End conditions for interpolatory septic splines

In this paper, septic spline is defined for interpolation at equally spaced knots along with the end conditions required to complete the definition of spline. These conditions are in terms of given functional values at the knots and lead to uniform convergence of O(h ⁸), throughout the interval of interpolation. The main objective of defining the end conditions for septic spline is to use the septic spline for the solutions of sixth-order boundary value problems, being discussed in the subsequent paper.     

An automatic adjustment procedure for rational splines

This paper presents the development of an adjustment method for the rational spline. The adjustment of the spline is completed by tensioning each internal section until the spline conforms to user defined data in a way which is both accurate and esthetically pleasing. The tensioning rationale for the rational spline is developed for use in an interactive mode with minicomputer and graphics terminal, but may also be used in a batch environment.     

An algorithm for computing smoothing splines in tension

A characterization of smoothing splines is derived which leads to procedures using either locally defined bases or small support bases. Difficulties in trying to compute the latter for splines in tension are discussed. A smoothing algorithm which avoids these difficulties by using locally defined bases is presented.     

An algorithm for constructing convexity and monotonicity-preserving splines in tension

In this paper we consider the problem of constructing an interpolatory spline in tension that matches the convexity and monotonicity properties of the data. In this connection, an algorithm is presented relying on the asymptotic properties of the splines in tension and making use of the generalized Newton-Raphson methods developed by Ben-Israel. The numerical performance of the proposed algorithm is tested and discussed for several data sets.     

Local control of interval tension using weighted splines

Cubic spline interpolation and B-spline sums are useful and powerful tools in computer aided design. These are extended by weighted cubic splines which have tension controls that allow the user to tighten or loosen the curve on intervals between interpolation points. The weighted spline is a C¹ piecewise cubic that minimizes a variational problem similar to one that a C² cubic spline minimizes. A B-spline like basis is constructed for weighted splines where each basis function is nonnegative and nonzero only on four intervals. The basis functions sum up identically to one, thus curves generated by summing control points multiplied by the basis functions have the convex hull property. Different weights are built into the basis functions so that the control point curves are piecewise cubics with local control of interval tension. If all weights are equal, then the weighted spline is the C² cubic spline and the basis functions are B-splines.     

Automatic feedrate adjustment for pocket machining

As high-speed machining and unmanned machining become common, the demand for cutting-load regularization increases, so NC machining can be more efficient. To be presented is a simple cutting-load regularization method for pocket machining. As the conventional off-line approaches where cutting-load is predicted and cutting parameters are adjusted before actual cutting, the proposed method requires a cutting force model, which is quite simplified with the function of two independent variables. One is the geometric measure so called 2D chip-load (cutter-engagement angle or effective cutting depth), and the other is the feedrate. Based on the 2D chip-load analysis for the concave line-line segment of the NC tool path, the adjusted feedrate is calculated by using the simplified-cutting force model (SCFM) obtained by the cutting experiment with a tool dynamometer. The concept of the automatic feedrate adjustment (AFA) method to be proposed is very simple, and the implementation requires little effort. Furthermore, the proposed method does not need much calculation time because there are no complex calculations or cutting simulation.     

Automatic model construction and pose estimation from photographsusing triangular splines

This paper addresses the automatic construction of complex spline object models from a few photographs. Our approach combines silhouettes from registered images to construct a G¹-continuous triangular spline approximation of an object with unknown topology. We apply a similar optimization procedure to estimate the pose of a modeled object from a single image. Experimental examples of model construction and pose estimation are presented for several complex objects     

Automatic computation of hierarchical biquadratic smoothing splines with minimum GCV

A computationally efficient numerical strategy for fitting approximate minimum GCV bivariate thin plate smoothing splines to large noisy data sets was developed. The procedure discretises the bivariate thin plate smoothing spline equations using biquadratic B-splines and uses a nested grid SOR iterative strategy to solve the discretised system. For efficient optimisation, the process incorporates a double iteration that simultaneously updates both the discretised solution and the estimate of the minimum GCV smoothing parameter. The GCV was estimated using a minimum variance stochastic estimator of the trace of the influence matrix associated with the fitted spline surface. A Taylor series expansion was used to estimate the smoothing parameter that minimises the GCV estimate. The computational cost of the procedure is optimal in the sense that it is proportional to the number of grid points supporting the fitted biquadratic spline. Convergence was improved by adding a first order correction to the solution estimate after each smoothing parameter update. The algorithm was tested on several simulated data sets with varying spatial complexity and noise level. An accurate approximation to the analytic minimum GCV thin plate smoothing spline was obtained in all cases.     

硬件测试中自动控制板卡电压拉偏的方法

在硬件板卡设计中,对硬件板卡的供电电压范围都有一定的要求,为了方便验证同一批次产品中每张板卡供电电压范围的一致性及板卡跌落电压检测电路是否能有效工作,利用TPS5430DDA开关电源芯片提出了一种板卡拉偏电压自动控制的方法。该方法易于实现,电路简单,可应用于整个生产测试环节。     

Control vectors for splines

Traditionally, modelling using spline curves and surfaces is facilitated by control points. We propose to enhance the modelling process by the use of control vectors. This improves upon existing spline representations by providing such facilities as modelling with local (semi-sharp) creases, vanishing and diagonal features, and hierarchical editing. While our prime interest is in surfaces, most of the ideas are more simply described in the curve context. We demonstrate the advantages provided by control vectors on several curve and surface examples and explore avenues for future research on control vectors in the contexts of geometric modelling and finite element analysis based on splines, and B-splines and subdivision in particular.     

Refinable interpolatory and quasi-interpolatory operators

In this paper, we present a new class of operators, which are refinable, quasi-interpolatory and satisfy some interpolation conditions. The refinability is achieved by using as functional bases the B-bases corresponding to totally positive refinable functions. We analyze the main properties of the constructed refinable operators and give some convergence results. Some examples are also displayed.     

Exact and interpolatory quadratures for curvature tensor estimation

The computation of the curvature of smooth surfaces has a long history in differential geometry and is essential for many geometric modeling applications such as feature detection. We present a novel approach to calculate the mean curvature from arbitrary normal curvatures. Then, we demonstrate how the same method can be used to obtain new formulae to compute the Gaussian curvature and the curvature tensor. The idea is to compute the curvature integrals by a weighted sum by making use of the periodic structure of the normal curvatures to make the quadratures exact. Finally, we derive an approximation formula for the curvature of discrete data like meshes and show its convergence if quadratically converging normals are available.     

Automatic knot adjustment using an artificial immune system for B-spline curve approximation

Reverse engineering transforms real parts into engineering concepts or models. First, sampled points are mapped from the object’s surface by using tools such as laser scanners or cameras. Then, the sampled points are fitted to a free-form surface or a standard shape by using one of the geometric modeling techniques. The curves on the surface have to be modeled before surface modeling. In order to obtain a good B-spline curve model from large data, the knots are usually respected as variables. A curve is then modeled as a continuous, nonlinear and multivariate optimization problem with many local optima. For this reason it is very difficult to reach a global optimum. In this paper, we convert the original problem into a discrete combinatorial optimization problem like in Yoshimoto et al. [F. Yoshimoto, M. Moriyama, T. Harada, Automatic knot placement by a genetic algorithm for data fitting with a spline, in: Proceedings of the International Conference on Shape Modeling and Applications, IEEE Computer Society Press, 1999, pp. 162-169] and Sarfraz et al. [M. Sarfraz, S.A. Raza, Capturing outline of fonts using genetic algorithm and splines, in: Fifth International Conference on Information Visualisation (IV’01), 2001, pp. 738-743]. Then, we suggest a new method that solves the converted problem by artificial immune systems. We think the candidates of the locations of knots as antibodies. We define the affinity measure benefit from Akaike’s Information Criterion (AIC). The proposed method determines the appropriate location of knots automatically and simultaneously. Furthermore, we do not need any subjective parameter or good population of initial location of knots for a good iterative search. Some examples are also given to demonstrate the efficiency and effectiveness of our method.     

Bivariate splines for fluid flows

We discuss numerical approximations of the 2D steady-state Navier-Stokes equations in stream function formulation using bivariate splines of arbitrary degree d and arbitrary smoothness r with r<d. We derive the discrete Navier-Stokes equations in terms of B-coefficients of bivariate splines over a triangulation, with curved boundary edges, of any given domain. Smoothness conditions and boundary conditions are enforced through Lagrange multipliers. The pressure is computed by solving a Poisson equation with Neumann boundary conditions. We have implemented this approach in MATLAB and our numerical experiments show that our method is effective. Numerical simulations of several fluid flows will be included to demonstrate the effectiveness of the bivariate spline method.     

A 4-point interpolatory subdivision scheme for curve design

A 4-point interpolatory subdivision scheme with a tension parameter is analysed. It is shown that for a certain range of the tension parameter the resulting curve is C¹. The role of the tension parameter is demonstrated by a few examples. The application to surfaces and some further potential generalizations are discussed.     

New symmetric interpolatory quadrature formulas

We introduce two families of symmetric, interpolatory integration formulas on the interval [−1,1]. These formulas, related to the class of recursive monotone stable (RMS) formulas, allow the application of higher order or compound rules with an efficient reuse of computed function values. One family (SM) uses function values computed outside the integration interval, the other one (HR) uses derivative data. These formulas are evaluated using a practical test based on a tecnique for comparing automatic quadrature routine introduced by Lyness and Kaganove and improved by the authors. Work supported by CNR, Grant no. 93.00570, CT01     

Local splines

The paper presents local formulas of smooth continuation and interpolation for functions whose values are given at the nodal points of a nonuniform rectangular net.     

Quasi-nodal splines

Most spline interpolation operators such as the nodal spline operator introduced by de Villiers and Rohwer in 1987 interpolate data at spline knots. In this paper, we are going to present a method of how to construct a local spline interpolation operator that interpolates at sites that are different from the spline knots. These considerations result in the quasi-nodal spline interpolation operator of degree n that reproduces all polynomials of degree not exceeding n.