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.

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.

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.

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.

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.

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.

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 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.

一种二次均匀B样条曲线的精简拟合方法

使用简单直观的方法把小线段拟合成二次均匀B样条曲线,基于嵌入式数控系统有限的主CPU的数据处理能力,严格地控制了拟合的运算量,在保证插补计算实时性的同时,有效控制了拟合曲线的误差,实现中高档数控系统的高精高速运动控制。

反求标准型有理二次Bezier曲线的参数与内权因子

给定不共线的三个控制顶点和位于这些顶点的凸包内在二次曲线上的一点，可以决定一条标准型有理二次Ｂｅｚｉｅｒ曲线。本文提供了一些简单有效的方法，反求曲线上该点的参数和内权因子，避免了数值计算的不稳定性。

三次T-Bézier曲线的光顺延拓

将曲线的最小物理变形能量作为目标函数,提出了一种三次T-Bézier曲线曲面的光顺延拓算法。利用G2连续性作为约束条件,则延拓的曲线具有两个自由度,并取其中的一个自由度为零;基于延拓曲线的最小物理变形能量确定第二个自由度及延拓曲线的控制点,进而确定延拓曲线,重新参数化所延拓的曲线可以与原曲线在拼接点处C3连续拼接;此外,将该方法应用于双三次T-Bézier曲面的延拓。实例表明,该方法构造的曲线曲面具有较好的光顺性。

边向量表示多边形实现2－d动画Bézier曲线Shape Blending

提出了边向量表示多边形的方法，该表示法能克服顶点位置表示法及几何内在参数表示法的一些缺陷，具有几何特性明显、计算机输人和处理容易等优点，给出了一些结论．还实现了边向量表示法在关键帧动画ＳｈａｐｅＢｌｅｎｄｉｎｇ方法中的运用，它能弥补线性插值ＳｈａｐｅＢｌｅｎｄｉｎｇ边长变化不单调而萎缩造成动画效果失真等缺陷，也能避免几何内在参数插值ＳｈａｐｅＢｌｅｎｄｉｎｇ因三角运算造成计算量较大等缺陷．

带形状参数的三阶三角Bézier多项式曲线

分析讨论两类三阶三角Bézier多项式基函数的构造方法和基本性质,给出两类三阶三角Bézier多项式曲线的定义.利用含调节参数的控制点的变换构造带四个形状参数的三阶三角Bézier多项式曲线并且研究该曲线与两类三阶三角Bézier多项式曲线的关系.这种曲线实质上是根据已知的四个控制点的位置生成6个带有调节参数的新的控制点,利用参数的调节来改变控制点的位置从而达到影响曲线的形状的目的.

Bézier曲线到AH-Bézier曲线的升阶算法

关于曲线升阶,已有的结论往往限于同类曲线之间。为了突破这一限制,考虑不同类曲线间的升阶,关注代数多项式空间中的Bézier曲线到代数双曲多项式空间中的AH-Bézier曲线的升阶。研究从基函数入手,利用Bézier和AH-Bézier共有的求导降阶的特点,结合矩阵分块的思想,先给出AH-Bézier基到Bernstein基的转换矩阵,进而推出控制顶点的升阶公式,最后给出升阶算法。结果表明,任意n次Bézier曲线可以通过该算法升到n+3阶(等同于n+2次)的AH-Bézier曲线。算法实现了Bézier到AH-Bézier曲线模型的精确转换。