一类二次曲面上的参数样条

在空间四个有序数据点所确定的一个二次曲面上,可以构造一类特殊的曲线。给出了四个形状控制因子的有理基函数,以及通过研究其参数间的函数关系定义函数集,构造一类样条曲线,使得通过改变控制因子能任意精确地逼近控制多边形。这类样条曲线端点处满足一定切线方向和有界曲率,容易将它们拼接成一条逼近样条曲线。利用这些样条构造出逼近样条曲面,具有更多的自由度。     

一类有理逼近参数样条

在二次曲面上构造一种带有形状因子的有理参数样条曲线,该样条曲线能逼近所在的控制多边形,且有较好的几何特性,并且可以作升阶和降阶处理。分析其端点性质,便于拼接成光滑曲线,如果选取合适的形状因子,可以使得曲线连接成G2连续。     

局部调整插值点的三次样条曲线表示

给出了带局部形状参数的三次样条曲线生成方法．所给方法以Hermite型插值曲线和非均匀三次B样条曲线为特殊情形，将插值于控制点的曲线和逼近于控制多边形的非均匀B样条曲线统一起来．一个形状参数只影响两条曲线段，曲线表达式保持了三次Bezier曲线表达式的简单结构．改变形状参数的值或调整Bezier控制点，可以局部调整曲线的形状．基于所给样条曲线，给出了带局部形状参数的双三次样条曲面．     

两种带形状参数的曲线

本文构造了两种带参数的三角样条基,基于这两组基定义了两种三角样条曲线。与二次B样条曲线类似,这两种曲线的每一段都由相继的三个控制顶点生成。这两种曲线具有许多与二次B样条曲线类似的性质,但它们的连续性都比二次B样条曲线更好。对于等距节点,在一般情况下,这两种曲线都整体C3连续,在特殊条件下,它们都可达C5连续。两种曲线中的形状参数均有明确的几何意义,参数越大,曲线越靠近控制多边形。另外,当形状参数满足一定条件时,这两种曲线都具有比二次B样条曲线更好的对控制多边形的逼近性。运用张量积方法,将这两种曲线推广后所得到的曲面也具有较好的连续性。     

带参的三次三角多项式样条曲线

给出了带有参数λ的三次三角多项式样条曲线.与二次B样条曲线类似,曲线的每一段由相继的三个控制顶点生成;对于等距节点,在一般情形下,曲线达到了C1连续;而当λ=1时,曲线达到了C3连续.λ有明确的几何意义,λ越大,曲线越逼近控制多边形.实例表明,所给曲线为曲线/曲面的设计提供了一种有效的方法.     

二次均匀B样条曲线的双圆弧逼近方法*

提出了一种用双圆弧对二次均匀B样条曲线的分段逼近方法。首先,对一条具有n 1个控制顶点的二次均匀B样条曲线按照相邻两节点界定的区间分成n-1段只有三个控制顶点的二次均匀B样条曲线段;然后对每一曲线段构造一条双圆弧进行逼近。所构造的双圆弧满足端点及端点切向量条件,即双圆弧的两个端点分别是所逼近的曲线段的端点,而且双圆弧在两个端点处的切向量是所逼近的曲线段在端点处的单位切向量。同时,双圆弧的连接点是双圆弧连接点轨迹圆与其所逼近的曲线段的交点。这些新构造出来的双圆弧连接在一起构成了一条圆弧样条曲线,即二次均匀B样条曲线的逼近曲线。另外给出了逼近误差分析和实例说明。     

第二类均匀CB样条曲线

当α→0时,均匀CB样条曲线逼近相应阶的均匀B样条曲线,改变α的取值时,生成的均匀CB样条曲线逼近控制多边形的极限位置是相应阶的均匀B样条曲线.为了突破这种逼近极限,构造一种新的均匀CB样条曲线.先构造一种三次均匀CB样条基,再运用积分递归定义出任意阶的均匀CB样条基.由这些基构造的均匀CB样条曲线与原CB样条曲线具有类似的性质:凸包性、几何不变性、局部性、对称性等,但随着α取值的不同生成相应阶的均匀B样条两侧曲线,能够更好的逼近控制多边形.最后给出了用新的均匀CB样条曲线精确表示椭圆,圆,抛物线,螺旋线等.定义的新均匀CB样条拓展了均匀CB样条曲线曲面的造型能力.     

代数曲线的有理二次B样条逼近

基于代数曲线的合理分割,给出了曲线段的三角形凸包的描述.提出了以曲线段端点的两条切线确定控制多边形的方案.详细地讨论了代数曲线的分段有理二次B样条逼近算法.逼近曲线保持了原始曲线的一些重要几何性质,如单调性,凹凸性,G1连续性.数值实验表明,该算法提供了代数曲线近似参数化的一条有效途径.     

带形状参数的C~2连续类三次三角样条曲线

传统的三次均匀B样条曲线在给定控制顶点时其形状不能调整,以及不能精确表示圆锥曲线。针对三次均匀B样条曲线的不足,提出了一种带形状参数的C2连续的类三次三角样条曲线。该曲线不仅与三次均匀B样条曲线具有相似的性质,而且在控制顶点保持不变时其形状可通过形状参数的取值进行调整。在适当条件下,类三次三角样条曲线比三次均匀B样条曲线更能逼近于控制多边形,且能精确表示圆、椭圆、抛物线等圆锥曲线。     

带形状控制的自由曲线曲面参数样条

目的 样条曲线曲面的构造是工程制图中的一个重要部分。针对双曲抛物面上参数样条曲线的构造,在已有的研究基础上提出了一种样条方法使曲线曲面可以任意地逼近一个多边形或者一个网格。方法 在标准四面体内构造一个双曲抛物面,在该曲面上以基函数参数化的方法定义一种带形状参数的参数样条曲线曲面,样条基函数通过将双曲抛物面的有理参数化进行限定,生成单参数有理样条基函数。详细研究了样条的保形性及其端点性质。结果 样条曲线具有一个可变的形状控制因子,可以对曲线进行调整,能以任意精度逼近这个控制四边形或网格。对空间节点列,利用该样条可以生成G²-连续空间曲线,同样对于空间网格可以构造G²-连续的拟合曲面,它所对应的基函数可以是有理形式。结论 实验结果表明,本文在笔者已有的研究基础上提出的参数样条曲线可以通过重心坐标系变换适应为任意的四边形,除了空间四面体内的样条曲线,四面体退化成四边形同样可实现。     

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.     

Hierarchical B-splines on regular triangular partitions

Multivariate splines have a wide range of applications in function approximation, finite element analysis and geometric modeling. They have been extensively studied in the last several decades, and specially the theory on bivariate B-splines over regular triangular partition is well developed. However, the above mentioned splines do not have local refinement property – a property that is very important in adaptive function approximation and level of detailed representation of geometric models. In this paper, we introduce the concept of hierarchial bivariate splines over regular triangular partitions and construct basis functions of such spline space that satisfy some nice properties. We provide some examples of hierarchical splines over triangular partitions in surface fitting and in solving numerical PDEs, and the results turn out to be promising.     

Optimal Spline Approximation via ℓ0‐Minimization

Splines are part of the standard toolbox for the approximation of functions and curves in ?^d. Still, the problem of finding the spline that best approximates an input function or curve is ill‐posed, since in general this yields a “spline” with an infinite number of segments. The problem can be regularized by adding a penalty term for the number of spline segments. We show how this idea can be formulated as an ?₀‐regularized quadratic problem. This gives us a notion of optimal approximating splines that depend on one parameter, which weights the approximation error against the number of segments. We detail this concept for different types of splines including B‐splines and composite Bézier curves. Based on the latest development in the field of sparse approximation, we devise a solver for the resulting minimization problems and show applications to spline approximation of planar and space curves and to spline conversion of motion capture data.     

Reconstruction of volume data with quadratic super splines

We propose a new approach to reconstruct nondiscrete models from gridded volume samples. As a model, we use quadratic trivariate super splines on a uniform tetrahedral partition. We discuss the smoothness and approximation properties of our model and compare to alternative piecewise polynomial constructions. We observe, as a nonstandard phenomenon, that the derivatives of our splines yield optimal approximation order for smooth data, while the theoretical error of the values is nearly optimal due to the averaging rules. Our approach enables efficient reconstruction and visualization of the data. As the piecewise polynomials are of the lowest possible total degree two, we can efficiently determine exact ray intersections with an isosurface for ray-casting. Moreover, the optimal approximation properties of the derivatives allow us to simply sample the necessary gradients directly from the polynomial pieces of the splines. Our results confirm the efficiency of the quasi-interpolating method and demonstrate high visual quality for rendered isosurfaces.     

Boosting additive models using component-wise P-Splines

An efficient approximation of L₂ Boosting with component-wise smoothing splines is considered. Smoothing spline base-learners are replaced by P-spline base-learners, which yield similar prediction errors but are more advantageous from a computational point of view. A detailed analysis of the effect of various P-spline hyper-parameters on the boosting fit is given. In addition, a new theoretical result on the relationship between the boosting stopping iteration and the step length factor used for shrinking the boosting estimates is derived.     

A Bivariate Spline Method for Second Order Elliptic Equations in Non-divergence Form

A bivariate spline method is developed to numerically solve second order elliptic partial differential equations in non-divergence form. The existence, uniqueness, stability as well as approximation properties of the discretized solution will be established by using the well-known Ladyzhenskaya–Babuska–Brezzi condition. Bivariate splines, discontinuous splines with smoothness constraints are used to implement the method. Computational results based on splines of various degrees are presented to demonstrate the effectiveness and efficiency of our method.     

Family of quadratic spline difference schemes for a convection–diffusion problem

We consider the finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. It is discretized using quadratic splines as approximation functions, equations with various piecewise constant coefficients as collocation equations and a piecewise uniform mesh of Shishkin type. The family of schemes is derived using the collocation method. The numerical methods developed here are non-monotone and therefore apart from the consistency error we use Green's grid function analysis to prove uniform convergence. We prove the almost first order of convergence and furthermore show that some of the schemes have almost second-order convergence. Numerical experiments presented in the paper confirm our theoretical results.     

Two approximation methods to synthesize the power spectrum of fractional Gaussian noise

The simplest models with long-range dependence (LRD) are self-similar processes. Self-similar processes have been formally considered for modeling packet traffic in communication networks. The fractional Gaussian noise (FGN) is a proper example of exactly self-similar processes. Several numeric approximation methods are considered and reviewed, two methods are found that are able to provide a better accuracy and less running time than previous approximation methods for synthesizing the power spectrum of FGN. The first method is based on a second-order approximation. It is demonstrated that a parabolic curve can be indirectly used to approximate the power spectrum of FGN. The second method is based on cubic splines. Despite the fact that splines cannot be used directly to approximate the power spectrum of FGN, they can, however, considerably simplify the calculations while maintaining high accuracy. Both of the methods proposed can be used to estimate the Hurst parameter using Whittle's estimator. Additionally, they can be used on synthesis of LRD sequences.     

Algorithm by hybrid splines

Hybrid splines whose bases are constructed by multi-order B-splines are presented. B-splines of order 2 are induced for representing corners, and those of orders 3 or 4 for representing curved parts. The hybrid splines can represent not only smooth parts (class C^m?22:m≥3) but also corners (class C°): Hybrid splines of order 2 and 4 cannot be represented by conventional splines with multiple knots. Satisfactory approximation results are obtained by hybrid splines of order 2 and 4 determined by a least-squares method.     

Isotropic Energies, Filters and Splines for Vector Field Regularization

The aim of this paper is to propose new regularization and filtering techniques for dense and sparse vector fields, and to focus on their application to non-rigid registration. Indeed, most of the regularization energies used in non-rigid registration operate independently on each coordinate of the transformation. The only common exception is the linear elastic energy, which enables cross-effects between coordinates. Cross-effects are yet essential to give realistic deformations in the uniform parts of the image, where displacements are interpolated.In this paper, we propose to find isotropic quadratic differential forms operating on a vector field, using a known theorem on isotropic tensors, and we give results for differentials of order 1 and 2. The quadratic approximation induced by these energies yields a new class of vectorial filters, applied numerically in the Fourier domain. We also propose a class of separable isotropic filters generalizing Gaussian filtering to vector fields, which enables fast smoothing in the spatial domain. Then we deduce splines in the context of interpolation or approximation of sparse displacements. These splines generalize scalar Laplacian splines, such as thin-plate splines, to vector interpolation. Finally, we propose to solve the problem of approximating a dense and a sparse displacement field at the same time. This last formulation enables us to introduce sparse geometrical constraints in intensity based non-rigid registration algorithms, illustrated here on intersubject brain registration.