构造曲率单调的组合二次h-Bézier曲线

h-Bézier曲线(h>0)又被称为Pólya曲线,它具有与经典Bézier曲线(h=0)一致的诸多优良性质。为此,文中研究了二次h-Bézier曲线具有单调曲率的充要条件及其构造算法。首先,讨论二次h-Bézier曲线曲率极值的存在性,得到曲线具有单调曲率的充要条件;通过引入曲率临界圆,给出判断二次h-Bézier曲线曲率单调性的几何方法,即检查二次h-Bézier曲线的中间控制点是否在曲率临界圆上或圆内;并由此得到构造具有单调曲率的二次h-Bézier曲线的两种算法,通过调节形状参数h可保证曲线具有单调递减或单调递增的曲率。其次,研究两条二次h-Bézier曲线的光滑拼接,基于对二次h-Bézier曲线性质的分析,选择第二条曲线在肩点处与第一条曲线的端点实现拼接,得到G~2拼接的充要条件;讨论参数对拼接曲线形状的影响。最后,构造出同时满足G~2拼接、曲率单调递减(或单调递增)的组合二次h-Bézier曲线。数值实例显示了组合二次h-Bézier曲线的造型优势和灵活性。

Bézier curves with shape parameter

In this paper, Bézier basis with shape parameter is constructed by an integral approach. Based on this basis, we define the Bézier curves with shape parameter. The Bézier basis curves with shape parameter have most properties of Bernstein basis and the Bézier curves. Moreover the shape parameter can adjust the curves’ shape with the same control polygon. As the increase of the shape parameter, the Bézier curves with shape parameter approximate to the control polygon. In the last, the Bézier surface with shape parameter is also constructed and it has most properties of Bézier surface.

A Novel Contour Tracing Algorithm for Object Shape Reconstruction Using Parametric Curves

Parametric curves such as Bézier and B-splines, originally developedfor the design of automobile bodies, are now also used in image processing andcomputer vision. For example, reconstructing an object shape in an image,including different translations, scales, and orientations, can be performedusing these parametric curves. For this, Bézier and B-spline curves can be generatedusing a point set that belongs to the outer boundary of the object. Theresulting object shape can be used in computer vision fields, such as searchingand segmentation methods and training machine learning algorithms. Theprerequisite for reconstructing the shape with parametric curves is to obtainsequentially the points in the point set. In this study, a novel algorithm hasbeen developed that sequentially obtains the pixel locations constituting theouter boundary of the object. The proposed algorithm, unlike the methods inthe literature, is implemented using a filter containing weights and an outercircle surrounding the object. In a binary format image, the starting point ofthe tracing is determined using the outer circle, and the next tracing movementand the pixel to be labeled as the boundary point is found by the filter weights.Then, control points that define the curve shape are selected by reducing thenumber of sequential points. Thus, the Bézier and B-spline curve equationsdescribing the shape are obtained using these points. In addition, differenttranslations, scales, and rotations of the object shape are easily provided bychanging the positions of the control points. It has also been shown that themissing part of the object can be completed thanks to the parametric curves.

A synthetic aperture radar sea surface distribution estimation by n-order Bézier curve and its application in ship detection

To dates,most ship detection approaches for single-pol synthetic aperture radar（SAR） imagery try to ensure a constant false-alarm rate（CFAR）.A high performance ship detector relies on two key components：an accurate estimation to a sea surface distribution and a fine designed CFAR algorithm.First,a novel nonparametric sea surface distribution estimation method is developed based on n-order Bézier curve.To estimate the sea surface distribution using n-order Bézier curve,an explicit analytical solution is derived based on a least square optimization,and the optimal selection also is presented to two essential parameters,the order n of Bézier curve and the number m of sample points.Next,to validate the ship detection performance of the estimated sea surface distribution,the estimated sea surface distribution by n-order Bézier curve is combined with a cell averaging CFAR（CA-CFAR）.To eliminate the possible interfering ship targets in background window,an improved automatic censoring method is applied.Comprehensive experiments prove that in terms of sea surface estimation performance,the proposed method is as good as a traditional nonparametric Parzen window kernel method,and in most cases,outperforms two widely used parametric methods,K and G0 models.In terms of computation speed,a major advantage of the proposed estimation method is the time consuming only depended on the number m of sample points while independent of imagery size,which makes it can achieve a significant speed improvement to the Parzen window kernel method,and in some cases,it is even faster than two parametric methods.In terms of ship detection performance,the experiments show that the ship detector which constructed by the proposed sea surface distribution model and the given CA-CFAR algorithm has wide adaptability to different SAR sensors,resolutions and sea surface homogeneities and obtains a leading performance on the test dataset.

Rational offset approximation of rational Bézier curves

The problem of parametric speed approximation of a rational curve is raised in this paper. Offset curves are widely used in various applications. As for the reason that in most cases the offset curves do not preserve the same polynomial or rational polynomial representations, it arouses difficulty in applications. Thus approximation methods have been introduced to solve this problem. In this paper, it has been pointed out that the crux of offset curve approximation lies in the approximation of parametric speed. Based on the Jacobi polynomial approximation theory with endpoints interpolation, an algebraic rational approximation algorithm of offset curve, which preserves the direction of normal, is presented.

Improvement of the termination criterion for subdivision of the rational Bézier curves

By using some elementary inequalities, authors in this paper makes further improvement for estimating the heights of Bézier curve and rational Bézier curve. And the termination criterion for subdivision of the rational Bézier curve is also improved. The conclusion of the extreme value problem is thus further confirmed.

Multi-Degree Reduction of Bézier Curves with Distance and Energy Optimization

In this paper, we propose a new approach to the problem of degree reduction of Bézier curves based on the given endpoint constraints. A differential term is added for the purpose of controlling the smoothness to a certain extent. Considering the adjustment of second derivative in curve design, a modified objective function including two parts is constructed here. One part is a kind of measure of the distance between original high order Bézier curve and degree-reduced curve. The other part represents the second derivative of degree-reduced curve. We tackle two kinds of conditions which are position vector constraint and tangent vector constraint respectively. The explicit representations of unknown points are presented. Some examples are illustrated to show the influence of the differential terms to approximation and smoothness effect.

Unifying representation of Bézier curve and two kinds of generalized ball curves

This paper presents a new basis, the WSB basis, which unifies the Bemstein basis, Wang-Ball basis and Said-Ball basis, and therefore the Bézier curve, Wang-Ball curve and Said-Ball curve are the special cases of the WSB curve based on the WSB basis In addition, the relative degree elevation formula, recursive algorithm and conversion formula between the WSB basis and the Bern- stein basis are given.

Sine Transformation and the Reparametrization of Bézier Curves

A solution to the reparametrization of Bézier curves by sine transformation of Bemstein basis is presented. The new effective reparametrization method is given through the following procedures： educing Sine Bemstein-Bézier Class-SBBC function, defining SBBC curve and discussing the relation between SBBC and Bézier curve.

PH-spline approximation for Bézier curve and rendering offset

In this paper, a G1, C1, C2 PH-spline is employed as an approximation for a given Bzier curve within error bound and further renders offset which can be regarded as an approximate offset to the Bzier curve. The errors between PH-spline and the Bzier curve, the offset to PH-spline and the offset to the given Bzier curve are also estimated. A new algorithm for constructing offset to the Bzier curve is proposed.

A class of quasi Bézier curves based on hyperbolic polynomials

This paper presents a basis for the space of hyperbolic polynomials Гm=span { 1, sht, cht, sh2t, ch2t shmt, chmt} on the interval [0,a] from an extended Tchebyshev system, which is analogous to the Bernstein basis for the space of polynomial used as a kind of well-known tool for free-form curves and surfaces in Computer Aided Geometry Design. Then from this basis, we construct quasi Bézier curves and discuss some of their properties. At last, we give an example and extend the range of the parameter variable t to arbitrary close interval [r, s] （r〈s）.

Shape modification of Bézier curves by constrained optimization

The Bézier curve is one of the most commonly used parametric curves in CAGD and Computer Graphics and has many good properties for shape design. Developing more convenient techniques for designing and modifying Bézier curve is an im- portant problem, and is also an important research issue in CAD/CAM and NC technology fields. This work investigates the optimal shape modification of Bézier curves by geometric constraints. This paper presents a new method by constrained optimi- zation based on changing the control points of the curves. By this method, the authors modify control points of the original Bézier curves to satisfy the given constraints and modify the shape of the curves optimally. Practical examples are also given.

A New Fractional Rational Bézier Curve

Based on rational Bézier curves given by Ron Goldman, a new fractional rational Bézier curve was first defined in terms of fractional Bernstein bases. Moreover, some basic properties were dicussed and a theorem connected to Poisson curves was obtained. Some examples in this paper were given by the visual results.

An optimization method for rational Bézier curve design

Adjusting weights as a shape control tool in rational B6zier curve design is not easy because the weights have a global in- fluence. The curve could not approximate control polygon satisfactorily by an interactive manner. In order to produce a curve close enough to control polygon at every control vertex, an optimization model is established to minimize the distance between rational B6zier curve and its control points. This optimization problem is converted to a quadratic programming problem by separating and recombining the objective function. The new combined multi-objective optimization problem is reasonable and easy to solve. With an optimal parameter, the computing process is discussed. Comparative examples show that the designed curve is closer to control polygon and preserves the shape of the control polygon well.

COMPUTING AREAS BOUNDED BYRATIONAL BEZIER CURVES

An approach is presented for computing integral values, such as areas and volumes of revo-lution . of regions bounded by rational plane B zier curves. The method approximates rational curveswith polynomial curves, and then computes the integral values on those polynomial curves. Errorbounds are provided. For high precision, this new algorithm performs much more quickly than con-ventional numerical methods.

三次H-Bézier曲线的分割、拼接及其应用

为了拓展曲线曲面的表示方法,提出一种曲线造型工具——H-Bzier曲线.在讨论三次H-Bzier曲线性质的基础上,提出了三次H-Bzier曲线的任意分割算法,即对三次H-Bzier曲线上任意一点p(t*)(0≤t*≤α),求该点把曲线分成的2个子曲线段pt*(t)(0≤t≤t*)与pα-t*(t)(0≤t≤α-t*)的控制参数和控制顶点;给出了三次H-Bzier曲线与三次Bzier曲线的拼接条件,以及三次H-Bzier曲线在曲面造型中应用的例子.采用该算法所得结果简单、直观,有效地增强了三次H-Bzier方法控制及表达曲线形状的能力.

H-Bézier曲线的降多阶逼近

国内外对参数曲线降阶,尤其是对Bézier曲线降阶的研究已渐趋成熟,但尚缺少对超越曲线降阶的研究.为此以能精确表示指数曲线、悬链线等超越曲线的H-Bézier曲线为载体,运用H-Bézier曲线的升阶公式,结合广义逆矩阵理论给出了H-Bézier曲线一次降多阶的逼近方法;同时估计了降阶的误差界,并建立了与Bézier曲线降阶的关系.实验结果表明,采用该方法可取得较好的逼近效果,有效地丰富了H-Bézier曲线的理论体系.

三次平面H-Bézier螺线

基于光顺曲线设计需求,一条曲率单调且曲率正负不变的三次平面H-Bézier螺线段被构造。由于此螺线具有起点曲率为零的特性,它可以替代回旋曲线作为道路设计中的缓和曲线。同时螺线还含有形状参数,故它具有曲线形状可调性的优点。最后,利用此H-Bézier螺线,构造了两直线间满足G2连续的缓和曲线。

H-Bézier曲面的分割与拼接

首先在研究H-Bézier曲线性质的基础上,给出了H-Bézier曲面在u向和v向两个方向的任意分割算法,并对曲面所具有的特性进行了分析;同时,研究了两片H-Bézier曲面在不同方向G1连续的拼接条件,并通过合理选取控制参数,简化了拼接条件。

三次TC-Bézier与H-Bézier曲线曲面的光滑拼接

文章研究了带两个形状参数的三次TC-Bézier曲线曲面与三次H-Bézier曲线曲面的光滑拼接问题.在对这两种曲线的基函数、端点性质和形状分析的基础上,给出了TC-Bézier与H-Bézier曲线曲面G1光滑拼接的条件,以及特定情况下TC-Bézier与H-Bézier曲线G2光滑拼接的条件.实例表明,通过合理选择参数来调整曲线曲面的形状,可以把带形状参数TC-Bézier曲线曲面与H-Bézier曲线曲面的拼接技术应用于复杂曲线曲面工程造型中.