Bézier 曲线的多项式重新参数化检测

研究了一种用于精确检测一条Bézier 曲线的次数是否可以通过多项式重新参数化 降低的算法。该算法对任意一条Bézier 曲线,将重新参数化前后的基函数的关系用方程组的形 式表达,但不需要解方程,而是通过系数表示的金字塔算法直接计算,可以精确求出用于重新 参数化的多项式和降低次数后的Bézier 曲线的控制顶点,并且该重新参数化的多项式在相差一 个线性变换的前提下是唯一的。通过实例应用,该算法运算速度较之前的算法快。     

参数曲线的最优参数化

利用有理重新参数化的自由度求解参数曲线的最优参数化问题,提出一种度量曲线的参数速度与弧长参数化接近程度的方法.利用该方法求得的最优参数化在曲线的重新参数化曲线族中,参数速度偏离单位速度的最大值达到最小.最后,通过计算实例对该方法与其他算法得到的最优参数化的参数速度进行了比较.     

Semiparametric indirect utility and consumer demand

A semiparametric model of consumer demand is considered. In the model, the indirect utility function is specified as a partially linear, where utility is nonparametric in expenditure and parametric (with fixed- or varying-coefficients) in prices. Because the starting point is a model of indirect utility, rationality restrictions like homogeneity and Slutsky symmetry are easily imposed. The resulting model for expenditure shares (as functions of expenditures and prices) is locally given by a fraction whose numerator is partially linear, but whose denominator is nonconstant and given by the derivative of the numerator. The basic insight is that given a local polynomial model for the numerator, the denominator is given by a lower order local polynomial. The model can thus be estimated using modified versions of local polynomial modeling techniques. For inference, a new asymmetric version of the wild bootstrap is introduced. Monte Carlo evidence that the proposed technique’s work is provided as well as an implementation of the model on Canadian consumer expenditure and price micro-data.     

Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials

In a recent article, Ge et al. (1997) identify a special class of rational curves (Harmonic Rational Bézier (HRB) curves) that can be reparameterized in sinusoidal form. Here we show how this family of curves strongly relates to the class of p-Bézier curves, curves easily expressible as single-valued in polar coordinates. Although both subsets do not coincide, the reparameterization needed in both cases is exactly the same, and the weights of a HRB curve are those corresponding to the representation of a circular arc as a p-Bézier curve. We also prove that a HRB curve can be written as a combination of its control points and certain Bernstein-like trigonometric basis functions. These functions form a normalized totally positive B-basis (that is, the basis with optimal shape preserving properties) of the space of trigonometric polynomials {1, sint, cost, …. sinmt, cosmt} defined on an interval of length < π.     

A framework for improving uniformity of parameterizations of curves

We defne quasi-speed as a generalization of linear speed and angular speed for parameterizations of curves and use the uniformity of quasi-speed to measure the quality of the parameterizations.With such conceptual setting,a general framework is developed for studying uniformity behaviors under reparameterization via proper parameter transformation and for computing reparameterizations with improved uniformity of quasispeed by means of optimal single-piece,C0piecewise,and C1piecewise M¨obius transformations.Algorithms are described for uniformity-improved reparameterization using diferent Mo¨bius transformations with diferent optimization techniques.Examples are presented to illustrate the concepts,the framework,and the algorithms.Experimental results are provided to validate the framework and to show the efciency of the algorithms.     

Admissible digit sets

We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “digit set” yields an admissible representation of [0,+∞]$[0,+\infty ]$. Furthermore, we establish the productivity and correctness of the homographic algorithm for such “admissible” digit sets. We present the Stern–Brocot representation and a modification of same as a working example throughout.     

Covariant-Conics Decomposition of Quartics for 2D Shape Recognition and Alignment

This paper outlines a new geometric parameterization of 2D curves where parameterization is in terms of geometric invariants and parameters that determine intrinsic coordinate systems. This new approach handles two fundamental problems: single-computation alignment, and recognition of 2D shapes under Euclidean or affine transformations. The approach is model-based: every shape is first fitted by a quartic represented by a fourth degree 2D polynomial. Based on the decomposition of this equation into three covariant conics, we are able, in both the Euclidean and the affine cases, to define a unique intrinsic coordinate system for non-singular bounded quartics that incorporates usable alignment information contained in the polynomial representation, a complete set of geometric invariants, and thus an associated canonical form for a quartic. This representation permits shape recognition based on 11 Euclidean invariants, or 8 affine invariants. This is illustrated in experiments with real data sets.     

Qubit Geometry and Conformal Mapping

Identifying the Bloch sphere with the Riemann sphere (the extended complex plane), we obtain relations between single qubit unitary operations and Möbius transformations on the extended complex plane. PACS: 03.67.-a, 03.67.Lx, 03.67.Hk     

Sparse interpolation of multivariate rational functions

Consider the black box interpolation of a τ-sparse, n-variate rational function f, where τ is the maximum number of terms in either numerator or denominator. When numerator and denominator are at most of degree d, then the number of possible terms in f is O(dⁿ) and explodes exponentially as the number of variables increases. The complexity of our sparse rational interpolation algorithm does not depend exponentially on n anymore. It still depends on d because we densely interpolate univariate auxiliary rational functions of the same degree. We remove the exponent n and introduce the sparsity τ in the complexity by reconstructing the auxiliary function’s coefficients via sparse multivariate interpolation.The approach is new and builds on the normalization of the rational function’s representation. Our method can be combined with probabilistic and deterministic components from sparse polynomial black box interpolation to suit either an exact or a finite precision computational environment. The latter is illustrated with several examples, running from exact finite field arithmetic to noisy floating point evaluations. In general, the performance of our sparse rational black box interpolation depends on the choice of the employed sparse polynomial black box interpolation. If the early termination Ben-Or/Tiwari algorithm is used, our method achieves rational interpolation in O(τd) black box evaluations and thus is sensitive to the sparsity of the multivariate f.     

一种DCT变换的三维网格物体盲水印方法

提出了一种基于DCT变换的3D网格物体鲁棒性盲水印方案。首先将3D物体模型转换到仿射不变空间,抽取三维物体重心到顶点的距离生成一个一维的离散信号,将该离散信号进行DCT变换,改变其系数以嵌入水印。然后经过逆DCT变换生成带水印的3D物体模型。在仿射不变空间下,实现了3D物体模型对平移、旋转、比例变换的鲁棒性,采用DCT变换使3D模型具有很强的水印不可见性,而且具有一定的噪声鲁棒性。试验结果也表明该方法不仅对于旋转、平移、比例变换具有很强的鲁棒性,而且具有良好的水印不可见性。     

Dual representation of spatial rational Pythagorean-hodograph curves

In this paper, the dual representation of spatial parametric curves and its properties are studied. In particular, rational curves have a polynomial dual representation, which turns out to be both theoretically and computationally appropriate to tackle the main goal of the paper: spatial rational Pythagorean-hodograph curves (PH curves). The dual representation of a rational PH curve is generated here by a quaternion polynomial which defines the Euler–Rodrigues frame of a curve. Conditions which imply low degree dual form representation are considered in detail. In particular, a linear quaternion polynomial leads to cubic or reparameterized cubic polynomial PH curves. A quadratic quaternion polynomial generates a wider class of rational PH curves, and perhaps the most useful is the ten-parameter family of cubic rational PH curves, determined here in the closed form.     

Node-pancyclicity and edge-pancyclicity of hypercube variants

Twisted cubes, crossed cubes, Möbius cubes, and locally twisted cubes are some of the widely studied hypercube variants. The 4-pancyclicity of twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes and the 4-edge-pancyclicity of twisted cubes, crossed cubes, Möbius cubes are proven in [C.P. Chang, J.N. Wang, L.H. Hsu, Topological properties of twisted cube, Inform. Sci. 113 (1999) 147-167; C.P. Chang, T.Y. Sung, L.H. Hsu, Edge congestion and topological properties of crossed cubes, IEEE Trans. Parall. Distr. 11 (1) (2000) 64-80; J. Fan, Hamilton-connectivity and cycle embedding of the Möbius cubes, Inform. Process. Lett. 82 (2002) 113-117; X. Yang, G.M. Megson, D.J. Evans, Locally twisted cubes are 4-pancyclic, Appl. Math. Lett. 17 (2004) 919-925; J. Fan, N. Yu, X. Jia, X. Lin, Embedding of cycles in twisted cubes with edge-pancyclic, Algorithmica, submitted for publication; J. Fan, X. Lin, X. Jia, Node-pancyclic and edge-pancyclic of crossed cubes, Inform. Process. Lett. 93 (2005) 133-138; M. Xu, J.M. Xu, Edge-pancyclicity of Möbius cubes, Inform. Process. Lett. 96 (2005) 136-140], respectively. It should be noted that 4-edge-pancyclicity implies 4-node-pancyclicity which further implies 4-pancyclicity. In this paper, we outline an approach to prove the 4-edge-pancyclicity of some hypercube variants and we prove in particular that Möbius cubes and locally twisted cubes are 4-edge-pancyclic.     

有理三角曲面的分片线性逼近

有理三角曲面的分片线性逼近在参数曲面的求交、绘制等方面有着重要应用.已有研究主要采用曲面的二阶导矢界来估计逼近误差,而有理曲面的导矢界估计是一项困难的工作.为解决上述问题,利用齐次坐标,给出了一种定义域为任意三角形的有理三角曲面的分片线性逼近算法.该算法有效地避免了有理三角曲面的导矢界估计,并且离散段数可先验地给出.此外,通过重新参数化技术来缩小有理三角Bézier曲面的权因子之间的比值,进一步提高了算法的效率.     

Computation of optimal composite re-parameterizations

Rational re-parameterizations of a polynomial curve that preserve the curve degree and [0,1] parameter domain are characterized by a single degree of freedom. The “optimal” re-parameterization in this family (that comes closest under the L₂ norm to arc-length parameterization) can be identified by solving a quadratic equation, but may exhibit too much residual parametric speed variation for motion control and other applications. Closer approximations to arc-length parameterizations require more flexible re-parameterization functions, such as piecewise-polynomial/rational forms. We show that, for fixed nodes, the optimal piecewise-rational parameterization of the same degree is defined by a simple recursion relation, and we analyze its convergence to the arc-length parameterization. With respect to the new curve parameter, this representation is only of C⁰ continuity, although the smoothness and geometry of the curve are unchanged. A C¹ parameterization can be obtained by using continuity conditions, rather than optimization, to fix certain free parameters, but the objective function is then highly non-linear and does not admit a closed-form optimization. Empirical results from implementations of these methods are presented.     

基于SIFT特征的小波域数字图像鲁棒水印方法*

利用数字图像SIFT(scale invariant feature transform)特征的稳定性和小波变换的特性,提出了一种抗仿射变换和剪切的鲁棒水印算法。水印信息通过量化调制方法嵌在小波变换的低频域。水印检测时,利用匹配的SIFT关键点的位置信息计算仿射变换参数和边缘剪切参数,然后对被检测图像进行逆变换和重定位,恢复水印的同步信息。实验结果表明该算法可以抗击仿射变换和剪切攻击,对常见的图像处理也有很强的鲁棒性。     

Degenerate parametric curves

This paper discusses two degenerate cases of polynomial parametric curves for which the degrees of the defining polynomials can be reduced without altering the curve. The first case is the improperly parametrized curve for which each point on the curve corresponds to several parameter values. The second case, which can only occur for rational polynomial parametric curves, exists when the defining polynomials all have a common factor.

This paper describes how to detect and correct each type of degeneracy. Examples are given which demonstrate that seemingly innocuous Bézier curves may suffer from either of these degeneracies.     

Construction of Bézier surface patches with Bézier curves as geodesic boundaries

Given four polynomial or rational Bézier curves defining a curvilinear rectangle, we consider the problem of constructing polynomial or rational tensor-product Bézier patches bounded by these curves, such that they are geodesics of the constructed surface. The existence conditions and interpolation scheme, developed in a general context in earlier studies, are adapted herein to ensure that the geodesic-bounded surface patches are compatible with the usual polynomial/rational representation schemes of CAD systems. Precise conditions for four Bézier curves to constitute geodesic boundaries of a polynomial or rational surface patch are identified, and an interpolation scheme for the construction of such surfaces is presented when these conditions are satisfied. The method is illustrated with several computed examples.     

Edge-pancyclicity of Möbius cubes

The Möbius cube Mn is a variant of the hypercube Qn and has better properties than Qn with the same number of links and processors. It has been shown by Fan [J. Fan, Hamilton-connectivity and cycle-embedding of Möbius cubes, Inform. Process. Lett. 82 (2002) 113-117] and Huang et al. [W.-T. Huang, W.-K. Chen, C.-H. Chen, Pancyclicity of Möbius cubes, in: Proc. 9th Internat. Conf. on Parallel and Distributed Systems (ICPADS'02), 17-20 Dec. 2002, pp. 591-596], independently, that Mn contains a cycle of every length from 4 to n². In this paper, we improve this result by showing that every edge of Mn lies on a cycle of every length from 4 to n² inclusive.     

Curve implicitization using moving lines

It is a classical result that two corresponding pencils of lines intersect in a conic section, and likewise any conic section can be expressed as the intersection of two pencils of lines. We here extend the idea of pencils to higher degree families lines, and show that any planar rational curve can be expressed as the intersection of two families of lines. This extension leads to a more efficient implicitization algorithm for curves, in which, for example, the implicit equation of a degree four rational curve can generally be expressed as the determinant of a 2 × 2 matrix (Bezout's resultant produces a 4 × 4 matrix and Sylvester's resultant an 8 × 8 matrix).     

C 1 NURBS representations of G 1 composite rational Bézier curves

This paper is concerned with the re-representation of a G ¹ composite rational Bézier curve. Although the rational Bézier curve segments that form the composite curve are G ¹ continuous at their joint points, their homogeneous representations may not be even C ⁰ continuous in the homogeneous space. In this paper, an algorithm is presented to convert the G ¹ composite rational Bézier curve into a NURBS curve whose nonrational homogeneous representation is C ¹ continuous in the homogeneous space. This re-representation process involves reparameterization using Möbius transformations, smoothing multiplication and parameter scaling transformations. While the previous methods may fail in some situations, the method proposed in this paper always works.