具有任意自由度的B样条非均匀细分*

为了便于工程实际应用,非均匀细分方法现在已经成为计算机图形学和几何建模中的热点问题。本文提出一个具有任意自由度的B样条非均匀细分算法,其实现与B样条均匀细分即Lane–Riesenfeld细分方法相似。该算法包含了非均匀d环结构生成的双重控制点,其中d环相似于d度均匀B样条曲线的Lane-Riesenfeld算法中均匀的d环结构。Lane-Riesenfeld算法是由B样条曲线基函数的连续卷积公式直接得出的,而本文的算法是blossoming方法的一个扩展。对于非均匀B样条曲线来说,本文的节点插入方法比之前的方法更简单更有效。     

一种n次均匀B样条曲线细分算法

利用 次均匀B样条细分的掩模与Pascal三角形关系,并借助控制多边形在每次加细过程中新旧控制顶点对应的几何位置关系,给出一种新的 次均匀B样条曲线细分算法,基于该算法构造出带有形状参数的局部插值约束的奇次均匀B样条细分曲线。通过理论和算例说明,该算法几何直观性强、新旧点对应明确、应用灵活且能保持良好的参数连续性。     

一种对称非均匀细分曲面算法

为了得到能更好应用于CAD系统的细分曲面造型方法,提出一种基于B-样条的对称非均匀细分算法,其中的思想和均匀Lane-Riesenfeld节点插入算法相似。基于B-样条的节点插入算法,以Blossoming为工具,计算出细分后的新控制顶点。细分后得到的极限曲面由张量积样条曲面组成,在奇异点达到2C连续。与传统的细分曲面算法相比,该细分曲面算法具有良好的局部支撑性,大大降低了算法的复杂度,而且该算法是对称的,不用考虑定向问题。     

自由曲线曲面的任意次非均匀细分

提出一种有效的建模自由曲线曲面的非均匀细分算法。首先在节点插入技术基础上推导出任意次自由曲线的非均匀细分规则,然后把它推广到张量积曲面得到任意次自由曲面的非均匀细分规则,最后对奇异点附近曲面采用类Doo-Sabin和Catmull-Clark的细分规则,从而使该算法可以实现建模任意次具有任意拓扑基网格的非均匀细分曲面。此外,该方法也实现了对传统细分格式的统一,例如,当次数为2并采用均匀节点矢量便转化为Doo-Sabin细分,当次数为3并采用均匀节点矢量便转化为Catmull-Clark细分。     

一种三次非均匀B样条曲线的细分算法

近几年来,以B样条曲线为代表的曲线细分已成为计算机图形学领域的一项重要研究内容。提出一种基于对分方式的细分算法,能均匀地细分曲线,并用较少的细分次数得到对曲线较好的逼近效果。采用该细分算法,方便而快速地在计算机上绘制B样条曲线,对给定参数做出更加优良的控制动作,并提高控制系统的运动速度和曲线的显示速度,实例表明了该算法的有效性。     

B样条曲线同时插入多个节点的快速算法

基于离散B样条的一个新的递推公式，提出B样条曲线同时插入多个节点的新算法。不同于Cohen等插入节点的Oslo算法，本算法用新的方法离算离散B样条，求每个离散B样条的值只需O(1)的运算量，从而使本算法高效，其时间复杂性为O(sk n)，其中k为B样条曲线的阶，n k 1为原节点数，s为新插入节点的个数,本算法的通用性强，适用于端点插值的和非端点插值的B样条曲线，可同时在曲线定义域内外的任意位置上插入任意个节点。     

带形状参数的指数均匀B样条模型

本文给出了k(k≥2)阶带形状参数指数多项式的均匀B样条模型.该类模型具有很多与B样条模型相同的性质,并且具有一个可调节的形状参数.由该模型构造的曲线,通过改变形状参数的取值,可以调整曲线接近其控制多边形的程度.该模型可以应用于CAD/CAM领域,作为几何造型一种新的有效模型.     

带形状参数的三角多项式均匀B样条

该文给出了n阶带形状参数的三角多项式均匀B样条基函数．由带形状参数的三角多项式均匀B样条基组成的样条曲线可通过改变形状参数的取值而调整曲线的形状,并且可以精确表示圆、椭圆、螺旋线等曲线．随着阶数的升高,形状参数的取值范围将扩大．     

非均匀三次B样条曲线插值的Jacobi-PIA算法

为了求解非均匀三次B样条曲线插值问题,基于解线性方程组的Jacobi迭代方法提出一种渐进迭代插值算法——Jacobi-PIA算法.该算法以待插值点为初始控制多边形得到第0层的三次B样条曲线,递归地求得插值给定点集的三次B样条曲线;在每个迭代过程中,定义待插值点与第k层的三次B样条曲线上对应点的差向量乘以该点对应的B样条系数的倒数为偏移向量,第k层的控制顶点加上对应的偏移向量得到第k+1层的三次B样条曲线的控制顶点.由于Jacobi-PIA算法在更新控制顶点时减少了一个减法运算,因而运算量更少.理论分析表明该算法是收敛的.数值算例结果表明,Jacobi-PIA算法的收敛速度优于经典的渐进迭代插值算法,与最优权因子对应的带权渐进迭代插值算法基本相同.     

细分生成B样条的几何作图法

研究均匀B样条曲线细分生成的几何作图问题,给出了采用p-nary细分法细分生成任意次均匀B样条曲线的递归细分算法。在此基础上,研究了任意次均匀B样条曲线p-nary细分生成的几何作图方法。利用这种几何作图法,可以直观地在计算机上通过编程来快速准确地绘制B样曲线,更重要的是,可以使基于几何方法的任意次B样曲线的手工绘制成为可能。     

用Catmull-Clark细分及网格调整方法重构牙齿曲面

首先根据牙齿表面测量数据点,计算出其长方体包围盒;并据此构造细分曲面的初始网格;采用矩阵对角化方法,推导Catmull-Clark细分极限点的表达式,计算初始网格的顶点经过细分后的极限点;按照极限点逼近数据点的原则移动控制网格顶点,经过逐次再细分、再调整网格,使各级网格在数据点的“引导”下逐步变形,使网格逐步逼近牙齿表面的测量数据点集合,实现牙齿表面模型的三维重建。     

A Subdivision Scheme for Continuous-Scale B-Splines and Affine-Invariant Progressive Smoothing

Multiscale representations and progressive smoothing constitutean important topic in different fields as computer vision, CAGD,and image processing. In this work, a multiscale representationof planar shapes is first described. The approach is based oncomputing classical B-splines of increasing orders, andtherefore is automatically affine invariant. The resultingrepresentation satisfies basic scale-space properties at least ina qualitative form, and is simple to implement.The representation obtained in this way is discrete in scale,since classical B-splines are functions in , where k isan integer bigger or equal than two. We present a subdivisionscheme for the computation of B-splines of finite support atcontinuous scales. With this scheme, B-splines representationsin are obtained for any real r in [0, ), andthe multiscale representation is extended to continuous scale.The proposed progressive smoothing receives a discrete set ofpoints as initial shape, while the smoothed curves arerepresented by continuous (analytical) functions, allowing astraightforward computation of geometric characteristics of theshape.     

C-B样条曲线的分割和拼接

曲线曲面造型中设计复杂的自由曲线时,单段曲线已不能满足外形设计的要求,因而在实际造型中,经常采用曲线的分割和拼接。C-B样条理论是曲线曲面造型的一项重要内容。在对C-B样条基函数及曲线端点特性分析的基础上,提出了C-B样条曲线的任意分割算法,并对C-B样条曲线间进行了G1拼接,给出了B样条曲线和C-B样条曲线G1和G2光滑拼接的几何条件。采用分割和拼接技术会增加C-B样条曲线的灵活性,所得结论具有明确的几何意义,并可以进一步推广到C-B样条曲面造型中。     

一种Loop细分小波的边界处理方法

细分小波近年来发展迅速,在计算机图形显示、渐进网格传输和网格多分辨率编辑等领域获得了广泛的应用。Bertram提出的Loop细分小波是基于提升格式的双正交细分小波的典型范例,它所针对的对象均为网格的内部顶点。目前尚未发现相关文献提及细分小波对于边界的处理。该文在Loop细分小波算法的基础上,给出了一种Loop细分小波边界处理的方法,经验证效果令人满意。     

针对非均匀问题的多目标进化算法

文章针对解集分布非均匀的问题,提出了一种新的多目标进化算法,称之为GNEA(带小生境的网格进化算法)。在算法中,针对分均匀问题的特点,采用了小生境技术来保持解集的局部非均匀分布,以及网格技术来保证整个解集的分布度。为了让GNEA运行效率更高,提出了用庄家法构造非支配集的方法。最后通过与其他算法进行比较,验证了算法具有较好的运行效率,且在解决非均匀问题上是一种有效的多目标进化算法。     

Subdivision models for varying-resolution and generalized perturbations

Subdivision schemes are multi-resolution methods used in computer-aided geometric design to generate smooth curves or surfaces. In this paper, we are interested in both smooth and non-smooth subdivision schemes. We propose two models that generalize the subdivision operation and can yield both smooth and non-smooth schemes in a controllable way:

• (1) The ‘varying-resolution’ model allows a structured access to the various resolutions of the refined data, yielding certain patterns. This model generalizes the standard subdivision iterative operation and has interesting interpretations in the geometrical space and also in creativity-oriented domains, such as music. As an infrastructure for this model, we propose representing a subdivision scheme by two dual rules trees. The dual tree is a permuted rules tree that gives a new operator-oriented view on the subdivision process, from which we derive an ‘adjoint scheme’.

• (2) The ‘generalized perturbed schemes’ model can be viewed as a special multi-resolution representation that allows a more flexible control on adding the details. For this model, we define the terms ‘template mask’ and ‘tension vector parameter’.

The non-smooth schemes are created by the permutations of the ‘varying-resolution’ model or by certain choices of the ‘generalized perturbed schemes’ model. We then present procedures that integrate and demonstrate these models and some enhancements that bear a special meaning in creative contexts, such as music, imaging and texture. We describe two new applications for our models: (a) data and music analysis and synthesis, which also manifests the usefulness of the non-smooth schemes and the approximations proposed, and (b) the acceleration of convergence and smoothness analysis, using the ‘dual rules tree’.     

Knot Insertion Algorithms for ECT B-spline Curves

Knot insertion algorithm is one of the most important technologies of B-spline method. By inserting a knot the local prop- erties of B-spline curve and the control flexibility of its shape can be fiu＇ther improved, also the segmentation of the curve can be rea- lized. ECT spline curve is drew by the multi-knots spline curve with associated matrix in ECT spline space; Muehlbach G and Tang Y and many others have deduced the existence and uniqueness of the ECT spline function and developed many of its important properties .This paper mainly focuses on the knot insertion algorithm of ECT B-spline curve.It is the widest popularization of B-spline Behm algorithm and theory. Inspired by the Behm algorithm, in the ECT spline space, structure of generalized P61ya poly- nomials and generalized de Boor Fix dual functional, expressing new control points which are inserted after the knot by linear com- bination of original control vertex the single knot, and there are two cases, one is the single knot, the other is the double knot. Then finally comes the insertion algorithm of ECT spline curve knot. By application of the knot insertion algorithm, this paper also gives out the knot insertion algorithm of four order geometric continuous piecewise polynomial B-spline and algebraic trigonometric spline B-spline, which is consistent with previous results.