基于最优控制理论的微分算子插值样条构造性质的新证法

本文将最优控制理论用于微分算子广义插值样条构造性质的研究.通过将微分算子插值样条描述成线性最优控制问题,用带状态约束的一类最优控制的必要条件推导出微分算子插值样条的构造与连续性质.这一方法不仅较容易地导出了微分算子插值样条熟知的构造和连续性质,而且还得到了样条经过微分算子作用后在节点处的跃度公式.进一步揭示了微分算子插值样条与最优控制理论的联系,为带障碍的算子插值样条构造性质的研究提供了新的方法.     

基于三次样条插值的硅压阻式压力传感器的温度补偿

传感器的零点温度漂移、灵敏度温度漂移和非线性误差是影响传感器性能的主要因素,如何能使该类误差得到有效补偿对于提高其性能有重要意义。提出了基于三次样条曲线插值的温度补偿方法,改进了传统三次样条曲线插值的补偿方法,分别对传感器的零点、灵敏度以及非线性进行补偿,用这种方法对测压范围为1.0140×105 Pa～3.0140×105 Pa,温度范围为-20℃～+60℃的硅压阻式压力传感器的实验标定结果进行了温度补偿。通过比较传统三次样条插值补偿后的传感器输出信号,验证了使用改进后的三次样条曲线插值法的补偿效果更好。这种方法为高精度压力传感器的温度补偿提供了一种有价值的理论依据。     

Subdivision algorithms for the generation of box spline surfaces

This paper proposes a general approach to subdivision algorithms used in interactive computer aided design for splines which are linear combinations of translates of any box splines. We show how these algorithms can be used for efficient generation of the corresponding spline surfaces. Our results extend several known special cases.     

双变量Box样条小波在图像处理中的应用

将单变量样条扩展到双变量Box样条,并给出Box样条在图像处理中极为有用的几条性质,然后,提出了一个使用一类双变量Box样条来计算滤波器库的方法,这样得到的滤波器库在4频带分析/综合框架中具有完全重构的性质,可用来十分有效地分解合成图像。最后,给出应用此方法的一个实例。     

Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics

This paper compares two techniques for the approximation of the offsets to a given planar curve. The two methods are based on approximate conversion of the planar curve into circular splines and Pythagorean hodograph (PH) splines, respectively. The circular splines are obtained using a novel variant of biarc interpolation, while the PH splines are constructed via Hermite interpolation of C¹ boundary data.We analyze the approximation order of both conversion procedures. As a new result, the C¹ Hermite interpolation with PH quintics is shown to have approximation order 4 with respect to the original curve, and 3 with respect to its offsets. In addition, we study the resulting data volume, both for the original curve and for its offsets. It is shown that PH splines outperform the circular splines for increasing accuracy, due to the higher approximation order.     

利用Box样条构造完全重构模型的研究

针对图象处理中的图象重构问题,结合多频带DFT滤波器库和双变量Box样条构造了一个完全重构模型,在介绍了DFT滤波器库后,给出了形成完全重构模型的条件,通过双变量Box样条及其在图象处理极为有用的几条性质的分析,提出了一类双变量Box样条,使用该样条即可构造分解/重构模型,实验证明,该模型能满足完全重构条件,并可有效分解合成图象,最后,给出了应用此模型的实验结果。     

Polycube splines

This paper proposes a new concept of polycube splines and develops novel modeling techniques for using the polycube splines in solid modeling and shape computing. Polycube splines are essentially a novel variant of manifold splines which are built upon the polycube map, serving as its parametric domain. Our rationale for defining spline surfaces over polycubes is that polycubes have rectangular structures everywhere over their domains, except a very small number of corner points. The boundary of polycubes can be naturally decomposed into a set of regular structures, which facilitate tensor-product surface definition, GPU-centric geometric computing, and image-based geometric processing. We develop algorithms to construct polycube maps, and show that the introduced polycube map naturally induces the affine structure with a finite number of extraordinary points. Besides its intrinsic rectangular structure, the polycube map may approximate any original scanned data-set with a very low geometric distortion, so our method for building polycube splines is both natural and necessary, as its parametric domain can mimic the geometry of modeled objects in a topologically correct and geometrically meaningful manner. We design a new data structure that facilitates the intuitive and rapid construction of polycube splines in this paper. We demonstrate the polycube splines with applications in surface reconstruction and shape computing.     

Parametric and semi-parametric approaches in the analysis of short-term effects of air pollution on health

Generalized additive models (GAMs) have become the standard tool for the analysis of short-term effects of air pollution on human health. Usually, the confounding effect of seasonality and long-term trend is described by flexible parametric or non-parametric functions of calendar time. Two different modeling strategies, i.e. GAM with penalized regression splines and GAM with regression splines, were compared by means of a simulation study, addressing attention to the inference on air pollutant effect. Simulation results indicated that GAM with regression splines provides negligibly biased estimates of air pollutant effect and it is robust to misspecification of the degrees of freedom of the spline. GAM with penalized regression splines requires a certain amount of undersmoothing in order to reduce the bias of the estimates and to improve the coverage of confidence intervals. These findings agree with asymptotic results developed in the context of partially splined models.     

Some smoothness conditions and conformality conditions for bivariate quartic and quintic splines

This paper is concerned with a study of some new formulations of smoothness conditions and conformality conditions for multivariate splines in terms of B-net representation. In the bivariate setting, a group of new parameters of bivariate quartic and quintic polynomials over a planar simplex is introduced, new formulations of smoothness conditions of bivariate quartic C ¹ splines and quintic C ² splines are given, and the conformality conditions of bivariate quartic C ¹ splines are simplified. Received: February 1998 / Accepted: August 1998     

带参数的四次Hermite插值样条

为了克服标准三次Hermite插值样条的不足,给出了一种带参数的四次Hermite插值样条,具有标准三次Hermite插值样条完全相同的性质。在插值条件给定时,四次Hermite插值样条的形状可通过改变参数的取值进行调控。通过选择合适的参数,四次Hermite曲线能达到C2连续,而且其整体逼近效果要好于标准三次Hermite插值样条。所提出的新样条进一步丰富了Hermite插值样条理论,也为工程中插值曲线曲面的构造提供了一种新方法。     

Splines in the parameter domain of surfaces and their application in filament winding

For non-axisymmetric filament winding in the composite industry, it is difficult to automatically design fiber paths with uniform distribution, full coverage, and non-slippage. Geodesics and semi-geodesics are commonly used as non-slippage trajectories in the fiber path design. However, the uniqueness of these curves from the initial conditions severely restricts the freedom of design, which plays an important role in producing high performance parts. This paper presents a new class of trajectories with more freedom by generalizing splines, and gives the conditions to make these splines on cylinders and cones stable. The proposed trajectory has been applied to filament winding on elbows. Practical results show that the fiber distribution is excellent, and the finished product has extremely uniform thickness.     

一种基于B样条的文字自由变形算法

本文提出了一种以 B样条为基础的平面文字自由变形算法 .变形后文字的闭包是由若干 B样条围成的区域 .该算法通过 B样条对目标区域进行合理分割 ,实现了文字的均匀变形 ,得到了较好的变形效果     

Fitting smoothing splines to time-indexed,noisy points on nonlinear manifolds

We address the problem of estimating full curves/paths on certain nonlinear manifolds using only a set of time-indexed points, for use in interpolation, smoothing, and prediction of dynamic systems. These curves are analogous to smoothing splines in Euclidean spaces as they are optimal under a similar objective function, which is a weighted sum of a fitting-related (data term) and a regularity-related (smoothing term) cost functions. The search for smoothing splines on manifolds is based on a Palais metric-based steepest-decent algorithm developed in Samir et al. [38]. Using three representative manifolds: the rotation group for pose tracking, the space of symmetric positive-definite matrices for DTI image analysis, and Kendall's shape space for video-based activity recognition, we demonstrate the effectiveness of the proposed algorithm for optimal curve fitting. This paper derives certain geometrical elements, namely the exponential map and its inverse, parallel transport of tangents, and the curvature tensor, on these manifolds, that are needed in the gradient-based search for smoothing splines. These ideas are illustrated using experimental results involving both simulated and real data, and comparing the results to some current algorithms such as piecewise geodesic curves and splines on tangent spaces, including the method by Kume et al. [24].     

A new approach to linear regression with multivariate splines

A new methodology for creating highly accurate, static nonlinear maps from scattered, multivariate data is presented. This new methodology uses the B-form polynomials of multivariate simplex splines in a new linear regression scheme. This allows the use of standard parameter estimation techniques for estimating the B-coefficients of the multivariate simplex splines. We present a generalized least squares estimator for the B-coefficients, and show how the estimated B-coefficient variances lead to a new model quality assessment measure in the form of the B-coefficient variance surface. The new modeling methodology is demonstrated on a nonlinear scattered bivariate dataset.     

Calculating with box splines

Box splines are multivariate splines over regular grids. Two recursion formulas for box splines are developed: (1) a Mansfield-de Boor-like expression of box splines as linear combinations of box splines of lower degree and (2) a deBoor-like reduction of the net of box spline control points. The ideas follow those from the paper by deBoor in 1972. The proofs are geometrical and simple.     

Manifold splines with a single extraordinary point

This paper develops a novel computational technique to define and construct manifold splines with only one singular point by employing the rigorous mathematical theory of Ricci flow. The central idea and new computational paradigm of manifold splines are to systematically extend the algorithmic pipeline of spline surface construction from any planar domain to an arbitrary topology. As a result, manifold splines can unify planar spline representations as their special cases. Despite its earlier success, the existing manifold spline framework is plagued by the topology-dependent, large number of singular points (i.e., |2g−2| for any genus-g surface), where the analysis of surface behaviors such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we devise new mathematical tools so that manifold splines can now be constructed with only one singular point, reaching their theoretic lower bound of singularity for real-world applications. Our new algorithm is founded upon the concept of discrete Ricci flow and associated techniques. First, Ricci flow is employed to compute a special metric of any manifold domain (serving as a parametric domain for manifold splines), such that the metric becomes flat everywhere except at one point. Then, the metric naturally induces an affine atlas covering the entire manifold except this singular point. Finally, manifold splines are defined over this affine atlas. The Ricci flow method is theoretically sound, and practically simple and efficient. We conduct various shape experiments and our new theoretical and algorithmic results alleviate the modeling difficulty of manifold splines, and hence, promote the widespread use of manifold splines in surface and solid modeling, geometric design, and reverse engineering.     

Generalized taylor splines

A generalization of polynomial spline theory is presented which allows the construction of splines of any order. The kernel is based on a Taylor series expansion of the governing equations and Gaussian elimination is used for the numerical calculation of the splines coefficients. The general steps are described and tests are performed for the classical cubic and quintic splines and also for an unorthodox 25th order spline.     

Sampling of signals and their reconstruction

This paper gives error bounds when a stochastic process or a deterministic signal is sampled and reconstruction is done either by piecewise straight lines or cubic splines. In the case of cubic splines the error bounds derived here take into account the number of samples. Using the results presented here it is possible to select the sampling interval for a given error.     

Algorithms for smoothing data on the sphere with tensor product splines

Algorithms are presented for fitting data on the sphere by using tensor product splines which satisfy certain boundary constraints. First we consider the least-squares problem when the knots are given. Then we discuss the construction of smoothing splines on the sphere. Here the knots are located automatically. A Fortran IV implementation of these two algorithms is described.     

Subdivision surfaces for CAD—an overview

Subdivision surfaces refer to a class of modelling schemes that define an object through recursive subdivision starting from an initial control mesh. Similar to B-splines, the final surface is defined by the vertices of the initial control mesh. These surfaces were initially conceived as an extension of splines in modelling objects with a control mesh of arbitrary topology. They exhibit a number of advantages over traditional splines. Today one can find a variety of subdivision schemes for geometric design and graphics applications. This paper provides an overview of subdivision surfaces with a particular emphasis on schemes generalizing splines. Some common issues on subdivision surface modelling are addressed. Several key topics, such as scheme construction, property analysis, parametric evaluation and subdivision surface fitting, are discussed. Some other important topics are also summarized for potential future research and development. Several examples are provided to highlight the modelling capability of subdivision surfaces for CAD applications.