隐式T样条实现封闭曲面重建

为了简化法向偏差约束条件和优化光滑能量项,提出一种隐式T样条曲面重建算法.首先利用八叉树及其细分过程从采样点集构造三维T网格,以确定每个控制系数对应的混合函数;然后基于隐式T样条曲面建立目标函数,利用偏移曲面点集控制法向,采用广义交叉检验(GCV)方法估计最优光滑项系数,并依据最优化原理将该问题转化为线性方程组求解得到控制系数,从而实现三角网格曲面到光滑曲面的重建.在误差较大的区域插入控制系数进行T网格局部修正,使得重建曲面达到指定精度.该算法使重建曲面C1连续条件得到松弛,同时给出最优的光顺项系数估计,较好地解决了封闭曲面的重建问题.实例结果表明,文中算法逼近精度高,运算速度快,仿真结果逼真.     

残缺网格模型的快速B样条曲面重建

针对残缺的三角网格模型,提出一种将网格模型的散乱数据点转化为有序阵列点再进行B样条曲面快速重建的算法.首先确定最小二乘平面上的一个矩形参数域,再构造出一个平面阵列点列,并部分映射到三维网格上;然后利用空间阵列点的邻域信息估计4个角点的空间坐标,并构造径向基函数曲面,用于补充空间阵列点列中残缺的数据;最后利用有序点列拟合的高效性构造B样条曲面.实验结果表明:该算法速度快、拟合精度高、鲁棒性强,重建的曲面具有良好的光顺性和可延伸性,适用于逆向工程中对经过数据分割后的网格模型的自由曲面重建.     

多个B样条曲面光滑重建技术研究

提出一种对多个B样条曲面光滑重建的拟合算法，并通过对牙模的例子进行重建给出了计算结果。说明新的拟合算法既能够处理大量测量点，又能够提高重建速度，很好地解决了多个曲面的光滑重建问题。     

T样条曲面在B样条曲面局部拼接中的应用

为解决相邻B样条曲面在局部拼接时由于曲面细分而加入冗余控制点的问题,利用T样条曲面良好的局部细分性,提出一种更有效的T样条曲面局部拼接法。将相邻B样条曲面精确转换为两张全行列T样条曲面;利用T样条曲面的局部均匀细分算法得到局部拼接边界上的一致性,在曲面上加入局部控制点;将局部细分后的曲面连续拼接为一张控制点更少的T样条组合曲面。与B样条曲面的细分拼接法相比,该方法在曲面细分时不加入任何冗余控制点,在局部拼接后能够得到一张更精简的组合曲面。     

T样条调配函数线性无关性的判定算法

针对T样条调配函数的公开问题——T样条曲面表达式中的调配函数是否具有线性无关性,提出一种T样条调配函数线性无关性的验证算法.首先分析了T样条调配函数的数学性质,给出T样条调配函数线性无关的充要条件和充分条件,通过充分条件可以较容易地对一些特殊的T样条调配函数的线性无关性进行判定;然后结合充要条件和充分条件给出一种T样条调配函数线性无关性的验证算法.最后通过实例验证了该算法的有效性.文中结果对于开展T样条造型算法方面的研究有重要意义.     

基于隐式T样条的曲面重构算法

提出隐式T样条曲面，将T网格从二维推广到三维情形，同时利用八叉树及其细分过程，从无结构散乱点数据集构造T网格，利用曲面拟合模型将曲面重构问题转化为最优化问题；然后基于隐式T样条曲面将最优化问题通过矩阵形式表述，依据最优化原理将该问题转化成线性方程组，通过求解线性方程组解决曲面重构问题；最后结合计算实例进行讨论．该方法能较好地解决曲面重构问题，与传统张量B样条函数相比，能效地减少未知控制系数与计算量．     

任意散乱点集的B-样条曲面重建

为了解决工业设计中复杂形体的曲面造型问题,提出了一种张量积型的低阶B-样条曲面重建算法。先将采集到的任意拓扑形状的散乱数据点进行三次不同的参数化得到四边形控制网格,然后再采用张量积型的双二次、双三次B-样条进行拟合,在拟合的过程中采用距离函数来控制拟合误差,得到光滑的曲面。运用该方法,直接对初始散乱点集进行重建,方法简单易实施,重建效率高并且重建后的样条曲面自然满足切平面连续。与以往的方法相比,该方法在逆向工程中可以在保证连续性的情况下,得到精准的结果曲面,提高了曲面造型的质量和效率。     

误差有界的低扭曲非结构T样条曲面拟合

为了计算对于任意复杂拓扑拟合域的低扭曲、满足拟合误差阈值和较少控制点的非结构T样条拟合曲面,提出了一种逐步求解的方法。首先,生成与拟合域具有相同拓扑的多立方体作为参数域,通过多次重参数化过程优化待拟合表面和参数域之间的对应关系,得到一个适用于获得低拟合误差样条曲面的低扭曲映射。与此同时,利用非结构T样条局部细分的性质对不满足拟合误差阈值的区域进行自适应局部细分,得到满足拟合误差阈值的低扭曲样条曲面。接下来,提出一种删除冗余控制顶点的拟合曲面简化策略,在满足拟合误差阈值和低扭曲的基础上删除冗余的控制顶点,得到控制顶点数量较少的误差有界的低扭曲非结构T样条拟合曲面。在各种复杂模型上证明了此方法的有效性。与最新的方法相比,该方法以更少的控制顶点实现了更低的参数化扭曲。     

逆向工程中平面轮廓线数据的B样条曲面拟合

利用统计学的知识,对空间数据点参数值分布情况进行了分析,给出了曲面拟合过程中节点矢量的确定方法。在此基础上给出了曲面轮廓线数据的B样条曲面拟合算法以及应用该方法对平面轮廓线扫描数据进行B样条曲面拟合的算例。     

带法向约束的隐式T样条曲线重构

目的 隐式曲线能够描述复杂的几何形状和拓扑结构,而传统的隐式B样条曲线的控制网格需要大量多余的控制点满足拓扑约束。有些情况下,获取的数据点不仅包含坐标信息,还包含相应的法向约束条件。针对这个问题,提出了一种带法向约束的隐式T样条曲线重建算法。方法 结合曲率自适应地调整采样点的疏密,利用二叉树及其细分过程从散乱数据点集构造2维T网格;基于隐式T样条函数提出了一种有效的曲线拟合模型。通过加入偏移数据点和光滑项消除额外零水平集,同时加入法向项减小曲线的法向误差,并依据最优化原理将问题转化为线性方程组求解得到控制系数,从而实现隐式曲线的重构。在误差较大的区域进行T网格局部细分,提高重建隐式曲线的精度。结果 实验在3个数据集上与两种方法进行比较,实验结果表明,本文算法的法向误差显著减小,法向平均误差由10^-3数量级缩小为10^-4数量级,法向最大误差由10^-2数量级缩小为10^-3数量级。在重构曲线质量上,消除了额外零水平集。与隐式B样条控制网格相比,3个数据集的T网格的控制点数量只有B样条网格的55.88%、39.80%和47.06%。结论 本文算法能在保证数据点精度的前提下,有效降低法向误差,消除了额外的零水平集。与隐式B样条曲线相比,本文方法减少了控制系数的数量,提高了运算速度。     

Free-form deformation with weighted T-spline

A new method of free-form deformation, w-TFFD, is proposed, for which an original shape is deformed using weighted T-spline volumes. We generalize T-splines to weighted T-spline volumes that also permit T-junctions. Weighted T-spline volumes are a natural generalization of NURBS volumes but permit more flexible control lattices. Thus, w-TFFD holds many virtues of traditional FFDs and is more adaptive to objects with arbitrary topology or complex shape. The lattices can be automatically generated and approximate the shape of the object arbitrarily close by octree subdivision.Besides constructing and deforming a multiresolution lattice, users can also sculpt specific local details to their required shape by modifying weights. A set of direct-acting tools that are similar to previously proposed techniques can be applied to w-TFFD.     

FasTFit: A fast T-spline fitting algorithm

T-spline has been recently developed to represent objects of arbitrary shapes using a smaller number of control points than the conventional NURBS or B-spline representations in computer aided design, computer graphics, and reverse engineering. However, existing methods for fitting a T-spline over a point cloud are slow. By shifting away from the conventional iterative fit-and-refine paradigm, we present a novel split-connect-fit algorithm to more efficiently perform the T-spline fitting. Through adaptively dividing a point cloud into a set of B-spline patches, we first discover a proper topology of T-spline control points, i.e., the T-mesh. We then connect these B-spline patches into a single T-spline surface with different continuity options between neighboring patches according to the data. The T-spline control points are initialized from their correspondences in the B-spline patches, which are refined by using a conjugate gradient method. In experiments using several types of large-sized point clouds, we demonstrate that our algorithm is at least an order of magnitude faster than state-of-the-art algorithms while provides comparable or better results in terms of quality and conciseness.     

Modified T-splines

T-splines are a generalization of NURBS surfaces, the control meshes of which allow T-junctions. T-splines can significantly reduce the number of superfluous control points in NURBS surfaces, and provide valuable operations such as local refinement and merging of several B-splines surfaces in a consistent framework. In this paper, we propose a variant of T-splines called Modified T-splines. The basic idea is to construct a set of basis functions for a given T-mesh that have the following nice properties: non-negativity, linear independence, partition of unity and compact support. Due to the good properties of the basis functions, the Modified T-splines are favorable both in adaptive geometric modeling and isogeometric analysis.     

An algorithm of determining T-spline classification

T-splines are a new surface modeling technology, whose theoretical framework is not still well founded. Aiming at the problem that how to classify T-splines this paper gives a mathematical analysis, and a sufficient condition of standard T-splines is also given. Then this paper presents an algorithm of determining the T-spline classification, and we can get the inherently mathematical properties of T-splines by the algorithm. At last, the experimental results verify the effectiveness of our algorithm. The results in this paper play an important role in the research on T-spline theory and T-spline modeling algorithms.     

Automatic reconstruction of B-spline surfaces with constrained boundaries

The aim of this study is to present an automatic surface reconstruction method that can take practical restrictions on scanned points into consideration and efficiently and reliably output a group of G¹ surfaces. The proposed method is mainly composed of three phases: quadrangle frame generation, point and curve networks planning, and surface patches reconstruction. In the first phase, the original triangle mesh is reduced and converted into a quadrangle mesh, the edges of which serve as the frame of the surfaces. In the second phase, the boundary data of the surfaces are prepared. These include a network of serial points, frame curves and surface normals which are also expressed as curves. In the final phase, surface initialization, harmonization mapping and surface warping are presented to yield the desired surfaces. The main advantage of the proposed method is that it can relax the pre-processing of a scanned triangle mesh, and hence, increase the efficiency and quality of the surface reconstruction. Several examples of various types of air bags are presented to demonstrate the feasibility of the proposed method.     

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.     

Reconsideration of T-spline data models and their exchanges using STEP

T-spline is a new approach to define freeform surfaces with relatively less control points than NURBS and is able to represent a model using a single surface without joining errors. Whereas, the complexity of T-spline data models leads numerous difficulties in its programming, which hinders the research and development of T-spline technologies. In addition, the data exchange of T-spline models still remains on a primitive level, and no standardized data format has been published so far. This article gives a reconsideration to the existing T-spline definitions, and proposes a set of redesigned data models which have much more understanding conveniences to both human and computer. Moreover, STEP-compliant data models are designed using the proposed T-spline models to standardize their data exchange between different CAx systems. The combination of T-spline with other product models in ISO 10303 makes it convenient to exchange the versatile resource data in a hybrid neutral file. A prototype system is developed for the validation purpose, which consists of a TSM-to-STEP convertor, a STEP parser and a T-spline kernel. Using the developed prototype system, one can automatically convert a Rhino system exported TSM file to a STEP file in the P21 format, which can be then parsed using the STEP reader and processed by the T-spline kernel. Some testing examples show that the proposed data models are much more efficient in processing and exchanging the T-spline data.     

3D shape metamorphosis based on T-spline level sets

We propose a new method for 3D shape metamorphosis, where the in-between objects are constructed by using T-spline scalar functions. The use of T-spline level sets offers several advantages: First, it is convenient to handle complex topology changes without the need of model parameterization. Second, the constructed objects are smooth (C² in our case). Third, high quality meshes can be easily obtained by using the marching triangulation method. Fourth, the distribution of the degrees of freedom can be adapted to the geometry of the object. Given one source object and one target object, we firstly find a global coordinate transformation to approximately align the two objects. The T-spline control grid is adaptively generated according to the geometry of the aligned objects, and the initial T-spline level set is found by approximating the signed distance function of the source object. Then we use an evolution process, which is governed by a combination of the signed distance function of the target object and a curvature-dependent speed function, to deform the T-spline level set until it converges to the target shape. Additional intermediate objects are inserted at the beginning/end of the sequence of generated T-spline level sets, by gradually projecting the source/target object to the initial/final T-spline level set. A fully automatic algorithm is developed for the above procedures. Experimental results are presented to demonstrate the effectiveness of our method.