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。     

均匀B样条曲线的降阶

文中给出均匀 B样条曲线退化的充要条件 ,证明了满足退化条件的 k阶均匀 B样条曲线可以表示为 (k- 1)阶均匀 B样条曲线 ,并给出相应的表示方法 .在此基础上 ,利用约束优化方法 ,提出均匀 B样条曲线的一种降阶方法 .而传统方法从升阶的反问题考虑降阶 ,要求被降阶的 B样条曲线的所有节点都是多重节点 ,从而无法处理均匀B样条曲线 .     

非均匀B样条曲线升阶的新算法

实践证明，传统的Ｂ样条曲线升阶算法只能解决端点插值Ｂ样条曲线的升阶问题，当用于其它非均匀Ｂ样条曲线以及均匀Ｂ样条曲线的升阶进均会出现严重错误，本文基于一个新的Ｂ样条恒等式，提出了一个Ｂ样条曲线升阶的新算法，该算法可用于任何均匀和非均匀的Ｂ样条曲线的升阶，当用于一段均匀Ｂ样条曲线的升阶时，不需要的节点矢量中间插入任何节点，升阶后仍为一条均匀Ｂ样条曲线，其计算简便、速度快。本文最后还得到两个新结论：（     

B样条曲线降阶新方法

首先导出了 B样条曲线退化的条件 ,然后根据 B样条升阶恒等式提出了 B样条曲线降阶的新算法 .最后 ,对结果进行了简要的误差分析 .如果结合节点插入技术 ,还可以将降阶后的误差限定在给定的容差之内 .实践表明 ,该算法容易实现、效率高、逼近效果好 .     

基于双阶样条的代数双曲B样条升阶及割角算法

考虑代数双曲B样条曲线的升阶问题,从理论上证明了曲线的升阶可以理解为控制顶点的割角过程.为了实现代数双曲B样条曲线的升阶,文中构造了一组基函数——双阶代数双曲B样条基函数,这组基函数并不具有统一的阶数,而具有"双阶"性质.代数双曲B样条基函数与双阶样条基函数之间的变换公式可以导出曲线升阶的割角算法.     

B样条乘积计算的区间跳跃算法及应用

利用B样条基函数节点区间的对应关系,首先给出了B样条基函数间的转换矩阵的计算方法,进而给出了计算B样条乘积的区间跳跃算法。该算法仅需计算部分节点区间上的转换矩阵,因此称其为区间跳跃算法。这一方法解决了分段多项式与B样条曲线乘积的计算问题,可应用到B样条曲线的升阶、曲面间光滑拼接等问题中。通过算例验证了该方法计算简捷、易于实现。     

NURBS曲线曲面的显式矩阵表示及其算法

从 B样条的差商定义出发 ,提出差商展开系数的概念 ,通过差商展开系数显式解析表示式的导出 ,得到任意次 NU RBS曲线曲面系数矩阵的显式解析表示式 ,并给出了求差商展开系数和 NURBS曲线曲面系数矩阵的数值算法 .文中给出的方法适用于一切 NU RBS曲线曲面 ,包括有理和非有理的 Bézier、均匀和非均匀的 B样条曲线曲面 .相应的数值算法计算简单 ,易于实现 .差商展开系数解析表示式为 NU RBS曲线曲面的表示、转换和节点插入、升阶等基本运算以及与差商相关的问题的研究提供了一个统一的构造性工具和应用方法 .     

二次曲线的多项式逼近

研究用B啨ｚｉｅｒ曲线或样条逼近任意长二次曲线弧的方法 对不同曲线类型 ,均得到具有 6阶逼近精度的误差函数 并且相邻的B啨ｚｉｅｒ曲线间ＧＣ1连续 最后给出任意二次曲线弧近似多项式或多项式样条参数化的算法     

一种基于节点插入技术的B样条曲线平滑方法

三次均匀B样条曲线是工程上广泛采用的曲线绘制方法。提高节点插入算法的效率,实现B样条曲线的平滑效果,是广大学者所关注的问题。文章给出了一种基于Tailor级数展开的三次均匀B样条曲线节点插入的生成算法,并利用该算法提供的节点插入技术来实现三次均匀B样条曲线的平滑效果。与Oslo算法的递推过程相比,该节点插入算法在一定程度上简化了运算过程,提高了算法的生成效率。     

B样条曲线递推升阶方法的快速算法

改进了Cohen等的B样条曲线递推升阶方法,推导出了一个新的递推公式,并基于该递推公式和Cohen等的递推公式,给出了一种快速的B样条曲线升阶算法,其时间复杂性为O（nk),在目前已有的升阶算法中,O（nk)为最低阶的时间复杂性。     

Knot calculation for spline fitting via sparse optimization

Curve fitting with splines is a fundamental problem in computer-aided design and engineering. However, how to choose the number of knots and how to place the knots in spline fitting remain a difficult issue. This paper presents a framework for computing knots (including the number and positions) in curve fitting based on a sparse optimization model. The framework consists of two steps: first, from a dense initial knot vector, a set of active knots is selected at which certain order derivative of the spline is discontinuous by solving a sparse optimization problem; second, we further remove redundant knots and adjust the positions of active knots to obtain the final knot vector. Our experiments show that the approximation spline curve obtained by our approach has less number of knots compared to existing methods. Particularly, when the data points are sampled dense enough from a spline, our algorithm can recover the ground truth knot vector and reproduce the spline.     

结点插值新算法

结点插值算法广泛应用开发系统样条曲线、曲面的生成表示和求交分类。本文给出的结点插值新算法不仅可以统一表示已知的Boehm算法和Oslo算法,而且算法效率上优于它们。本算法已用于三维几何造型系统GEMS中。     

Single-knot wavelets for non-uniform B-splines

We propose a flexible and efficient wavelet construction for non-uniform B-spline curves and surfaces. The method allows to remove knots in arbitrary order minimizing the displacement of control points when a knot is re-inserted. Geometric detail subtracted from a shape by knot removal is represented by an associated wavelet coefficient replacing one of the control points at a coarser level of detail. From the hierarchy of wavelet coefficients, perfect reconstruction of the original shape is obtained. Both knot removal and insertion have local impact. Wavelet synthesis and analysis are both computed in linear time, based on the lifting scheme for biorthogonal wavelets. The method is perfectly suited for multiresolution surface editing, progressive transmission, and compression of spline curves and surfaces.     

Spline multiresolution and wavelet-like decompositions

Splines are useful tools to represent, modify and analyze curves and they play an important role in various practical applications. We present a multiresolution approach to spline curves with arbitrary knots that provides good feature detection and localization properties for non-equally distributed geometric data. In addition, we show how equidistributed data and knot sequences can be efficiently handled using signal processing techniques.     

An Extension of Chaiken′s Algorithm to B-Spline Curves with Knots in Geometric Progression

Chaiken′s algorithm is a procedure for inserting new knots into uniform quadratic B-spline curves by doubling the control points and taking two successive averages. Lane and Riesenfeld showed that Chaiken′s algorithm extends to uniform B-spline curves of arbitrary degree. By generalizing the notion of successive averaging, we further extend Chaiken′s algorithm to B-spline curves of arbitrary degree for knot sequences in geometric and affine progression.     

一种B样条表面插值控制点的缩减方法

研究了从给定节点向量中选择节点进行B样条曲线插值的方法，并将此方法应用到行数据点不相同的B样条曲面插值，得到了一个通过对行节点矢量调整传递的曲面插值方法，理论分析和实验表明该方法可大量减少曲面控制点的数目．     

Creases and boundary conditions for subdivision curves

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work for arbitrary degrees for B-splines, these methods introduce unnecessary (ghost) control points.The situation is not so simple in modifying subdivision rules. Based on subdivision and subspace selection matrices, a novel approach to finding boundary and sharp subdivision rules that generalises to any degree is presented. Our approach leads to new higher-degree polynomial subdivision schemes with crease control without introducing new control points.     

Minimizing the Distortion of Affine Spline Motions

In a simple approach to the affine motion interpolation problem, an affine spline motion is generated that interpolates a given sequence of keyframes and approximately satisfies rigidity constraints and certain optimization criteria. An affine spline motion is generated so as to interpolate the given keyframes, and then it is progressively refined by knot insertion and degree elevation using an iterative optimization procedure.