Reconstruction from contour lines based on bi-cubic Bézier spline surface

A novel reconstruction method from contours lines is provided. First, we use a simple method to get rid of redundant points on every contour, then we interpolate them by using cubic Bézier spline curve. For corresponding points of different con- tours, we interpolate them by the cubic Bézier spline curve too, so the whole surface can be reconstructed by the bi-cubic Bézier spline surface. The reconstructed surface is smooth because every Bézier surface is patched with G2 continuity, the reconstruction speed is fast because we can use the forward elimination and backward substitution method to solve the system of tridiagonal equations. We give some reconstruction examples at the end of this paper. Experiments showed that our method is applicable and effective.

A new algorithm for designing developable Bézier surfaces

A new algorithm is presented that generates developable Bézier surfaces through a Bézier curve called a directrix. The algorithm is based on differential geometry theory on necessary and sufficient conditions for a surface which is developable, and on degree evaluation formula for parameter curves and linear independence for Bernstein basis. No nonlinear characteristic equations have to be solved. Moreover the vertex for a cone and the edge of regression for a tangent surface can be obtained easily. Aumann’s algorithm for developable surfaces is a special case of this paper.

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.

Computation of lower derivatives of rational triangular Bézier surfaces and their bounds estimation

By introducing the homogenous coordinates, degree elevation formulas and combinatorial identities, also by using multiplication of Bernstein polynomials and identity transformation on equations, this paper presents some explicit formulas of the first and second derivatives of rational triangular Bézier surface with respect to each variable (including the mixed derivative) and derives some estimations of bound both on the direction and magnitude of the corresponding derivatives. All the results above have value not only in surface theory but also in practice.

Visible region extraction from a sequence of rational Bézier surfaces

A method for computing the visible regions of free-form surfaces is proposed in this paper. Our work is focused on accurately calculating the visible regions of the sequenced rational Bézier surfaces forming a solid model and having coincident edges but no inner-intersection among them. The proposed method calculates the silhouettes of the surfaces without tessellating them into triangle meshes commonly used in previous methods so that arbitrary precision can be obtained. The computed sil- houettes of visible surfaces are projected onto a plane orthogonal to the parallel light. Then their spatial relationship is applied to calculate the boundaries of mutual-occlusion regions. As the connectivity of the surfaces on the solid model is taken into account, a surface clustering technique is also employed and the mutual-occlusion calculation is accelerated. Experimental results showed that our method is efficient and robust, and can also handle complex shapes with arbitrary precision.

Convexity and Positivity of Bivariate Quadratic Bézier Surfaces over a Rectangle

With the help of several discriminants about the zero points of a quartic polynomial, the sufficient and necessary conditions for the positivity and nonnegativity of the quartic polynomial over an interval I(-∞,+∞) was derived. Based on these conclusions, the sufficient and necessary conditions for the positivity and convexity of the 2×2 Bézier surface over a rectangle were obtained. A simple sufficient condition was deduced also and finally several examples were given.

Real-time accurate Free-Form Deformation in terms of triangular Bézier surfaces

We implemented accurate FFD in terms of triangular Bezier surfaces as matrix multiplications in CUDA and rendered them via OpenGL. Experimental results show that the proposed algorithm is more efficient than the previous GPU acceleration algorithm and tessel- lation shader algorithms.

A Sufficient Convexity Condition for Parametric Bézier Surface over Rectangle

Surface convexity is a key issue in computer aided geometric design, which is widely applied in geometric modeling field, such as physical models, industrial design, automatic manufacturing, etc. In this paper, a sufficient convexity condition of the parametric Bézier surface over rectangles is proposed, which is firstly considered as a sufficient convexity condition for the Bézier control grid. The condition is proved by De Casteljau surface subdivision arithmetic, in which the recursive expressions elaborate that the control grid eventually converges to the surface. At last, two examples for the modeling of interpolation-type surface are discussed, one of which is a general surface and the other is a degenerate surface.

Scattered Data Interpolation Using Cubic Trigonometric Bézier

This paper discusses scattered data interpolation using cubic trigonometric Bézier triangular patches with C1 continuity everywhere.We derive the C1 condition on each adjacent triangle.On each triangular patch,we employ convex combination method between three local schemes.The final interpolant with the rational corrected scheme is suitable for regular and irregular scattered data sets.We tested the proposed scheme with 36,65,and 100 data points for some well-known test functions.The scheme is also applied to interpolate the data for the electric potential.We compared the performance between our proposed method and existing scattered data interpolation schemes such as Powell–Sabin(PS)and Clough–Tocher(CT)by measuring the maximum error,root mean square error(RMSE)and coefficient of determination(R^(2)).From the results obtained,our proposed method is competent with cubic Bézier,cubic Ball,PS and CT triangles splitting schemes to interpolate scattered data surface.This is very significant since PS and CT requires that each triangle be splitting into several micro triangles.

区间Bézier曲面的降阶

论文讨论了用低阶的区间Bzier曲面来界定高阶的区间Bzier曲面的问题 ,提出了两种不同的解决方法———线性规划及最佳一致逼近 ,最后提供的实例结果表明线性规划方法能得到一个界 。

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。

Cubic B样条旋转曲面的自由绘制技术

随着基于样条曲线的模型逐渐成为CAGD的主要研究对象,进一步带动了高度自由曲面建模技术的发展.本文描述一个新颖、直观的旋转曲面绘制技术,它支持三次B样条的自由输入作为轮廓线,并通过提取的样点矢量以及最小方差逼近方法形成旋转轴,利用轴上测量点的切线方向确定各个圆平面,从而建立了平面上旋转坐标矩阵,最终插值空间球面样点生成双三次B样条曲面.实验结果有效地验证了此技术的高效性.

利用Bézier曲面的面部表情和口型几何造型

本文主要介绍-汉语文本驱动的基本表情和口型的合成系统中的面部表情和口型几何造型．通过对面部曲面结构和面部各器官分布结构的分析，建立了面部曲面的几何模型和模拟人脸面部曲面的分片二次直积形Bezier参数曲面计算格式．应用此模型和算法，依据面部表情和汉语口型特征，实现基本表情和口型的造型．

用Bézier曲线设计可展曲面

利用n次Bzier曲线作为设计曲线，给出了一种构造可展曲面的方法，允许Bzier控制顶点有较多的定位自由，有益于自由型可展曲面的设计。

三角域上B-B曲面和矩形域上BZIER曲面之间的几何连续条件

利用 KAHMAN R关于曲面片之间的 GC2连续条件得到了当三角域上 B- B曲面已知时 ,三角域上 B- B曲面和矩形域上 BE、ZIER曲面之间的 GC2连续条件。

CONSTRUCTION OF BEZIER SURFACES FROM PRESCRIBED BOUNDARY

In this paper,we present a method for generating Bézier surfaces from the boundary information based on a general second order functional and a third order functional associated with the triharmonic equation.By solving simple linear equations,the internal control points of the resulting Bézier surface can be obtained as linear combinations of the given boundary control points.This is a generalization of previous works on Plateau-Bezier problem,harmonic,biharmonic and quasi-harmonic Bézier surfaces.Some representative examples show the effectiveness of the presented method.

Constructing C^1 Continuous Surface on Irregular Quad Meshes

A new method is proposed for surface construction on irregular quad meshes as extensions to uniform B-spline surfaces. Given a number of control points, which form a regular or irregular quad mesh, a weight function is constructed for each control point. The weight function is defined on a local domain and is C1 continuous. Then the whole surface is constructed by the weighted combination of all the control points. The property of the new method is that the surface is defined by piecewise Cl bi-cubic rational parametric polynomial with each quad face. It is an extension to uniform B-spline surfaces in the sense that its definition is an analogy of the B-spline surface, and it produces a uniform bi-cubic B-spline surface if the control mesh is a regular quad mesh. Examples produced by the new method are also included.

The Effects of Shape Parameters for Surface Joining with High Order Geometric Continuity

The necessary and sufficient conditions and an algorithm to reach continuity between adjacent Bézier patches are presented. The effects of shape parameters for surface connection of G4 geometric continuity are investigated in detail. The algorithm can be generalized directly to the case of surface joining with higher order geometric continuity. It has important applications in surface modeling and surface joining.

参数三次B样条曲线的一种整体光顺方法

本文在能量法的基础上，提出了一种新的目标函数，给出了参数三次Ｂ样条曲线的一种新的整体光顺方法。利用这种方法得到的曲线不仅具有较小的应变能，而且曲率变化比较均匀，具有很好的光顺效果。该方法能推广到对曲央的光顺。

局部构造C^2连续的三次B样条插值曲线和双三次插值曲面

为了避免一般的局部插值算法生成的B样条曲线和曲面在段点处达不到理想的连续性以及出现多重内节点的问题,一种局部构造C2连续的三次B样条插值曲线和双三次插值曲面的方法被介绍。该方法借助节点插入算法逐步地迭代出样条控制顶点,其思想简单、几何直观、算法速度快,在曲线中夹直线段、尖点以及在曲面中夹棱边和平面都能比较容易实现。生成的曲线光滑度高、无重节点。文章最后还利用这种构造方法给出了一种在指定范围内按规定变形曲线的方法。