Modifying CAD/CAM surfaces according to displacements prescribed at a finite set of points

This article presents a method for modifying CAD/CAM surfaces automatically in accordance with displacements prescribed at a finite set of points in R³, such as node displacements predicted by finite-element analysis. The method is based on the ‘morphing’ approach introduced by Sederberg and Parry in 1986. The input to the process consists of (a) a CAD/CAM model containing trimmed polynomial B-spline surfaces and (b) a set of points and associated displacement vectors in R³. These points are assumed to be close to, but not necessarily on, the objects of the CAD/CAM model. A rectangular volume, enclosing the CAD/CAM model and the input points in R³, is represented as a volume spline, i.e. a trivariate tensor-product spline. A modified volume spline is computed using (a) a least-squares fit based on the given point displacements, and (b) a smoothing functional. The modified CAD/CAM objects are defined as compositions of the original parametric functions and the modified volume spline (i.e. a morphing). In order to ensure compatibility with standard commercial CAD/CAM systems, the modified surfaces are fitted with appropriate splines using any standard, reasonably shape-preserving, fitting procedure applied in the parameter domains of the original surfaces.     

A new family of convex splines for data interpolation

This paper develops a new family of convexity-preserving splines of order n, hereby entitled the CPn-spline, that preserves convexity when derivatives at the data points satisfy some reasonable conditions. The spline comprises four components: a constant term, a first order term, and two nth order binomials. A slope-averaging-method is proposed for the general implementation of the new spline. Numerical results that allow for an assessment of the new spline are provided. In particular, a comparative analysis of the CPn-spline, the cubic spline, and of the Carnicer '92 spline is performed. By varying two parameters, the spline shape can be controlled at the local level, while other conventional means can be used to control the shape at the global level. The CPn-spline has no singularities in the case where inflection points are present. Additionally, a less general form of the CPn-spline that applies to most practical cases can be implemented with extreme ease.     

Curve fitting and fairing using conic splines

We present an efficient geometric algorithm for conic spline curve fitting and fairing through conic arc scaling. Given a set of planar points, we first construct a tangent continuous conic spline by interpolating the points with a quadratic Bézier spline curve or fitting the data with a smooth arc spline. The arc spline can be represented as a piecewise quadratic rational Bézier spline curve. For parts of the G¹ conic spline without an inflection, we can obtain a curvature continuous conic spline by adjusting the tangent direction at the joint point and scaling the weights for every two adjacent rational Bézier curves. The unwanted curvature extrema within conic segments or at some joint points can be removed efficiently by scaling the weights of the conic segments or moving the joint points along the normal direction of the curve at the point. In the end, a fair conic spline curve is obtained that is G² continuous at convex or concave parts and G¹ continuous at inflection points. The main advantages of the method lies in two aspects, one advantage is that we can construct a curvature continuous conic spline by a local algorithm, the other one is that the curvature plot of the conic spline can be controlled efficiently. The method can be used in the field where fair shape is desired by interpolating or approximating a given point set. Numerical examples from simulated and real data are presented to show the efficiency of the new method.     

A numerical investigation of errors arising in applying the galerkin method of the solution of nonlinear partial differential equations

The Galerkin method is applied to the solution of Burgers' equation and a nonlinear wave equation, using expansions of B-splines of increasing number of terms and order of spline. The accuracy of the solutions obtained numerically is compared with analytical solutions, and the effect upon accuracy of increasing the order of spline and number of terms in the expansion is considered for a variety of initial conditions corresponding to waves having a range of wavelengths.Burgers' equation is used as a model for the hydrodynamic shallow water equations, and results illustrate the importance of using a sufficient number of functions in the expansion to accurately model the distortion of a wave progressing into a shallow water region where shorter waves contribute appreciably to the total wave profile.     

Spline approximation of offset curves

This paper deals with the approximation of (regular) offset curves (of a given spline curve of degree n) by a spline curve of arbitrarily chosen degree m. The approximating spline curves are determined by geometric continuity conditions and by parameter optimization for minimizing the range of the approximation error.     

A class of general quartic spline curves with shape parameters

With a support on four consecutive subintervals, a class of general quartic splines are presented for a non-uniform knot vector. The splines have C² continuity at simple knots and include the cubic non-uniform B-spline as a special case. Based on the given splines, piecewise quartic spline curves with three local shape parameters are given. The given spline curves can be C²∩G³ continuous by fixing some values of the curve?s parameters. Without solving a linear system, the spline curves can also be used to interpolate sets of points with C² continuity. The effects of varying the three shape parameters on the shape of the quartic spline curves are determined and illustrated.     

Uniform convergence of interpolation by cubic splines

For a sequence of meshes on [0, 1] sufficient conditions are given to obtain uniform convergence of cubic spline interpolants for continous functions respectively for the third derivatives of cubic spline interpolants for functions fromC ³ [0, 1].     

Quasi-circular Splines: A Shape-Preserving Approximation

The "quasi-circular spline" is introduced as a new method for approximating closed, smooth planar shapes from curvature information. A current application is the measurement of shapes of solid rocket booster cross-sections. Because of the efficiency of the algorithm and its desirable geometric properties, it is also particularly appropriate for computer graphics. The simplicity and efficiency of the quasi-circular spline compare well with previously proposed schemes which are important in graphical applications. It is invariant under the transformations of the Euclidean group. Furthermore, it is shape-preserving in that the quasi-circular spline approximation to a convex planar curve is also convex. Sufficient conditions for convergence are described, and O(h²) approximation to sufficiently smooth curves is demonstrated.     

Functional linear regression after spline transformation

Functional linear regression has been widely used to model the relationship between a scalar response and functional predictors. If the original data do not satisfy the linear assumption, an intuitive solution is to perform some transformation such that transformed data will be linearly related. The problem of finding such transformations has been rather neglected in the development of functional data analysis tools. In this paper, we consider transformation on the response variable in functional linear regression and propose a nonparametric transformation model in which we use spline functions to construct the transformation function. The functional regression coefficients are then estimated by an innovative procedure called mixed data canonical correlation analysis (MDCCA). MDCCA is analogous to the canonical correlation analysis between two multivariate samples, but is between a multivariate sample and a set of functional data. Here, we apply the MDCCA to the projection of the transformation function on the B-spline space and the functional predictors. We then show that our estimates agree with the regularized functional least squares estimate for the transformation model subject to a scale multiplication. The dimension of the space of spline transformations can be determined by a model selection principle. Typically, a very small number of B-spline knots is needed. Real and simulation data examples are further presented to demonstrate the value of this approach.     

Local and singularity-free G 1 triangular spline surfaces using a minimum degree scheme

We develop a scheme for constructing G ¹ triangular spline surfaces of arbitrary topological type. To assure that the scheme is local and singularity-free, we analyze the selection of scalar weight functions and the construction of the boundary curve network in detail. With the further requirements of interpolating positions, normals, and surface curvatures, we show that the minimum degree of such a triangular spline surface is 6. And we present a method for constructing boundary curves network, which consists of cubic Bézier curves. To deal with certain singular cases, the base mesh must be locally subdivided and we proposed an adaptive subdivision strategy for it. An application of our G ¹ triangular spline surfaces to the approximation of implicit surfaces is described. The visual quality of this scheme is demonstrated by some examples.     

Locating object contours in complex background using improved snakes

An active contour model, called snake, can adapt to object boundary in an image. A snake is defined as an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines or edges. The traditional snake model fails to locate object contours that appear in complex background. In this paper, we present an improved snake model associated with new regional similarity energy and a gravitation force field to attract the snake approaching the object contours efficiently. Experiment results show that our snake model works successfully for convex and concave objects in a variety of complex backgrounds.     

Numerical approximations for fractional diffusion equations via splines

A one-dimensional fractional diffusion model is considered, where the usual second order derivative gives place to a fractional derivative of order α, with 1<α≤2. We consider the Caputo derivative as the space derivative, which is a form of representing the fractional derivative by an integral operator. An implicit numerical method is derived which uses a spline approximation for the Caputo derivative. The consistency and stability of the method are examined and numerical results are presented.     

On the approximate solution of the delay integral equation of the statistical theory of radiation damage

In the statistical theory of radiation damage, the mean number of atoms displaced in the atomic cascade is given by a delay integral equation with specified initial conditions. Numerical procedures the use spline functions in conjuction with appropriate quadrature rules are presented for the construction of continuous approximations to the mean number of displaced atoms, represented by the delay integral. The methods presented are shown to be stable and to be of order (m + 1) for spline functions of degree m. Finally, the method for quadratic splines is used to compute the mean number of displaced atoms for atomic collisions with Firsov potentials, and with truncated Coulomb potentials.     

The approximate solution of initial-value problems for general Volterra integro-differential equations

We study the application of certain spline collocation methods to Volterra integro-differential equations of orderr where ther-th order derivative of the unknown solution occurs also in the kernel of the integral term. The analysis focuses on the question of the optimal discrete convergence order (at the knots of the approximating spline function).     

C2 interpolation of spatial data subject to arc-length constraints using Pythagorean–hodograph quintic splines

In order to reconstruct spatial curves from discrete electronic sensor data, two alternative C² Pythagorean–hodograph (PH) quintic spline formulations are proposed, interpolating given spatial data subject to prescribed constraints on the arc length of each spline segment. The first approach is concerned with the interpolation of a sequence of points, while the second addresses the interpolation of derivatives only (without spatial localization). The special structure of PH curves allows the arc-length conditions to be expressed as algebraic constraints on the curve coefficients. The C² PH quintic splines are thus defined through minimization of a quadratic function subject to quadratic constraints, and a close starting approximation to the desired solution is identified in order to facilitate efficient construction by iterative methods. The C² PH spline constructions are illustrated by several computed examples.     

Adaptive CAD model (re-)construction with THB-splines

Computer Aided Design (CAD) software libraries rely on the tensor-product NURBS model as standard spline technology. However, in applications of industrial complexity, this mathematical model does not provide sufficient flexibility as an effective geometric modeling option. In particular, the multivariate tensor-product construction precludes the design of adaptive spline representations that support local refinements. Consequently, many patches and trimming operations are needed in challenging applications. The investigation of generalizations of tensor-product splines that support adaptive refinement has recently gained significant momentum due to the advent of Isogeometric Analysis (IgA) [2], where adaptivity is needed for performing local refinement in numerical simulations. Moreover, traditional CAD models containing many small (and possibly trimmed) patches are not directly usable for IgA. Truncated hierarchical B-splines (THB-splines) provide the possibility of introducing different levels of resolution in an adaptive framework, while simultaneously preserving the main properties of standard B-splines. We demonstrate that surface fitting schemes based on THB-spline representations may lead to significant improvements for the geometric (re-)construction of critical turbine blade parts. Furthermore, the local THB-spline evaluation in terms of B-spline patches can be properly combined with commercial geometric modeling kernels in order to convert the multilevel spline representation into an equivalent – namely, exact – CAD geometry. This software interface fully integrates the adaptive modeling tool into CAD systems that comply with the current NURBS standard. It also paves the way for the introduction of isogeometric simulations into complex real world applications.     

Constructions of QC LDPC codes based on integer sequences

Based on difference sequence and Hoey sequence,three types of(2,F)and(3,F)quasi cyclic(QC)low-density parity-check(LDPC)codes are constructed.All 4-cycles and even 6-cycles are removed in the Tanner graph,and the girth is not less than six.The decoding complexity as well as extension to irregular case is analyzed.Simulation results show that in AWGN and Rayleigh fading channels,the codes can achieve the same error performance as their counterpart PEG codes,and outperform the corresponding MacKay codes and array codes.     

Analysis of switched quantizer based on the quadratic spline functions

In this paper, an approximation of the optimal compressor function using the quadratic spline functions with 2L?=?8 segments is described. Since the quadratic spline with 2L?=?8 segments provides better approximation of the optimal compression function than quadratic spline with 2L?=?4 segments, capitalizing on the benefits of the obtained spline approximation, quantizer designing process is firstly performed for the so assumed number of segments and the Laplacian source of a unit variance. Then, to enhance the usability of the proposed model, the switched quantization technique is applied and a beneficial analysis is derived, providing insight in the robustness of the proposed quantizer performances with respect to the mismatch in designed for and applied to variances. Reached quality has been compared to another model from the literature, and it has been shown that the proposed model outperforms the previous model by almost 1.3?dB.     

Efficient circular arc interpolation based on active tolerance control

In this paper, we present an efficient sub-optimal algorithm for fitting smooth planar parametric curves by G¹ arc splines. To fit a parametric curve by an arc spline within a prescribed tolerance, we first sample a set of points and tangents on the curve adaptively as well as with enough density, so that an interpolation biarc spline curve can be with any desired high accuracy. Then, we construct new biarc curves interpolating local triarc spirals explicitly based on the control of permitted tolerances. To reduce the segment number of fitting arc spline as much as possible, we replace the corresponding parts of the spline by the new biarc curves and compute active tolerances for new interpolation steps. By applying the local biarc curve interpolation procedure recursively and sequentially, the result circular arcs with no radius extreme are minimax-like approximation to the original curve while the arcs with radius extreme approximate the curve parts with curvature extreme well too, and we obtain a near optimal fitting arc spline in the end. Even more, the fitting arc spline has the same end points and end tangents with the original curve, and the arcs will be jointed smoothly if the original curve is composed of several smooth connected pieces. The algorithm is easy to be implemented and generally applicable to circular arc interpolation problem of all kinds of smooth parametric curves. The method can be used in wide fields such as geometric modeling, tool path generation for NC machining and robot path planning, etc. Several numerical examples are given to show the effectiveness and efficiency of the method.     

分数阶B样条小波域的图像变分去噪

分数阶B样条具有分数阶逼近,可以更好地刻画图像纹理部分。将分数阶B样条小波推广到二维领域,利用分数阶B样条小波进行图像阈值去噪,提出了分数阶B样条小波域图像去噪的变分模型。同传统小波函数与全变差结合模型比较,分数阶B样条小波在保持纹理和去噪方面得到了明显改进。