同构平面三角网格的保凸变形方法

对于具有不同凸边界的同构平面三角网格的变形,提出了一种简单、有效的方法．该方法结合了两种已有的算法．能够保证网格边界在变形过程中始终保持凸性,且任意时刻的中间网格与初末网格同构,即不产生自交现象;同时文中方法实现了两个凸多边形的保凸变形。     

基于直线和二次曲线的多回路平面曲线分割

提出了一种多回路平面曲线的分割算法。算法以直线段和二次曲线段作为拟合的基本单元，通过边缘跟踪把多回路的平面曲线分割成曲线段或单回路的封闭曲线，然后在曲线曲率不连续处继续分割。针对一次分割时可能会产生的迷向问题，提出了基于对偶原理的线段合并技术。该算法的结果能简单有效地描述场景中的物体，使高级视觉任务更简单。实验结果表明算法能取得较好的效果。     

Invariant Signatures of Closed Planar Curves

We prove that any subset of ℝ² parametrized by a C ¹ periodic function and its derivative is the Euclidean invariant signature of a closed planar curve. This solves a problem posed by Calabi et al. (Int. J. Comput. Vis. 26:107–135, 1998). Based on the proof of this result, we then develop some cautionary examples concerning the application of signature curves for object recognition and symmetry detection as proposed by Calabi et al.

Area-Based Medial Axis of Planar Curves

A new definition of affine invariant medial axis of planar closed curves is introduced. A point belongs to the affine medial axis if and only if it is equidistant from at least two points of the curve, with the distance being a minimum and given by the areas between the curve and its corresponding chords. The medial axis is robust, eliminating the need for curve denoising. In a dynamical interpretation of this affine medial axis, the medial axis points are the affine shock positions of the affine erosion of the curve. We propose a simple method to compute the medial axis and give examples. We also demonstrate how to use this method to detect affine skew symmetry in real images.     

截面轮廓曲线分段约束拟合

本文以线段和圆弧为逼近基元对封闭的截面曲线进行分段拟合,给出了曲线的拟合算法和逼近误差的分析表达式。在对截面曲线分界点进行迭代寻优后,得到了综合累积误差最小的分界点,并以此进行截面曲线的分段拟合;针对拟合所得的曲线,提出了具体的约束修正方法。通过实例分析表明,该算法简单有效,能较好地解决以线段和圆弧为基元的截面曲线拟合问题。     

Comparison of Distance Measures for Planar Curves

The Hausdorff distance is a very natural and straightforward distance measure for comparing geometric shapes like curves or other compact sets. Unfortunately, it is not an appropriate distance measure in some cases. For this reason, the Fréchet distance has been investigated for measuring the resemblance of geometric shapes which avoids the drawbacks of the Hausdorff distance. Unfortunately, it is much harder to compute. Here we investigate under which conditions the two distance measures approximately coincide, i.e., the pathological cases for the Hausdorff distance cannot occur. We show that for closed convex curves both distance measures are the same. Furthermore, they are within a constant factor of each other for so-called κ-straight curves, i.e., curves where the arc length between any two points on the curve is at most a constant κ times their Euclidean distance. Therefore, algorithms for computing the Hausdorff distance can be used in these cases to get exact or approximate computations of the Fréchet distance, as well.     

Comparison of Distance Measures for Planar Curves

The Hausdorff distance is a very natural and straightforward distance measure for comparing geometric shapes like curves or other compact sets. Unfortunately, it is not an appropriate distance measure in some cases. For this reason, the Fréchet distance has been investigated for measuring the resemblance of geometric shapes which avoids the drawbacks of the Hausdorff distance. Unfortunately, it is much harder to compute. Here we investigate under which conditions the two distance measures approximately coincide, i.e., the pathological cases for the Hausdorff distance cannot occur. We show that for closed convex curves both distance measures are the same. Furthermore, they are within a constant factor of each other for so-called -straight curves, i.e., curves where the arc length between any two points on the curve is at most a constant times their Euclidean distance. Therefore, algorithms for computing the Hausdorff distance can be used in these cases to get exact or approximate computations of the Fréchet distance, as well.     

平面隐式曲线的Hermite 插值逼近

隐式曲线在医学图像处理、地理信息系统、数值场可视化等领域中有着重要应用。 在分析点采样和曲线逼近理论的基础上,提出一种运用Hermite 插值方法逼近平面隐式曲线的 算法。首先将曲线绘制区域网格化,在网格单元各边中通过线性插值计算曲线采样点;其次通 过计算采样点精简前后构成的曲线段之间产生的误差优化采样点;最后通过Hermite 插值法逼 近隐函数曲线。实验表明,通过该算法绘制出的曲线在采样点数量较少的情况下,其光滑度和 准确度仍较高。     

Robust Plotting of Polar Algebraic Curves, Space Algebraic Curves, and Offsets of Planar Algebraic Curves

In this paper, based on modifications to the well-known subdivision scheme in combination with modified affine arithmetic method to guide the subdivision, we propose four robust and reliable algorithms for plotting polar algebraic curves, space algebraic curves and offsets of planar algebraic curves.     

Translational Covering of Closed Planar Cubic B-Spline Curves

Spline curves are useful in a variety of geometric modeling and graphics applications and covering problems abound in practical settings. This work defines a class of covering decision problems for shapes bounded by spline curves. As a first step in addressing these problems, this paper treats translational spline covering for planar, uniform, cubic B‐splines. Inner and outer polygonal approximations to the spline regions are generated using enclosures that are inside two different types of piecewise‐linear envelopes. Our recent polygonal covering technique is then applied to seek translations of the covering shapes that allow them to fully cover the target shape. A feasible solution to the polygonal instance provides a feasible solution to the spline instance. We use our recent proof that 2D translational polygonal covering is NP‐hard to establish NP‐hardness of our planar translational spline covering problem. Our polygonal approximation strategy creates approximations that are tight, yet the number of vertices is only a linear function of the number of control points. Using recent results on B‐spline curve envelopes, we bound the distance from the spline curve to its approximation. We balance the two competing objectives of tightness vs. number of points in the approximation, which is crucial given the NP‐hardness of the spline problem. Examples of the results of our spline covering work are provided for instances containing as many as six covering shapes, including both convex and nonconvex regions. Our implementation uses the LEDA and CGAL C++ libraries of geometric data structures and algorithms.     

平面三次NURBS曲线的自动光顺算法

针对平面三次NURBS曲线的光顺问题，基于节点插入，节点消法和重新确定权因子等技术，给出了平面三次NURBS曲线的一种同算法，算法根据给定的光顺准则，自动选择需要光顺的节点，局部修改控制顶点和权因子。     

平面简单闭合曲线离散采样与重建算法

提出一种鲁棒的平面简单闭合曲线离散采样与重建算法。算法分为采样过程和重 建过程两部分。采样部分首先对平面闭合曲线均匀取点,然后计算各点到曲线所围平面区域中 轴的最近距离,最后根据所求距离确定采样间隔,获取采样点集;重建部分首先构建采样点集 的Delaunay 三角剖分,然后从得到的三角形中选择边构建初始化图形,最后通过修改该图形获 得重建图形。实验表明算法得到的采样点较少且能反映曲线的局部几何特性,重建图形能够较 好地表示原闭合曲线的形状及走向。     

基于曲率圆的平面参数曲线求交算法

主要讨论了平面参数曲线求交的迭代算法,提出了迭代过程中迭代可信度的概念,并给出了计算方法.在此基础上,改进了MAF求交算法,给出了曲率圆迭代算法,即使用二次曲线对参数曲线的局部形状进行近似,进行迭代交点和迭代步长的计算.     

平面代数曲线间最近距离的计算

通过几何观察,指出一条曲线上的最近点是另一条曲线的等距曲线与该曲线的切点这一事实,同时提出基于等距思想的方法来求解2条平面代数曲线间的最近距离．该方法几何意义明显,可同时用来计算代数曲线／参数曲线间的最近距离．对于平面二次曲线,采用文中方法得到的单变量多项式方程次数比已有类似方法中结果方程的次数更低,从而可以降低方程求解的计算复杂度或提高求解的稳定性．     

由整体到局部的平面曲线部分匹配算法

在基于曲线匹配的检索系统中,提高曲线的匹配速度和精度具有重要的意义.提出一种平面曲线的部分匹配算法,该算法分为整体搜索和局部匹配2个阶段.首先整体搜索确定候选的匹配区域,然后在局部进行精确匹配和验证.对于特征点较少的曲线,根据曲率极值点将曲线划分为多条曲线段,采用局部线性搜索法实现曲线的部分匹配.实验结果说明了算法的有效性.     

平面参数曲线变形的仿射映射B样条方法

提出一种新的参数曲线变形方法:采用一种特殊的B样条展开式作为伸缩函数,构造了具有明确几何意义的变换矩阵,用它作用于待变形的曲线,可使曲线发生变形。此方法数学模型简单而变形效果良好。展开式的系数作为变形的控制参数,每个参数具有局部可控性,变形效果较丰富。可分别定量地控制变形的发生区间、变形区间界点处的连续性与光滑性、变形方向和变形幅度等。实验表明,该方法通过交互改变控制参数,可获得预期的、丰富的形状修改和变形效果,适用于几何造型、计算机动画、CAD等领域。     

一种新的平面开曲线形状距离的度量

平面曲线形状识别是最基本的模式识别问题，然而这个问题至今仍然未能很好地解决。其困难在于难以给出两条曲线的形状差别的定量描述。本文为基于曲率表示的两条平面开曲线的等形下了严格的数学定义，从而找到了一种新的形状距离度量，并且证明了这种形状距离的计算问题可以转化为一个泛函的极值问题，同时给出了求解形状距离的微分方程。由于解这个微分方程是困难的，实验中采用粗略的分段匹配法。本文还介绍了算法的程序实现，尤其是离散情况下的曲线的曲率表示，并且用基于动量守恒的高斯滤波解决了曲率法表示曲线的噪声敏感性问题。实验表明本文提出的形状距离度量方法是有效的。     

On Projective Invariant Smoothing and Evolutions of Planar Curves and Polygons

Several recently introduced and studied planar curve evolutionequations turn out to be iterative smoothing procedures that areinvariant under the actions of the Euclidean and affine groups ofcontinuous transformations. This paper discusses possible ways toextend these results to the projective group of transformations.Invariant polygon evolutions are also investigated.