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.     

Spherical DCB-spline surfaces with hierarchical and adaptive knot insertion

This paper develops a novel surface fitting scheme for automatically reconstructing a genus-0 object into a continuous parametric spline surface. A key contribution for making such a fitting method both practical and accurate is our spherical generalization of the Delaunay configuration B-spline (DCB-spline), a new non-tensor-product spline. In this framework, we efficiently compute Delaunay configurations on sphere by the union of two planar Delaunay configurations. Also, we develop a hierarchical and adaptive method that progressively improves the fitting quality by new knot-insertion strategies guided by surface geometry and fitting error. Within our framework, a genus-0 model can be converted to a single spherical spline representation whose root mean square error is tightly bounded within a user-specified tolerance. The reconstructed continuous representation has many attractive properties such as global smoothness and no auxiliary knots. We conduct several experiments to demonstrate the efficacy of our new approach for reverse engineering and shape modeling.     

Shape-preserving knot removal

Starting with a shape-preserving C¹ quadratic spline, we show how knots can be removed to produce a new spline which is within a specified tolerance of the original one, and which has the same shape properties. We give specific algorithms and some numerical examples, and also show how the method can be used to compute approximate best free-knot splines. Finally, we discuss how to handle noisy data, and develop an analogous knot removal algorithm for a monotonicity preserving surface method.     

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.     

B样条曲线节点插入和升阶的统一方法

利用 B样条基转换矩阵的有关结果 ,本文给出了 B样条曲线插入节点和升阶的统一方法及算法 .本文方法建立在严密的数学背景上 ,以简洁严谨的递推公式给出其算法的数学模型 ,相应的算法效率高且易实现 ,算法的时间复杂性为 O((k- k 1) kn) ,其中 k,k分别为升阶前后曲线的阶 ,n k 1为插入节点和升阶后的节点数 .而且 ,本文方法及算法使用灵活 ,适用范围广 ,可用于同时插入任意个相同的或不同的节点并升任意阶 ,也可用于只插入节点或只升阶 .Cohen等的 Oslo算法、升阶方法都是本文方法的特例 ,而且本文方法效率更高     

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．     

高精度三次参数样条曲线的构造

构造参数样条曲线的关键是选取节点，该文讨论了GC^2三次参数样条曲线需满足的连续性方程，提出了构造GC^2三次参数样条曲线的新方法，在讨论了平面有序五点确定一组三次多项式函数曲线，平面有序六点唯一确定一条三次多项式函数曲线的基础上，提出了计算相邻两区间上的节点的算法，构造的插值曲线具有三次多项式函数精,该文还以实例对新方法与其它方法构造的插值曲线的精度进行了比较。     

Smooth surface approximation to serial cross-sections

The reconstruction of the surface model of an object from 2D cross-sections plays an important role in many applications. In this paper, we present a method for surface approximation to a given set of 2D contours. The resulting surface is represented by a bicubic closed B-spline surface with C² continuity. The method performs the skinning of intermediate contour curves represented by cubic B-spline curves on a common knot vector, each of which is fitted to its contour points within a given accuracy. In order to acquire more compact representation for the surface, the method includes an algorithm for reducing the number of knots in the common knot vector. The proposed method provides a smooth and accurate surface model, yet realizes efficient data reduction. Some experimental results are given using synthetic and MRI data.     

Evolution of T-spline level sets for meshing non-uniformly sampled and incomplete data

Given a large set of unorganized point sample data, we propose a new framework for computing a triangular mesh representing an approximating piecewise smooth surface. The data may be non-uniformly distributed, noisy, and may contain holes. This framework is based on the combination of two types of surface representations, triangular meshes and T-spline level sets, which are implicit surfaces defined by refinable spline functions allowing T-junctions. Our method contains three main steps. Firstly, we construct an implicit representation of a smooth (C ² in our case) surface, by using an evolution process of T-spline level sets, such that the implicit surface captures the topology and outline of the object to be reconstructed. The initial mesh with high quality is obtained through the marching triangulation of the implicit surface. Secondly, we project each data point to the initial mesh, and get a scalar displacement field. Detailed features will be captured by the displaced mesh. Finally, we present an additional evolution process, which combines data-driven velocities and feature-preserving bilateral filters, in order to reproduce sharp features. We also show that various shape constraints, such as distance field constraints, range constraints and volume constraints can be naturally added to our framework, which is helpful to obtain a desired reconstruction result, especially when the given data contains noise and inaccuracies.     

Automatic knot adjustment using an artificial immune system for B-spline curve approximation

Reverse engineering transforms real parts into engineering concepts or models. First, sampled points are mapped from the object’s surface by using tools such as laser scanners or cameras. Then, the sampled points are fitted to a free-form surface or a standard shape by using one of the geometric modeling techniques. The curves on the surface have to be modeled before surface modeling. In order to obtain a good B-spline curve model from large data, the knots are usually respected as variables. A curve is then modeled as a continuous, nonlinear and multivariate optimization problem with many local optima. For this reason it is very difficult to reach a global optimum. In this paper, we convert the original problem into a discrete combinatorial optimization problem like in Yoshimoto et al. [F. Yoshimoto, M. Moriyama, T. Harada, Automatic knot placement by a genetic algorithm for data fitting with a spline, in: Proceedings of the International Conference on Shape Modeling and Applications, IEEE Computer Society Press, 1999, pp. 162-169] and Sarfraz et al. [M. Sarfraz, S.A. Raza, Capturing outline of fonts using genetic algorithm and splines, in: Fifth International Conference on Information Visualisation (IV’01), 2001, pp. 738-743]. Then, we suggest a new method that solves the converted problem by artificial immune systems. We think the candidates of the locations of knots as antibodies. We define the affinity measure benefit from Akaike’s Information Criterion (AIC). The proposed method determines the appropriate location of knots automatically and simultaneously. Furthermore, we do not need any subjective parameter or good population of initial location of knots for a good iterative search. Some examples are also given to demonstrate the efficiency and effectiveness of our method.     

Visualization of Seifert surfaces

The genus of a knot or link can be defined via Seifert surfaces. A Seifert surface of a knot or link is an oriented surface whose boundary coincides with that knot or link. Schematic images of these surfaces are shown in every text book on knot theory, but from these it is hard to understand their shape and structure. In this paper, the visualization of such surfaces is discussed. A method is presented to produce different styles of surface for knots and links, starting from the so-called braid representation. Application of Seifert's algorithm leads to depictions that show the structure of the knot and the surface, while successive relaxation via a physically based model gives shapes that are natural and resemble the familiar representations of knots. Also, we present how to generate closed oriented surfaces in which the knot is embedded, such that the knot subdivides the surface into two parts. These closed surfaces provide a direct visualization of the genus of a knot. All methods have been integrated in a freely available tool, called SeifertView, which can be used for educational and presentation purposes.     

Trivariate Biharmonic B‐Splines

In this paper, we formulate a novel trivariate biharmonic B‐spline defined over bounded volumetric domain. The properties of bi‐Laplacian have been well investigated, but the straightforward generalization from bivariate case to trivariate one gives rise to unsatisfactory discretization, due to the dramatically uneven distribution of neighbouring knots in 3D. To ameliorate, our original idea is to extend the bivariate biharmonic B‐spline to the trivariate one with novel formulations based on quadratic programming, approximating the properties of localization and partition of unity. And we design a novel discrete biharmonic operator which is optimized more robustly for a specific set of functions for unevenly sampled knots compared with previous methods. Our experiments demonstrate that our 3D discrete biharmonic operators are robust for unevenly distributed knots and illustrate that our algorithm is superior to previous algorithms.     

Adaptive knot placement using a GMM-based continuous optimization algorithm in B-spline curve approximation

One of the key problems in using B-splines successfully to approximate an object contour is to determine good knots. In this paper, the knots of a parametric B-spline curve were treated as variables, and the initial location of every knot was generated using the Monte Carlo method in its solution domain. The best km knot vectors among the initial candidates were searched according to the fitness. Based on the initial parameters estimated by an improved k-means algorithm, the Gaussian Mixture Model (GMM) for every knot was built according to the best km knot vectors. Then, the new generation of the population was generated according to the Gaussian mixture probabilistic models. An iterative procedure repeating these steps was carried out until a termination criterion was met. The GMM-based continuous optimization algorithm could determine the appropriate location of knots automatically. A set of experiments was then implemented to evaluate the performance of the new algorithm. The results show that the proposed method achieves better approximation accuracy than methods based on artificial immune system, genetic algorithm or squared distance minimization (SDM).     

B样条曲线曲面GC2扩展

提出了一个扩展B样条曲线曲面的新方法,扩展B样条曲线曲面的关键是为新增加的点确定节点值,新方法的基本思想是：首先,B样条曲线和扩展部分在连接点处满足GC^2连续,用能量极小化方法确定扩展部分的曲线形状,通过对曲线重新参数化使两部分曲线满足C^2连续,进而确定新增加点的节点值,新B样条曲线的控制点由一个显式递推公式计算,原B样条曲线和扩展后的部分合在一起形成一条新的B样条曲线,新的B样条曲线满足原B样条曲线和扩展的点,文章还讨论了运用该方法进行B样条曲面扩展,且以实例对新方法与其它方法进行了比较,结果表明新方法的光顺性得到了明显改善,曲率变化更平坦,且有较小的旋转数指标。     

A New Approach for Direct Manipulation of Free-Form Curve

There is an increasing demand for more intuitive methods for creating and modifying free-form curves and surfaces in CAD modeling systems. The methods should be based not only on the change of the mathematical parameters, such as control points, knots, and weights, but also on the user's specified constraints and shapes. This paper presents a new approach for directly manipulating the shape of a free-form curve, leading to a better control of the curve deformation and a more intuitive CAD modeling interface. The user's intended deformation of a curve is automatically converted into the modification of the corresponding NURBS control points and knot sequence of the curve. The algorithm for this approach includes curve elevation, knot refinement, control point repositioning, and knot removal. Several examples shown in this paper demonstrate that the proposed method can be used to deform a NURBS curve into the desired shape. Currently, the algorithm concentrates on the purely geometric consideration. Further work will include the effect of material properties.     

多结点样条插值及其多尺度细化算法

针对风线与曲面拟事问题，研究多结点样条插值方法。这类方法具有基数型，显式计算及局部性等优点。主要的新结果是：对多结点样条基本函数的构造给出了新的表述；提出了一类新的不带移动的混合形多结点样条基本函数；基于多尺度分析的思想，给出了一种自适应的细化算法，它对消减采样数据的相关性是简便有效的。     

Sub-pixel distance maps and weighted distance transforms

A new framework for computing the Euclidean distance and weighted distance from the boundary of a given digitized shape is presented. The distance is calculated with sub-pixel accuracy. The algorithm is based on a equal distance contour evolution process. The moving contour is embedded as a level set in a time varying function of higher dimension. This representation of the evolving contour makes possible the use of an accurate and stable numerical scheme, due to Osher and Sethian [22]. The relation between the classical shape from shading problem and the weighted distance transform is presented, as well as an algorithm that calculates the geodesic distance transform on surfaces.     

连续等距区间上积分值的二次样条插值

目的 在现实中,某些插值问题结点处的函数值往往是未知的,而仅仅已知一些区间上的积分值。为此提出一种给定已知函数在连续等距区间上的积分值构造二次样条插值函数的方法。方法 首先,利用二次B样条基函数的线性组合去满足给定的积分值和两个端点插值条件,该插值问题等价于求解n+2个方程带宽为3的线性方程组。然后,运用算子理论给出二次样条插值函数的误差估计,继而得到二次样条函数逼近结点处的函数值时具有超收敛性。最后,通过等距区间上积分值的线性组合逼近两个端点的函数值方法实现了不带任何边界条件的积分型二次样条插值问题。结果 选取低频率函数,对积分型二次样条插值方法和改进方法分别进行数值测试,发现这两种方法逼近效果都是良好的。同样,选取高频率函数对积分型二次样条插值方法进行数值实验,得到数值收敛阶与理论值相一致。结论 实验结果表明,本文算法相比已有的方法更简单有效,对改进前后的二次样条插值函数在逼近结点处的函数值时的超收敛性得到了验证。该方法对连续等距区间上积分值的函数重构具有普适性。     

B-spline surface fitting based on adaptive knot placement using dominant columns

By expanding the idea of B-spline curve fitting using dominant points (Park and Lee 2007 [13]), we propose a new approach to B-spline surface fitting to rectangular grid points, which is based on adaptive knot placement using dominant columns along u- and v-directions. The approach basically takes approximate B-spline surface lofting which performs adaptive multiple B-spline curve fitting along and across rows of the grid points to construct a resulting B-spline surface. In multiple B-spline curve fitting, rows of points are fitted by compatible B-spline curves with a common knot vector whose knots are computed by averaging the parameter values of dominant columns selected from the points. We address how to select dominant columns which play a key role in realizing adaptive knot placement and thereby yielding better surface fitting. Some examples demonstrate the usefulness and quality of the proposed approach.     

Knot placement for piecewise polynomial approximation of curves

For piecewise polynomial representation of curves, an algorithm to create knots is presented. The aim is to minimize the interpolation error for a given number of knots or, conversely, the number of knots needed to interpolate within a tolerance. The method used is a modification of de Boor's knot placement scheme. The algorithm described in this paper has been realized in the CADCAM system SYRKO, a Daimler-Benz development for car body design and manufacturing.