of the proposed method is decreased by about 70% compared with the second-order interpolation method.

Determination of Rhenium in Seawater from the Jiulong River Estuary and Taiwan Strait, China by Automated Flow Injection Inductively Coupled Plasma–Mass Spectrometry

ABSTRACTA novel and automated flow injection (FI) system using a commercially available anion-exchange resin was developed for the online determination of rhenium in seawater by inductively coupled plasma–mass spectrometry (ICP–MS). The flow rate, reagent concentration, and sample volume were optimized. The rhenium concentration was determined using the integrated peak area through interpolation of standard addition calibration curves. The relative standard deviation was less than 2.45% (n = 5), the recoveries were from 94.59 to 104.11%, and the linear dynamic range of the method was up to 998 ng kg−1. There was a good agreement between the rhenium concentrations in a standard measured in this work with results from previous studies. The limit of detection was 0.11 ng kg−1 for a sample volume of 7.5 mL and one cycle required 8.33 min. The developed FI–ICP–MS method was used to determine rhenium in seawater from the Jiulong River Estuary and Taiwan Strait.

Spatial prediction of soil penetration resistance using functional geostatistics

Knowledge of agricultural soils is a relevant factor for the sustainable development of farming activities. Studies on agricultural soils usually begin with the analysis of data obtained from sampling a finite number of sites in a particular region of interest. The variables measured at each site can be scalar (chemical properties) or functional (infiltration water or penetration resistance). The use of functional geostatistics (FG) allows to perform spatial curve interpolation to generate prediction curves (instead of single variables) at sites that lack information. This study analyzed soil penetration resistance (PR) data measured between 0 and 35 cm depth at 75 sites within a 37 ha plot dedicated to livestock. The data from each site were converted to curves using non-parametric smoothing techniques. In this study, a B-splines basis of 18 functions was used to estimate PR curves for each of the 75 sites. The applicability of FG as a spatial prediction tool for PR curves was then evaluated using cross-validation, and the results were compared with classical spatial prediction methods (univariate geostatistics) that are generally used for studying this type of information. We concluded that FG is a reliable tool for analyzing PR because a high correlation was obtained between the observed and predicted curves (R2 = 94 %). In addition, the results from descriptive analyses calculated from field data and FG models were similar for the observed and predicted values.

High quality local interpolation by composite parametric surfaces

In CAGD the design of a surface that interpolates an arbitrary quadrilateral mesh is definitely a challenging task. The basic requirement is to satisfy both criteria concerning the regularity of the surface and aesthetic concepts.With regard to aesthetic quality, it is well known that interpolatory methods often produce shape artifacts when the data points are unevenly spaced. In the univariate setting, this problem can be overcome, or at least mitigated, by exploiting a proper non-uniform parametrization, that accounts for the geometry of the data. Recently the same principle has been generalized and proven to be effective in the context of bivariate interpolatory subdivision schemes.In this paper, we propose a construction for parametric surfaces of good aesthetic quality and high smoothness interpolating quadrilateral meshes of arbitrary topology. In the classical tensor product setting the same parameter interval must be shared by an entire row or column of mesh edges. Conversely, in this paper, we assign a different parameter interval to each mesh edge. This particular structure, which we call an augmented parametrization, allows us to interpolate each section polyline at parameters values that prevent wiggling of the resulting curve or other interpolation artifacts and yields high quality interpolatory surfaces.The proposed method is generalization of the local univariate spline interpolants introduced in Beccari et al. (2013a) and Antonelli et al. (2014), that have arbitrary continuity and arbitrary order of polynomial reproduction. The generated surfaces retain the same smoothness of the underlying class of univariate splines in the regular regions of the mesh (where, locally, all vertices have valence 4). Mesh regions containing vertices of valence other than 4 are covered with suitably defined surface patches joining the neighboring regular ones with G1- or G2-continuity.

Density-Scaled Balanced Continuum Surface Force Model with a Level Set Based Curvature Interpolation Technique

We examine the recently-proposed density-scaled balanced continuum surface force (CSF) model with a level set-based curvature interpolation technique. The density-scaled balanced CSF model is combined with a numerical framework which is based on the coupled level set and volume-of-fluid (CLSVOF) method, the tangent of hyperbola for interface capturing/weighted line interface calculation (THINC/WLIC) scheme, the constrained interpolation profile conservative semi-Langrangian with rational function (CIP-CSLR) and the volume/surface integrated average-based multi-moment method (VSIAM3). The present CSF model is examined for various bench mark problems such as drop–drop collisions and drop splashing. Comparisons of the present model results with experimental observations and those from the other CSF models show that the present CSF model can minimize spurious current and capture complicated fluid phenomena with minimizing floatsam.

Smooth tool path generation for 5-axis machining of triangular mesh surface with nonzero genus

NC machining of a nonzero genus triangular mesh surface is being more widely confronted than before in the manufacturing field. At present, due to the complexity of geometry computation related to tool path generation, only one path pattern of iso-planar type is adopted in real machining of such surface. To improve significantly 5-axis machining of the nonzero genus mesh surface, it is necessary to develop a more efficient and robust tool path generation method. In this paper, a new method of generating spiral or contour-parallel tool path is proposed, which is inspired by the cylindrical helix or circle which are a set of parallel lines on the rectangular region obtained by unwrapping the cylinder. According to this idea, the effective data structure and algorithm are first designed to transform a nonzero genus surface into a genus-0 surface such that the conformal map method can be used to build the bidirectional mapping between the genus-0 surface and the rectangular region. In this rectangular region, the issues of spiral or contour-parallel tool path generation fall into the category of simple straight path planning. Accordingly, the formula for calculating the parameter increment for the guide line is derived by the difference scheme on the mesh surface and an accuracy improvement method is proposed based on the edge curve interpolation for determining the cutter contact (CC) point. These guarantee that the generated tool path can meet nicely the machining requirement. To improve further the kinematic and dynamic performance of 5-axis machine tool, a method for optimizing tool orientation is also preliminarily investigated. Finally, the experiments are performed to demonstrate the proposed method and show that it can generate nicely the spiral tool path or contour-parallel tool path on the nonzero genus mesh surface and also can guarantee the smooth change of tool orientation.

A Class of Cubic Trigonometric Automatic Interpolation Curves and Surfaces with Parameters

This paper presents the cubic trigonometric interpolation curves with two parameters generated over the space {1, sint, cost, sin2t, sin3t, cos3t}. The new curves can not only automatically interpolate the given data points without solving equation systems, but are also C2 and adjust their shape by altering values of the two parameters. The optimal interpolation curves can be determined by an energy optimization model. The corresponding interpolation surfaces have characteristics similar to the new curves.

Non-circular parametric curve and curved surface interpolation and tool compensation for WEDM based on unit arc length increment method

This paper proposes a novel non-circular parametric curve interpolation algorithm based on the unit arc length increment method (UALI). In this method, axial coordinates are taken as functions of arc length. The reference point moves by 1 basic length unit (BLU) along a curve in each interpolation period, and the corresponding projections on each coordinate are added into accumulators of each axis respectively. An axis will then move by 1 BLU when its accumulator exceeds 1 BLU. Theoretically, based on the UALI method, an interpolation algorithm is derived for parametric curves such as involutes, Archimedes spirals, cycloids, and parabolas. This interpolation algorithm is applicable for ruled surfaces of a workpiece with different parametric curves in the upper and lower planes. Tool compensation is considered for both non-tapered workpieces as well as tapered workpieces. Simulation results and experiment results demonstrate the advantages of this interpolation algorithm, and the accuracy of this interpolation algorithm is analyzed.