一种基于切线的曲线直线拟合的矢量化算法

提出一种在位图矢量化系统中用直线拟合曲线的新算法.算法利用曲线的一系列切线寻找曲线上与切线距离为门限距离的点作为直线拟合的特征点.算法包括两个步骤:尖点提取和直线逼近.第一步将曲线中具有局部曲率极大值的尖点提取出来,并利用尖点将曲线分割为多个曲线片断;第二步利用曲线片断的切线系找出特征点,再分别用直线依次将邻近两特征点相连,即完成拟合过程.经实验表明,算法既能准确提取边界形状的特征关键点,又能有效地实现对曲线的直线拟合.     

Efficient circular arc interpolation based on active tolerance control

In this paper, we present an efficient sub-optimal algorithm for fitting smooth planar parametric curves by G¹ arc splines. To fit a parametric curve by an arc spline within a prescribed tolerance, we first sample a set of points and tangents on the curve adaptively as well as with enough density, so that an interpolation biarc spline curve can be with any desired high accuracy. Then, we construct new biarc curves interpolating local triarc spirals explicitly based on the control of permitted tolerances. To reduce the segment number of fitting arc spline as much as possible, we replace the corresponding parts of the spline by the new biarc curves and compute active tolerances for new interpolation steps. By applying the local biarc curve interpolation procedure recursively and sequentially, the result circular arcs with no radius extreme are minimax-like approximation to the original curve while the arcs with radius extreme approximate the curve parts with curvature extreme well too, and we obtain a near optimal fitting arc spline in the end. Even more, the fitting arc spline has the same end points and end tangents with the original curve, and the arcs will be jointed smoothly if the original curve is composed of several smooth connected pieces. The algorithm is easy to be implemented and generally applicable to circular arc interpolation problem of all kinds of smooth parametric curves. The method can be used in wide fields such as geometric modeling, tool path generation for NC machining and robot path planning, etc. Several numerical examples are given to show the effectiveness and efficiency of the method.     

3-D curve matching using splines

A machine vision algorithm to find the longest common subcurve of two 3-D curves is presented. The curves are represented by splines fitted through sequences of sample points extracted from dense range data. The approximated 3-D curves are transformed into 1-D numerical strings of rotation and translation invariant shape signatures, based on a multiresolution representation of the curvature and torsion values of the space curves. The shape signature strings are matched using an efficient hashing technique that finds longest matching substrings. The results of the string matching stage are later verified by a robust, least-squares, 3-D curve matching technique, which also recovers the Euclidean transformation between the curves being matched. This algorithm is of average complexity O(n) where n is the number of the sample points on the two curves. The algorithm has applications in assembly and object recognition tasks. Results of assembly experiments are included.     

Multiple View Geometry of General Algebraic Curves

We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the recovery of camera geometry from matching curves. We first show how one can compute, without any knowledge on the camera, the homography induced by a single planar curve. Then we continue with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus.We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation in which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d – 1). Moreover using this representation, we address the problem of homography recovery for planar curves, (ii) dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of curve degree alone.     

基于双参数的几何细分法

提出一种基于两个参数的几何细分方法。首先,借助于标准型的二次有理Bézier 曲 线公式,以相邻的两个初始控制点及其切向量所在直线的交点作为该二次有理Bézier 曲线的控制 顶点;同时,选取分点参数值t  0.5,并以该曲线的权因子作为控制顶点的参数λ,计算新增控 制顶点。其次,定义每个顶点的临时切向量,以每点及其相邻两点确定该点的圆切向;引入切向 量的控制参数,从而确定该顶点新切向量的计算公式。然后,从理论上证明了该方法的保凸性 与收敛性。取定切向量参数=0,重新定义每步的权因子参数λ,其极限曲线是C1连续的分段二 次有理Bézier 曲线;令=1,在每一步骤中采用不同的权因子参数λ 求新增点,具有保圆性。最 后,通过一些实例说明了该方法的有效性。     

A geometric constraint on curve networks suitable for smooth interpolation

A key problem when interpolating a network of curves occurs at vertices: an algebraic condition, called the vertex enclosure constraint, must hold wherever an even number of curves meet. This paper recasts the vertex enclosure constraint in terms of the local geometry of the curve network. This allows formulating a new geometric constraint, related to Euler’s Theorem on local curvature. The geometric constraint implies the vertex enclosure constraint and is equivalent to it where four curve segments meet without forming an X. Also the limiting case of collinear curve tangents is analyzed.     

Automatic construction of a cubic B-spline representation for a general curve

This paper describes an algorithm which, when given a digitized curve of general shape, without a direct analytical representation, automatically finds suitable knot points and the cubic B-spline polygon vertices corresponding to that curve. That B-spline polygon is used to regenerate the curve graphically. The method is generally applicable in many fields, including computer aided design, where a need exists to represent arbitrary curves with a given accuracy. In this paper results applied to pictures of archeological vessels are shown, with a discussion of accuracy and compactness of the representation.The algorithm will represent a curve accurately, using less than 10% of the storage required if all the digitized points of the curve were stored.     

一种基于模糊线段生长的离散曲率估计

针对Z2空间中的8连通边界曲线,提出一种基于模糊线段生长的离散曲率估计方法。该方法引入“序”为d的模糊线段生长算法,将曲线上生长出的最长模糊线段作为切线的近似,并计算出了离散曲率。实验表明,该方法不仅较好地反映了曲线点的局部特性,而且增加了对离散曲线噪声的适应能力,离散曲率估计性能获得了显著的提高。     

Approximate merging of B-spline curves via knot adjustment and constrained optimization

This paper addresses the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. The basic idea of the approach is to find the conditions for precise merging of two B-spline curves, and perturb the control points of the curves by constrained optimization subject to satisfying these conditions. To obtain a merged curve without superfluous knots, we present a new knot adjustment algorithm for adjusting the end k knots of a kth order B-spline curve without changing its shape. The more general problem of merging curves to pass through some target points is also discussed.     

Möbius变换下四次有理抛物-PH曲线的C2 Hermite插值

目的 曲线插值问题在机器人设计、机械工业、航天工业等诸多现代工业领域都有广泛的应用，而已知端点数据的Hermite插值是计算机辅助几何设计中一种常用的曲线构造方法，本文讨论了一种偶数次有理等距曲线，即四次抛物-PH曲线的C² Hermite插值问题。方法 基于M bius变换引入参数，利用复分析的方法构造了四次有理抛物-PH曲线的C² Hermite插值，给出了具体插值算法及相应的Bézier曲线表示和控制顶点的表达式。结果 通过给出"合理"的端点插值数据，以数值实例表明了该算法的有效性，所得12条插值曲线中，结合最小绝对旋转数和弹性弯曲能量最小化两种准则给出了判定满足插值条件最优曲线的选择方法，并以具体实例说明了与其他插值方法的对比分析结果。结论 本文构造了M bius变换下的四次有理抛物-PH曲线的C² Hermite插值，在保证曲线次数较低的情况下，达到了连续性更高的插值条件，计算更为简单，插值效果明显，较之传统奇数次PH曲线具有更加自然的几何形状，对偶数次PH曲线的相关研究具有一定意义。     

Data Approximation Using Biarcs

. An algorithm for data approximation with biarcs is presented. The method uses a specific formulation of biarcs appropriate for parametric curves in Bézier or NURBS formulation. A base curve is applied to obtain tangents and anchor points for the individual arcs joining in G 1 continuity. Data sampled from circular arcs or straight line segments is represented precisely by one biarc. The method is most useful in numerical control to drive the cutter along straight line or circular paths.     

Euler Spiral for Shape Completion

In this paper we address the curve completion problem, e.g., the geometric continuation of boundaries of objects which are temporarily interrupted by occlusion. Also known as the gap completion or shape completion problem, this problem is a significant element of perceptual grouping of edge elements and has been approached by using cubic splines or biarcs which minimize total curvature squared (elastica), as motivated by a physical analogy. Our approach is motivated by railroad design methods of the early 1900's which connect two rail segments by transition curves, and by the work of Knuth on mathematical typography. We propose that in using an energy minimizing solution completion curves should not penalize curvature as in elastica but curvature variation. The minimization of total curvature variation leads to an Euler Spiral solution, a curve whose curvature varies linearly with arclength. We reduce the construction of this curve from a pair of points and tangents at these points to solving a nonlinear system of equations involving Fresnel Integrals, whose solution relies on optimization from a suitable initial condition constrained to satisfy given boundary conditions. Since the choice of an appropriate initial curve is critical in this optimization, we analytically derive an optimal solution in the class of biarc curves, which is then used as the initial curve. The resulting interpolations yield intuitive interpolation across gaps and occlusions, and are extensible, in contrast to the scale-invariant version of elastica. In addition, Euler Spiral segments can be used in other applications of curve completions, e.g., modeling boundary segments between curvature extrema or modeling skeletal branch geometry.     

B样条驱动纹理生成的构造式信息隐藏

目的 传统构造式图像信息隐藏算法通常直接将图像空域特征与秘密信息关联,对算法的安全性造成威胁。因此,本文将曲线绘制函数与信息隐藏相结合,提出一种以B样条控制点为特征,在图像空域间接隐藏信息的算法。方法 算法主要分为信息隐藏及信息提取两阶段。在信息隐藏阶段,发送方首先通过选取初始控制点、仿射变换及B样条曲线绘制生成多条参考曲线,然后利用曲线控制点的位置隐藏信息,最后为图像填充颜色,即完成含密纹理图像的构造。在信息提取阶段,提取方根据纹理曲线和图像颜色获得含密曲线及参考曲线,经对照计算即可提取出秘密信息。结果 本算法具有较高的隐藏容量、鲁棒性和安全性。实验结果表明,由本文算法生成的800×800像素图像,其最高隐藏容量可达2870bits,分别是另两种典型构造式信息隐藏算法的6.7和3.4倍,且在质量因子为10的JPEG（joint photographic experts group）压缩攻击下的提取误码率可低至0,优于鲁棒较强的选择式信息隐藏算法LDA-DCT（robust coverless image steganography based on DCT and LDA topic classification）以及与之类似的构造式信息隐藏算法。同时,抗隐写分析检测实验表明,在隐藏容量小于250bits时检测误差趋近于0.5。结论 本文以B样条曲线控制点为特征,在纹理图像的绘制过程中隐藏信息,有效提高了传统构造式图像信息隐藏算法的安全性、隐藏容量和鲁棒性。     

Incremental Labelling of Voronoi Vertices for Shape Reconstruction

We present an incremental Voronoi vertex labelling algorithm for approximating contours, medial axes and dominant points (high curvature points) from 2D point sets. Though there exist many number of algorithms for reconstructing curves, medial axes or dominant points, a unified framework capable of approximating all the three in one place from points is missing in the literature. Our algorithm estimates the normals at each sample point through poles (farthest Voronoi vertices of a sample point) and uses the estimated normals and the corresponding tangents to determine the spatial locations (inner or outer) of the Voronoi vertices with respect to the original curve. The vertex classification helps to construct a piece‐wise linear approximation to the object boundary. We provide a theoretical analysis of the algorithm for points non‐uniformly (ε‐sampling) sampled from simple, closed, concave and smooth curves. The proposed framework has been thoroughly evaluated for its usefulness using various test data. Results indicate that even sparsely and non‐uniformly sampled curves with outliers or collection of curves are faithfully reconstructed by the proposed algorithm.     

Canopy surface technique for parametric blending

A method based on canopy surfaces is presented for blending parametric surfaces. The blend designed using this method gives freedom to the designer in selecting i) the primitive surfaces to be blended, ii) the contact curves lying on them (in which the blend meets the primitives), iii) the endtangent directions along the contact curves (which are used to ensure tangentplane continuity), and iv) the shape of the cross-sectional curve. An important feature of this method, which is not seen in earlier methods, is the use of end tangents to ensure theC ¹ continuity of iso-parametric curves across the junctions between the blend and the primitives in addition to the tangent-plane continuity.     

利用距离与内能极小平滑链接Bézier曲线

目的 对于满足低阶连续的链接Bézier曲线,提高曲线之间的连续性以达到平滑的目的,需要对曲线的控制顶点进行相应调整。因此,可根据具体的目标对需要调整的控制顶点进行优化选取,使得平滑后的链接曲线满足相应的要求。针对这一问题,给出了3种目标下优化调整控制顶点的方法。方法 首先对讨论的问题进行描述,分别指出链接Bézier曲线从C⁰连续平滑为C¹连续和从C¹连续平滑为C²连续两种情形需调整的控制顶点;然后分别给出两种情形下,以新旧控制顶点距离极小为目标、曲线内能极小为目标、新旧控制顶点距离与曲线内能同时极小为目标,对链接Bézier曲线进行平滑的方法,最后对3种极小化方法进行对比,并指出了不同方法的适用场合。结果 数值算例表明,距离极小化方法调整后的控制顶点偏离原控制顶点的距离相对较小,适合于控制顶点取自于实物时的应用场合;内能极小化方法获得的链接曲线内能相对较小,适合于要求曲线能量尽可能小的应用场合;距离与内能同时极小化方法兼顾了新旧控制顶点的距离和链接曲线的内能,适合于对两个目标都有要求的应用场合。结论 提出的方法为链接Bézier曲线的平滑提供了3种有效手段,且易于实现,对其他类型链接曲线的平滑具有参考价值。     

C3连续的单位四元数插值样条曲线

目的 构造一类C³连续的单位四元数插值样条曲线,证明它的插值性和连续性,并把它应用于刚体关键帧动画设计中。方法 利用R³空间中插值样条曲线的5次多项式调配函数的累和形式构造了S³空间中单位四元数插值样条曲线,它不仅能精确通过一系列给定的方向,而且能生成C³连续的朝向曲线。结果 与Nielson的单位四元数均匀B样条插值曲线的迭代构造方法相比,所提方法避免了为获取四元数B样条曲线控制顶点对非线性方程组迭代求解的过程,提高了运算效率;与单位四元数代数三角混合插值样条曲线的构造方法（Su方法）相比,所提方法只用到多项式基,运算速度更快。本例中创建关键帧动画所需的时间与Nielson方法和Su方法相比平均下降了73%和33%。而且,相比前两种方法,所提方法产生的四元数曲线连续性更高,由C²连续提高到C³连续,这意味着动画中刚体的朝向变化更加自然。结论 仿真结果表明,本文方法对刚体关键帧动画设计是有效的,对实时性和流畅性要求高的动画设计场合尤为适用。     

A technique for smoothing scattered data with conic sections

Piecewise rational quadratic curves are frequently used in many fields of computer science to represent curved shapes. For a large number of applications of computer aided design, the representation of data with a piecewise parametric curve that lies close to the data is an important consideration. This paper presents a geometric solution to the problem of automatically generating piecewise parametric rational quadratic polynomial approximations to shapes from sampled data. The algorithm takes a set of sample points, automatically generates tangents at some points and derives a piecewise rational quadratic curve that lies close to the data points. Combining this algorithm with the biquadratic search to subdivide the data if it cannot be represented with a single arc, we have a very stable algorithm that gives good results over a range of shapes and applications.     

一种基于自适应模糊线段的离散曲率估计

针对Z 2 空间中8-连通离散曲线的噪声影响,提出"序"为d的模糊线段生长算法.将曲线点上生长出的最长模糊线段作为切线的近似,并根据曲线局部粗糙度自适应地选择序,在此基础上进行离散曲率估计.实验结果表明:通过自适应选择序值,最长离散模糊线段不仅较好地反映了曲线点的局部特性,而且增加了对离散曲线噪声的适应能力,离散曲率估计的性能明显提高.     

Computing medial axes of generic 3D regions bounded by B-spline surfaces

A new approach is presented for computing the interior medial axes of generic regions in R³ bounded by C⁽⁴⁾-smooth parametric B-spline surfaces. The generic structure of the 3D medial axis is a set of smooth surfaces along with a singular set consisting of edge curves, branch curves, fin points and six junction points. In this work, the medial axis singular set is first computed directly from the B-spline representation using a collection of robust higher order techniques. Medial axis surfaces are computed as a time trace of the evolving self-intersection set of the boundary under the the eikonal (grassfire) flow, where the bounding surfaces are dynamically offset along the inward normal direction. The eikonal flow results in special transition points that create, modify or annihilate evolving curve fronts of the (self-) intersection set. The transition points are explicitly identified using the B-spline representation. Evolution of the (self-) intersection set is computed by adapting a method for tracking intersection curves of two different surfaces deforming over generalized offset vector fields. The proposed algorithm accurately computes connected surfaces of the medial axis as well its singular set. This presents a complete solution along with accurate topological structure.